# Optimal data placements for triple replication111Supported by NSFC Grant
11971053.

Ruijing Liu, Junling Zhou
Department of Mathematics
Beijing Jiaotong University
Beijing 100044, P. R. China
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Given a set $V$ of $v$ servers along with $b$ files (data), each file is
replicated (placed) on exactly $k$ servers and thus a file can be represented
by a set of $k$ servers. Then we produce a data placement consisting of $b$
subsets of $V$ called blocks, each of size $k$. Each server has some
probability to fail and we want to find a placement that minimizes the
variance of the number of available files. It was conjectured that there
always exists an optimal data placement (with variance better than any other
placement for any value of the probability of failure). An optimal data
placement for triple replication with $b$ blocks (of size three) on a $v$-set
was proved to exist by Wei et al. if $v$ and $b$ are not excluded by two
conditions. This article concentrates on the parameters $v,b$ satisfying the
two conditions and characterizes the combinatorial properties of the
corresponding optimal data placements. Nearly well-balanced triple systems
(NWBTSs) are defined to produce optimal data placements. Many constructions
for NWBTSs are developed, mainly by constructing candelabra systems with
various desirable partitions. The main result of this article is that there
always exist optimal data placements for triple replication with $b$ blocks on
a $v$-set possibly except when $v\equiv 4$ (mod 24) or $v=50,74$, and
$\frac{\lambda v(v-1)}{6}-\frac{v}{6}<b<\frac{\lambda v(v-1)}{6}+\frac{v}{6}$
for an odd integer $\lambda$.

Keywords: data placement, optimal, triple system, $s$-fan design, candelabra
system, nearly well-balanced, partitionable

##  1 Introduction

Large scale distributed storage systems play a vital role in maintaining data
across storage locations globally. These systems are built using cheap
commodity hardware, that can be unreliable. Hence, it is essential for the
system to store data in replication to ensure reliability and fault-tolerance.
There is growing interest of using distributed systems for massive video
distribution. Documents replication and repartition can also solve some
performance problems for video distribution, and is an effective way to
improve the performance of video download time. Those problems have been
studied extensively and their connections to combinatorial objects have been
displayed [3, 4, 2, 13].

In this article we consider data placements for triple replication, which is
the default replication storage in Hadoop’s Distributed File System. Here, we
use the terminology of combinatorial design theory. We are given a set $V$ of
$v$ servers along with $b$ files (data, documents). Each file is replicated
(placed) on exactly $k$ servers (nodes). The set of servers containing file
$i$ is therefore a subset of size $k$, which will be called a _block_ and
denoted by $B_{i}$. Then the set system $(V,\mathcal{F})$ or the family
$\mathcal{F}$ of the $b$ blocks $B_{i}$, $1\leq i\leq b$, is called a _data
placement_. A data placement for triple replication is a triple system
$(V,{\cal F})$, also denoted by TS$(v;b)$, where every block of ${\cal F}$ has
cardinality three.

It is interesting to study the situation where some of the servers might not
be available when asked to provide a file. We assume that a server is
available (online) with some probability $\delta$, and so unavailable (off-
line, failed) with the probability $1-\delta$. The file $i$ is said to be
available if one of the servers containing it is available, or equivalently
the file is unavailable if all the servers containing it are unavailable. In
[4, 2, 13], the authors studied the random variable $\Lambda$, the number of
available files, and they proved that the expectation $E(\Lambda)$ is
independent of the data placement. However, they proved that the variance of
$\Lambda$ depends on the data placement $(V,\mathcal{F})$. Jean-Marie et al.
[13] showed that minimizing the variance of $\Lambda$ corresponds to
minimizing the polynomial

$P(\mathcal{F},x)=\sum^{k}_{j=0}v_{j}x^{j},$

where $x=\frac{1}{1-\delta}$ (so $x\geq 1$) and $v_{j}$ denotes the number of
ordered pairs of blocks intersecting in exactly $j$ elements of $V$.

A basic problem is to find an _optimal_ data placement $(V,\mathcal{F})$,
which minimizes $P(\mathcal{F},x)$ for all the values of $x\geq 1$. In [13],
the following conjecture is proposed.

######  Conjecture 1.1.

[13] For any nonnegative integers $v,k$ and $b$ there exists an optimal data
placement consisting of $b$ blocks of size $k$ on a $v$-set.

Bermond et al. [6] gave an equivalent formulation of the polynomial
$P(\mathcal{F},x)$. For a set system $(V,\mathcal{F})$, let
$\lambda^{\mathcal{F}}_{x_{1},\ldots,x_{j}}$ (or
$\lambda_{x_{1},\ldots,x_{j}}$) denote the number of blocks of $\mathcal{F}$
containing the $j$-subset $\\{x_{1},\ldots,x_{j}\\}$ of $V$.

######  Lemma 1.2.

[6] For a data placement $(V,\mathcal{F})$ consisting of $b$ blocks of size
$k$,

$P(\mathcal{F},x)=\sum^{k}_{j=1}\sum_{\\{x_{1},\ldots,x_{j}\\}\in{V\choose
j}}\lambda_{x_{1},\ldots,x_{j}}^{2}(x-1)^{j}-bx^{k}+b^{2},~{}x\geq 1,$

where ${V\choose j}$ denotes the collection of all $j$-subsets of $V$.

A set system $(V,\mathcal{F})$ is _$j$ -balanced_ if the
$\lambda_{x_{1},\ldots,x_{j}}$’s are all equal or almost equal, that is, if
$|\lambda_{x_{1},\ldots,x_{j}}-\lambda_{y_{1},\ldots,y_{j}}|\leq 1$ for any
two $j$-subsets $\\{x_{1},\ldots,x_{j}\\}$ and $\\{y_{1},\ldots,y_{j}\\}$ of
$V$. Furthermore, $(V,\mathcal{F})$ is _well-balanced_ if it is $j$-balanced
for all $1\leq j\leq k$, where $k$ is the size of each block.

######  Lemma 1.3.

[6] Let $\mathcal{F}$ consist of $b$ blocks of size $k$ on a $v$-set $V$. For
$1\leq j\leq k$, if $(V,\mathcal{F})$ is $j$ -balanced, then
$\sum_{\\{x_{1},\ldots,x_{j}\\}\in{V\choose
j}}\lambda^{2}_{x_{1},\ldots,x_{j}}$ is minimized.

Bermond et al. [6] used Lemmas 1.2 and 1.3 to prove that a well-balanced set
system gives rise to an optimal data placement. They proved the truth of
Conjecture 1.1 for $k=2$ and provided some positive instances for $k=3$.
Afterwards, Wei et al. [20] determined completely the spectrum of pairs
$(v,b)$ for which there exists a well-balanced triple system, simply denoted
WBTS$(v;b)$.

######  Theorem 1.4.

[20, Theorem 8.1] A WBTS$(v;b)$ exists for any positive integers $v,b$ and
$v\geq 3$, with definite exceptions that

(i) $b\in\\{\lfloor\frac{\lambda v(v-1)}{6}\rfloor,\lceil\frac{\lambda
v(v-1)}{6}\rceil\\}$ for an integer $\lambda$ such that $\lambda
v(v-1)\not\equiv 0\pmod{6}$; and

(ii) $v$ is even and $\frac{\lambda v(v-1)}{6}-\frac{v}{6}<b<\frac{\lambda
v(v-1)}{6}+\frac{v}{6}$ for an odd integer $\lambda$.

######  Corollary 1.5.

An optimal data placement TS$(v;b)$ exists for any positive integers $v,b$ and
$v\geq 3$, possibly except that they satisfy (i) or (ii) of Theorem 1.4.

######  Remark 1.6.

[6, Propositions 6, 7] When $(v,b)$ satisfies $(i)$ or $(ii)$ of Theorem 1.4,
a $2$-balanced TS$(v;b)$ does not exist.

By Theorem 1.4 and Corollary 1.5, the existence problem of well-balanced
triple systems has been resolved, while the existence of optimal data
placements for triple replication remains open. In this article we concentrate
on optimal data placements TS$(v;b)$s with $v,b$ meeting (i) or (ii) of
Theorem 1.4. For convenience, we redistribute these parameters by two
exclusive conditions as follows:

(C1) $v\equiv 2$ (mod 3) and $b\in\\{\lfloor\frac{\lambda
v(v-1)}{6}\rfloor,\lceil\frac{\lambda v(v-1)}{6}\rceil\\}$, where
$\lambda\equiv 1,2$ (mod 3) if $v\equiv 5$ (mod 6) and $\lambda\equiv 2,4$
(mod 6) if $v\equiv 2$ (mod 6).

(C2) $v$ is even and $\frac{\lambda v(v-1)}{6}-\frac{v}{6}<b<\frac{\lambda
v(v-1)}{6}+\frac{v}{6}$ for an odd $\lambda$.

The rest of this article is organized as follows. In Section 2, we
characterize the combinatorial properties of an optimal data placement
TS$(v;b)$ by defining a defect graph for a triple system and introducing the
concept of a nearly well-balanced triple system (NWBTS). We show that an
NWBTS$(v;b)$ gives rise to an optimal data placement. In Section 3 we
introduce a vital design named candelabra system and define its desirable
partitions to produce NWBTSs when $(v,b)$ satisfies (C1) or (C2). Then in
Section 4 we study the case of (C1) and completely prove the existence of
NWBTSs in this case. In Sections 5 and 6 we turn to (C2) and generalize the
concept of partitionable candelabra systems to treat this case. Effective
recursive constructions are developed; necessary small input designs are
constructed directly; a large portion of NWBTSs with parameters satisfying
(C2) is proved to exist. In Section 7 we summarize our main results and
present a promising approach to resolving entirely the existence problem of
NWBTSs and optimal data placements.

##  2 Combinatorial descriptions

In this section we study the combinatorial properties of a TS$(v;b)$
corresponding to an optimal data placement for triple replication when $(v,b)$
satisfies (C1) or (C2). We will define nearly well-balanced triple systems to
produce optimal data placements.

For a TS$(v;b)$, where $\frac{\lambda v(v-1)}{6}-\frac{v}{6}<b<\frac{\lambda
v(v-1)}{6}+\frac{v}{6}$ for some nonnegative integer $\lambda$, we may write

$\displaystyle 3b=\lambda{v\choose 2}+\varepsilon,\ {\rm where\
}-\frac{v}{2}<\varepsilon<\frac{v}{2}\ {\rm and\ }\varepsilon\ {\rm is~{}an\
integer}.$ (1)

In this article we always use the notion of $(\lambda,\varepsilon)$, which is
associated with the given parameters $v$ and $b$ by (1). In particular, for
(C1) we have $\varepsilon\in\\{-1,2\\}$ if $\lambda\equiv 1$ (mod 3) and
$\varepsilon\in\\{-2,1\\}$ if $\lambda\equiv 2$ (mod 3). We will study the
combinatorial properties of a TS$(v;b)$ which produces an optimal data
placement for (C1) or (C2). By Remark 1.6, the optimal data placement
TS$(v;b)$ cannot be 2-balanced. However, we will show that a little weaker
balance should hold for pairs of elements.

Let $(V,\mathcal{F})$ be a TS$(v;b)$ where $(v,b)$ associates with
$(\lambda,\varepsilon)$ as in (1). For any nonzero integer $i$, define the
_defect graph of $(V,\mathcal{F})$ at level $i$_ by

$D_{i}=\\{\\{x,y\\}\in\tbinom{V}{2}:\lambda_{x,y}=\lambda+i\\}$

and define the _defect_ of $(V,\mathcal{F})$ by the largest $\beta$ such that
$D_{\beta}\neq\emptyset$ or $D_{-\beta}\neq\emptyset$. Then the _defect graph_
of $(V,\mathcal{F})$ is defined to be $D=\cup_{i=1}^{\beta}(D_{i}\cup
D_{-i})$. A 2-balanced triple system has a defect graph $D=D_{1}\cup D_{-1}$
with $D_{1}=\emptyset$ or $D_{-1}=\emptyset$.

######  Lemma 2.1.

Let $(v,b)$ satisfy (C1) and they associate with $(\lambda,\varepsilon)$ by
(1). Then the minimum of
$\big{\\{}\sum_{\\{x,y\\}\in\binom{V}{2}}(\lambda_{x,y}^{\cal
F})^{2}:(V,\mathcal{F})$ is a TS$(v;b)\big{\\}}$ is achieved when
$(V,\mathcal{F})$ has a defect graph $D$ meeting one of the followings:

(i) $D=D_{1}\cup D_{-1}$ with
$(|D_{1}|,|D_{-1}|)=\begin{cases}(1+\varepsilon,1),&if\
\varepsilon\in\\{1,2\\},\\\ &\\\ (1,1-\varepsilon),&if\
\varepsilon\in\\{-1,-2\\};\end{cases}$

(ii) if $\varepsilon\in\\{\pm 2\\}$, we may also have $D=D_{\varepsilon}$ and
$|D|=1$.

###### Proof.

Suppose that $(V,\mathcal{F})$ is a TS$(v;b)$ with defect $\beta$ and its
defect graph $D=\cup_{i=1}^{\beta}(D_{i}\cup D_{-i})$. Denote $|D_{i}|=a_{i}$,
$-\beta\leq i\leq\beta~{}(i\neq 0)$. Further denote
$a_{0}=|\\{\\{x,y\\}\in{V\choose 2}:\lambda_{x,y}^{\cal F}=\lambda\\}|$. It is
immediate that

$\sum_{i=-\beta}^{\beta}a_{i}(\lambda+i)=3b~{}\text{and}~{}\sum_{i=-\beta}^{\beta}a_{i}={v\choose
2}.$

Since $3b=\lambda{v\choose 2}+\varepsilon$ where $\varepsilon\in\\{\pm 1,\pm
2\\}$, we have

$\displaystyle\sum_{i=-\beta}^{\beta}ia_{i}=\varepsilon,$ (2)
$\displaystyle\sum_{\\{x,y\\}\in{V\choose 2}}(\lambda_{x,y}^{\cal
F})^{2}=\sum_{i=-\beta}^{\beta}a_{i}(\lambda+i)^{2}=\binom{v}{2}\lambda^{2}+2\varepsilon\lambda+\sum_{i=-\beta}^{\beta}i^{2}a_{i}.$
(3)

If the defect $\beta\geq 2$, then (3) gives

$\displaystyle\sum_{\\{x,y\\}\in{V\choose 2}}(\lambda_{x,y}^{\cal
F})^{2}\geq\binom{v}{2}\lambda^{2}+2\varepsilon\lambda+4~{}~{}(\text{if}~{}\beta\geq
2).$ (4)

Next we let $\beta=1$. By (3) we have $\sum_{\\{x,y\\}\in{V\choose
2}}(\lambda_{x,y}^{\cal
F})^{2}=\binom{v}{2}\lambda^{2}+2\varepsilon\lambda+a_{1}+a_{-1}$. It follows
from (2) that $a_{1}-a_{-1}=\varepsilon$. Notice that $a_{1},a_{-1}\geq 1$;
otherwise $(V,\mathcal{F})$ is a 2-balanced triple system, which contradicts
Remark 1.6. Hence we have

$\displaystyle\sum_{\\{x,y\\}\in{V\choose
2}}(\lambda_{x,y}^{\mathcal{F}})^{2}\geq\binom{v}{2}\lambda^{2}+2\varepsilon\lambda+2+|\varepsilon|~{}~{}({\rm
if}~{}\beta=1).$ (5)

Comparing (4) and (5) yields that, if $\varepsilon\in\\{\pm 1\\}$, then
$\sum_{\\{x,y\\}\in{V\choose 2}}(\lambda_{x,y}^{\cal F})^{2}$ attains the
minimum in (5) with $(a_{1},a_{-1})$ as assumed in the lemma; and if
$\varepsilon\in\\{\pm 2\\}$, then it attains the minimum both in (4) and (5)
when the defect graph meets (i) or (ii) of the lemma. ∎

Now we continue considering the case that $(V,\mathcal{F})$ is a TS$(v;b)$
with $(v,b)$ satisfying (C1). Assume that $(V,\mathcal{F})$ has a defect graph
$D$ satisfying (i) of Lemma 2.1. It is easy to depict the graph $D$ in detail.
Denote by $d_{D}(x)$ the degree of a vertex $x$ in the graph $D$. For any
$x\in V$, we have

$\displaystyle\lambda_{x}^{\mathcal{F}}$
$\displaystyle=\frac{1}{2}((\lambda+1)d_{D_{1}}(x)+(\lambda-1)d_{D_{-1}}(x)+\lambda(v-1-d_{D_{1}}(x)-d_{D_{-1}}(x))$
$\displaystyle=\frac{1}{2}(d_{D_{1}}(x)-d_{D_{-1}}(x)+\lambda(v-1)).$

Then we get that $d_{D_{1}}(x)\equiv d_{D_{-1}}(x)\pmod{2}$ as $\lambda(v-1)$
is even. It follows from Lemma 2.1 that the defect graph $D=D_{1}\cup D_{-1}$
must be isomorphic to $G_{\varepsilon}$ in Figure 1, where an edge of $D_{1}$
or $D_{-1}$ is displayed in a solid line or dotted line, respectively.

$G_{-1}$ $G_{-2}$ $G_{1}$ $G_{2}$

Figure 1: Defect graphs with defect 1 for (C1)

######  Lemma 2.2.

Let $(v,b)$ satisfy (C2) and they associate with $(\lambda,\varepsilon)$ by
(1). Then the minimum of
$\big{\\{}\sum_{\\{x,y\\}\in\binom{V}{2}}(\lambda_{x,y}^{\cal
F})^{2}:(V,\mathcal{F})$ is a TS$(v;b)\big{\\}}$ is attained when
$(V,\mathcal{F})$ has a defect graph $D$ isomorphic to $H_{v,\varepsilon}^{0}$
if $\varepsilon\equiv\frac{v}{2}\pmod{2}$ or isomorphic to one of
$H_{v,\varepsilon}^{i}$ with $1\leq i\leq 4$ if
$\varepsilon\not\equiv\frac{v}{2}\pmod{2}$ (see Figure 2, also a solid line
stands for an edge of $D_{1}$ and a dotted line for an edge of $D_{-1}$).

$\underbrace{}$ $\underbrace{}$ $\underbrace{}$ $\underbrace{}$
$\underbrace{}$ $\underbrace{}$
$~{}~{}~{}~{}~{}~{}\frac{v+2\varepsilon}{4}$ $\frac{v-2\varepsilon}{4}$
$\frac{v-6+2\varepsilon}{4}$ $\frac{v-2-2\varepsilon}{4}$
$\frac{v-2+2\varepsilon}{4}$ $\frac{v-6-2\varepsilon}{4}$
$H_{v,\varepsilon}^{0}$ $H_{v,\varepsilon}^{1}$ $H_{v,\varepsilon}^{2}$

$\underbrace{}$ $\underbrace{}$ $\underbrace{}$ $\underbrace{}$
$\frac{v-10+2\varepsilon}{4}$ $\frac{v+2-2\varepsilon}{4}$
$\frac{v+2+2\varepsilon}{4}$ $\frac{v-10-2\varepsilon}{4}$
$H_{v,\varepsilon}^{3}$ $H_{v,\varepsilon}^{4}$

Figure 2: Defect graphs for (C2)

###### Proof.

Suppose that $(V,\mathcal{F})$ is a TS$(v;b)$ with defect $\beta$ and its
defect graph $D=\cup_{i=1}^{\beta}(D_{i}\cup D_{-i})$. Let
$D_{0}=\\{\\{x,y\\}\in\binom{V}{2}:\lambda_{x,y}^{\cal F}=\lambda\\}$. Denote
$|D_{i}|=a_{i}$, $-\beta\leq i\leq\beta$. For any $x\in V$, we have

$\displaystyle\lambda_{x}^{\cal F}$
$\displaystyle=\frac{1}{2}(\sum_{i=-\beta}^{\beta}(\lambda+i)d_{D_{i}}(x))=\frac{1}{2}(\lambda\sum_{i=-\beta}^{\beta}d_{D_{i}}(x)+\sum_{i=-\beta}^{\beta}id_{D_{i}}(x))$
$\displaystyle=\frac{1}{2}(\lambda(v-1)+\sum_{i=-\beta}^{\beta}id_{D_{i}}(x)).$

Since $v$ is even and $\lambda$ is odd, we have
$\sum_{i=-\beta}^{\beta}id_{D_{i}}(x)\equiv 1\pmod{2}$ to ensure that
$\lambda_{x}^{\cal F}$ is an integer. Hence we have $d_{D}(x)\geq 1$. It
follows from the arbitrariness of $x$ that $|D|\geq{v\over 2}$.

Similarly to the proof of Lemma 2.1, we have (2) and (3). It is obvious that

$\displaystyle\sum_{i=-\beta}^{\beta}i^{2}a_{i}$
$\displaystyle=\sum_{i=1}^{\beta}(a_{i}+a_{-i})+\sum_{i=2}^{\beta}(i^{2}-1)(a_{i}+a_{-i})$
$\displaystyle\geq\sum_{i=1}^{\beta}(a_{i}+a_{-i})=|D|,$

where the equality occurs if and only if $\beta=1$.

So we let $\beta=1$ next. Note that $|D|\geq\frac{v}{2}$ and
$a_{1}-a_{-1}=\varepsilon$ from (2). We have that $|D|$ is minimized to
$|D|=\frac{v}{2}$ if $\frac{v}{2}\equiv\varepsilon\pmod{2}$,
$(a_{1},a_{-1})=(\frac{v+2\varepsilon}{4},\frac{v-2\varepsilon}{4})$ and $D$
is a perfect matching isomorphic to the graph $H_{v,\varepsilon}^{0}$ in
Figure 2; and that $|D|$ is minimized to $|D|=\frac{v}{2}+1$ if
$\frac{v}{2}\not\equiv\varepsilon\pmod{2}$,
$(a_{1},a_{-1})=(\frac{v+2+2\varepsilon}{4},\frac{v+2-2\varepsilon}{4})$ so
that $D$ is isomorphic to one of the graphs $H_{v,\varepsilon}^{i}$, $1\leq
i\leq 4$, in Figure 2. From (3), when $|D|$ attains the minimum,
$\sum_{\\{x,y\\}\in{V\choose 2}}(\lambda_{x,y}^{\cal F})^{2}$ also does. This
proves the lemma. ∎

Recall that an optimal data placement TS$(v;b)$ minimizes the polynomial

$P(\mathcal{F},x)=\sum^{3}_{j=1}\sum_{\\{x_{1},\ldots,x_{j}\\}\in{V\choose
j}}(\lambda^{\cal F}_{x_{1},\ldots,x_{j}})^{2}(x-1)^{j}-bx^{3}+b^{2}$

for all $x\geq 1.$ From Lemma 1.3, the values $\sum_{x\in V}(\lambda_{x}^{\cal
F})^{2}$ and $\sum_{\\{x,y,z\\}\in{V\choose 3}}(\lambda_{x,y,z}^{\cal F})^{2}$
are minimized when $(V,{\cal F})$ is 1-balanced and 3-balanced, respectively.
Lemmas 2.1 and 2.2 describe the combinatorial property of an optimal TS$(v;b)$
in term of its defect graph when the property of 2-balance cannot hold. For
(C1) its defect graph $D$ with defect one must be isomorphic to
$G_{\varepsilon}$ and $D$ must be isomorphic to a graph in Figure 2 for (C2).
It is not difficult to see that if $D$ is isomorphic to
$H_{v,\varepsilon}^{3}$ or $H_{v,\varepsilon}^{4}$, the family $\mathcal{F}$
is not 1-balanced. For convenience, we then define $\mathcal{F}$ to be _nearly
$2$-balanced_ if for (C1) its defect graph $D$ is isomorphic to
$G_{\varepsilon}$, or for (C2) $D$ is isomorphic to $H_{v,\varepsilon}^{0}$ if
$\varepsilon\equiv\frac{v}{2}\pmod{2}$ or to one of $H_{v,\varepsilon}^{1}$
and $H_{v,\varepsilon}^{2}$ if $\varepsilon\not\equiv\frac{v}{2}\pmod{2}$.
Furthermore, a TS$(v;b)$ is said to be _nearly well-balanced_ , denoted by
NWBTS$(v;b)$, if it is nearly $2$-balanced (hence 1-balanced) and
$3$-balanced. With these preparations, we have the main result of this section
recorded in the following theorem.

######  Theorem 2.3.

An NWBTS$(v;b)$ yields an optimal data placement for triple replication when
$(v,b)$ satisfies (C1) or (C2).

In the following of the article when referring to a nearly well-balanced or a
nearly 2-balanced TS$(v;b)$ we always imply that the parameter pair $(v,b)$
satisfies (C1) or (C2) and the pair $(\lambda,\varepsilon)$ is associated with
$(v,b)$ by (1). By the definition of a nearly 2-balanced triple system, the
defect graph depends only on $\varepsilon$ for (C1) and on $v,\varepsilon$ for
(C2), which will be often used in later constructions.

A triple system $(V,\mathcal{F})$ is called _simple_ if it contains no
repeated blocks. Obviously a simple TS$(v;b)$ is 3-balanced and we have the
following lemma.

######  Lemma 2.4.

A nearly $2$-balanced simple TS$(v;b)$ is an NWBTS$(v;b)$.

######  Lemma 2.5.

Let $v,h,b$ be positive integers and $b<\binom{v}{3}$.

  1. (i)

If there exists an NWBTS$(v;b)$, then so does an NWBTS$(v;b_{1})$ where
$b_{1}=h\binom{v}{3}+b$.

  2. (ii)

If there exists an NWBTS$(v;b)$, then so does an NWBTS$(v;b_{2})$ where
$b_{2}={v\choose 3}-b$.

###### Proof.

Suppose that $(V,\mathcal{F})$ is an NWBTS$(v;b)$. Then $\mathcal{F}$ is
simple as $b<\binom{v}{3}$ and it is nearly 2-balanced. Let $\mathcal{F}$ have
a defect graph $D=D_{1}\cup D_{-1}$. Assume
$3b=\lambda\binom{v}{2}+\varepsilon$ as in (1).

(i) Let $\mathcal{F}_{1}$ consist of all blocks of $\mathcal{F}$ and all
triples of $V$ with each triple taken $h$ times. Obviously every triple of $V$
occurs $h$ or $h+1$ times in $\mathcal{F}_{1}$ and thus $\mathcal{F}_{1}$ is
3-balanced. We also learn that every pair of $D_{1}$ and $D_{-1}$ is
respectively contained in $\lambda_{1}+1$ and $\lambda_{1}-1$ blocks of ${\cal
F}_{1}$ and all other pairs in $\lambda_{1}$ blocks, where
$\lambda_{1}=\lambda+h(v-2)$. It follows that $D$ is also the defect graph of
$\mathcal{F}_{1}$, which fulfils the definition of nearly 2-balance as
$3b_{1}=3b+3h\binom{v}{3}=\lambda_{1}{v\choose 2}+\varepsilon$. Now that
$(V,\mathcal{F}_{1})$ is nearly 2-balanced and 3-balanced, it is nearly well-
balanced with $b_{1}$ blocks.

(2) Let $\mathcal{F}_{2}={V\choose 3}\setminus{\cal F}$. Clearly ${\cal
F}_{2}$ is 3-balanced with $b_{2}$ blocks. Denote the defect graph of ${\cal
F}_{2}$ by $D^{\prime}$. It is easy to see that $D^{\prime}$ has defect 1 and
$D^{\prime}=D_{-1}\cup D_{1}$, which means every pair of $D_{-1}$ and $D_{1}$
is respectively contained in $\lambda_{2}+1$ and $\lambda_{2}-1$ blocks of
${\cal F}_{2}$ and all other pairs in $\lambda_{2}$ blocks where
$\lambda_{2}=v-2-\lambda$. Noticing
$3b_{2}=3\binom{v}{3}-3b=\lambda_{2}{v\choose 2}-\varepsilon$, we can check
that ${\cal F}_{2}$ is also nearly 2-balanced in either case of
$\varepsilon\equiv{v\over 2}$ (mod 2) or not. Thus we have an
NWBTS$(v;b_{2})$. ∎

##  3 Candelabra systems with desirable partitions

In this section we introduce the definition of candelabra systems and then
define two kinds of partitions. The main effect is to produce NWBTSs with a
large range of parameters. Analogous constructions are widely used for large
sets, see for instance [7, 15].

A _candelabra $t$-system_ (or $t$-CS as in [19]) of order $v$ and block sizes
from $K$, denoted by CS$(t,K,v)$, is a quadruple
$(X,S,\mathcal{G},\mathcal{A})$ that satisfies the following properties:

(1) $X$ is a set of $v$ elements (_points_);

(2) $S$ is a subset of $X$ (the _stem_) of size $s$;

(3) $\mathcal{G}=\\{G_{1},G_{2},\ldots\\}$ is a set of nonempty subsets
(_groups_) of $X\backslash S$ that partition $X\backslash S$;

(4) $\mathcal{A}$ is a family of subsets (_blocks_) of $X$, each having
cardinality from $K$; and

(5) each $t$-subset $T$ of $X$ with $|T\cap(S\cup G_{i})|<t$ for all $i$ is
contained in a unique block and no $t$-subset of $S\cup G_{i}$ for any $i$ is
contained in any block.

The _type_ of a $t$-CS is the multiset list ($\\{|G|:G\in\mathcal{G}\\}:|S|$),
listing the group sizes and stem size. When a $t$-CS has $n_{i}$ groups of
size $g_{i}$, $1\leq i\leq r$, and stem size $s$, we use the notation
$(g^{n_{1}}_{1}g^{n_{2}}_{2}\cdots g^{n_{r}}_{r}:s)$. We primarily employ
candelabra systems with $t=3$ and $K=\\{3\\}$, which will be denoted by
CS$(g^{n_{1}}_{1}g^{n_{2}}_{2}\cdots g^{n_{r}}_{r}:s)$.

A _$t$ -wise balanced design_ ($t$BD) with parameters $t$-$(v,K,\lambda)$ is a
pair $(X,\mathcal{B})$ where $X$ is a set of $v$ points and $\mathcal{B}$ is a
collection of subsets (blocks) of $X$ with the property that the size of every
block is in the set $K$ and every $t$-subset of $X$ is contained in exactly
$\lambda$ blocks. A $t$-$(v,K,\lambda)$ design is also denoted by
S${}_{\lambda}(t,K,v)$. When $K=\\{k\\}$, we simply write as
S${}_{\lambda}(t,k,v)$.

A _group divisible $t$-design_ (or $t$-GDD) of order $v$, index $\lambda$, and
block sizes from $K$, denoted by GDD${}_{\lambda}(t,K,v)$, is a triple
$(X,\mathcal{G},\mathcal{B})$ such that

(1) $X$ is a set of $v$ elements (_points_);

(2) $\mathcal{G}$ is a set of nonempty subsets (_groups_) of $X$ that
partition $X$;

(3) $\mathcal{B}$ is a family of subsets (_blocks_) of $X$, each of
cardinality from $K$, such that each block intersects any given group in at
most one point; and

(4) each $t$-subset of points from $t$ distinct groups is contained in exactly
$\lambda$ blocks.

A GDD of type $g^{t_{1}}_{1}\cdots g^{t_{s}}_{s}$ is a GDD in which there are
exactly $t_{i}$ groups of cardinality $g_{i}$ for $1\leq i\leq s$. When
$\lambda$ is omitted in the notation, we mean $\lambda=1$.

A candelabra system CS$(g^{n}$ : $s)~{}(X,S,\mathcal{G},\mathcal{A})$ with
$s\geq 2$, is called partitionable and denoted by PCS$(g^{n}:s)$ (as in [7]),
if the block set $\mathcal{A}$ can be partitioned into $\mathcal{A}_{x}$,
$x\in G,G\in\mathcal{G},$ and
$\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{s-2}$ so that

(i) for each $x\in G$ and $G\in\mathcal{G}$, $\mathcal{A}_{x}$ is the block
set of a GDD$(2,3,gn+s)$ of type $1^{gn-g}(g+s)^{1}$ with $G\cup S$ as the
long group; and

(ii) for $1\leq i\leq s-2$, $(X\setminus S,\mathcal{G},\mathcal{A}_{i})$ is a
GDD$(2,3,gn)$ of type $g^{n}$.

Let $a,g$ be positive integers and let $(X,S,\mathcal{G},\mathcal{A})$ be a
CS$((2ag)^{n}:s)$. It is said to be $g$-partitionable with index two and
denoted by $g$-PCS${}_{2}((2ag)^{n}:s)$ if its block set $\mathcal{A}$
contains a subset $\mathcal{A}^{\prime}$ and $\mathcal{A}^{\prime}$ can be
partitioned into $gn$ pairwise disjoint parts $\mathcal{A}_{i}$, $0\leq i\leq
gn-1$, such that, for each group $G\in\mathcal{G}$, there are exactly $g$
$\mathcal{A}_{i}$’s such that each $\mathcal{A}_{i}$ is the block set of a
GDD${}_{2}(2,3,2agn+s)$ of type $1^{2ag(n-1)}(2ag+s)^{1}$ with $G\cup S$ as
its long group.

Making use of a PCS$(g^{n}:s)$ or a $g$-PCS${}_{2}((2ag)^{n}:s)$ as a master
design will enable us to produce NWBTSs. By this in Section 4 we determine the
existence of NWBTSs for (C1) and in Sections 5 and 6 we will define candelabra
systems with another several types of partitions to treat (C2). Before
presenting the constructions, we prove two auxiliary lemmas.

######  Lemma 3.1.

Suppose that $(V,\mathcal{F})$ is an NWBTS$(v;b)$ where $b<\binom{v}{3}$.
Further suppose that $(V,\mathcal{F}^{\prime})$ is a triple system comprising
of $q$ pairwise disjoint simple S${}_{p}(2,3,v)$s. If $\mathcal{F}$ and
$\mathcal{F}^{\prime}$ are disjoint, then the triple system
$(V,\mathcal{F}\cup\mathcal{F}^{\prime})$ forms an NWBTS$(v;c)$ where
$c=b+\frac{pqv(v-1)}{6}$.

###### Proof.

Clearly $(V,\mathcal{F})$ is a simple NWBTS$(v;b)$ and
$(V,\mathcal{F}\cup\mathcal{F}^{\prime})$ is a 3-balanced TS$(v;c)$ with
$c=b+\frac{pqv(v-1)}{6}$. Let $(v,b)$ associate with $(\lambda,\varepsilon)$.
It follows from the construction that the pairs of $V$ appear in
$pq+\lambda-1,pq+\lambda$, or $pq+\lambda+1$ blocks of
$\mathcal{F}\cup\mathcal{F}^{\prime}$ and that the defect graph of
$\mathcal{F}\cup\mathcal{F}^{\prime}$ is the same as that of $\mathcal{F}$.
Noticing $3c=\frac{(pq+\lambda)v(v-1)}{2}+\varepsilon$ yields that
$\mathcal{F}\cup\mathcal{F}^{\prime}$ is nearly 2-balanced. As a result,
$\mathcal{F}\cup\mathcal{F}^{\prime}$ is an NWBTS$(v;c)$. ∎

######  Lemma 3.2.

Let $v\equiv u\equiv 2,5\pmod{6}$, $c<\binom{u}{3}$, and $(u,c)$ satisfy (C1)
associating with $(\lambda,\varepsilon)$. Suppose that $(U,{\cal F})$ is an
NWBTS$(u;c)$ and $(V,\mathcal{F}^{\prime})$ is a simple
GDD${}_{\lambda}(2,3,v)$ of type $1^{v-u}u^{1}$ with $U\subset V$ as its long
group. Then $(V,{\cal F}\cup{\cal F}^{\prime})$ forms an NWBTS$(v;b)$ where
$3b=\lambda{v\choose 2}+\varepsilon$.

###### Proof.

Clearly $(V,{\cal F}\cup{\cal F}^{\prime})$ has $b$ blocks where

$3b=3c+\frac{\lambda(v(v-1)-u(u-1))}{2}=\frac{\lambda
u(u-1)}{2}+\varepsilon+\frac{\lambda(v(v-1)-u(u-1))}{2}=\lambda{v\choose
2}+\varepsilon.$

It is readily checked that $\lambda_{x,y}^{{\cal F}\cup{\cal
F}^{\prime}}\in\\{\lambda-1,\lambda,\lambda+1\\}$ for any pair $\\{x,y\\}$ of
$V$ and ${\cal F}\cup{\cal F}^{\prime}$ has the same defect graph as that of
${\cal F}$, which is isomorphic to $G_{\varepsilon}$ in Figure 1. As a result,
$\mathcal{F}\cup\mathcal{F}^{\prime}$ is nearly 2-balanced. It is also simple
and thus 3-balanced as $c<\binom{u}{3}$ and $\mathcal{F}^{\prime}$ is simple.
This proves that $(V,{\cal F}\cup{\cal F}^{\prime})$ forms an NWBTS$(v;b)$. ∎

######  Construction 3.3.

Suppose that there exists a PCS$(g^{n}:s)$ where $g\equiv 0\pmod{3}$, $s\equiv
2\pmod{3}$, and $g,n\geq 3$. Let $\lambda=1,2$. Further suppose that there
exist

  1. (i)

$l$ disjoint simple S${}_{3}(2,3,g+s)$s defined over a $(g+s)$-set $X$ with
each block not strictly contained in an $s$-subset $Y\subset X$, where $l=1$
if $g=3$ and $l=2$ otherwise; and

  2. (ii)

an NWBTS$(g+s;c)$ where $(g+s,c)$ satisfies (C1) associating with
$(\lambda,\varepsilon)$.

Then there exists an NWBTS$(gn+s;b)$ for any $b=q\binom{gn+s}{2}+b^{\prime}$
where $0\leq q\leq l(n-1)$ and
$3b^{\prime}=\lambda\binom{gn+s}{2}+\varepsilon$.

###### Proof.

Let $V=(Z_{n}\times Z_{g})\cup S$, $|S|=s$, and
$\mathcal{G}=\\{G_{0},G_{1},\ldots,G_{n-1}\\}$ where $G_{i}=\\{i\\}\times
Z_{g}$. Let $(V,S,\mathcal{G},\mathcal{A})$ be a PCS$(g^{n}:s)$. Then the
block set $\mathcal{A}$ can be partitioned into $gn$ pairwise disjoint parts
$\mathcal{A}(i,j)$, $0\leq i\leq n-1$ and $0\leq j\leq g-1$, such that each
$\mathcal{A}(i,j)$ is the block set of a GDD$(2,3,gn+s)$ of type
$1^{g(n-1)}(g+s)^{1}$ with long group $G_{i}\cup S$. For $\lambda=1,2$,
construct an NWBTS$(g+s;c)$ on $G_{0}\cup S$ with block set ${\cal
C}_{\lambda}$ where $3c=\lambda\binom{g+s}{2}+\varepsilon$, which exists by
assumption. Let $\mathcal{F}=\mathcal{A}(0,0)\cup{\cal C}_{1}$ if $\lambda=1$
and $\mathcal{F}=\mathcal{A}(0,0)\cup\mathcal{A}(0,1)\cup{\cal C}_{2}$ if
$\lambda=2$. Then by Lemma 3.2, $\mathcal{F}$ is an NWBTS$(gn+s;b^{\prime})$
where $3b^{\prime}=\lambda\binom{gn+s}{2}+\varepsilon$. This proves the case
of $q=0$.

For $1\leq i\leq n-1$, by assumption we may construct $l$ disjoint simple
S${}_{3}(2,3,g+s)$s over $G_{i}\cup S$ with block sets ${\cal C}(i,j),1\leq
j\leq l,$ and each block is not contained in $S$. Let

$\mathcal{F}(i,j)=\mathcal{A}(i,3(j-1))\cup\mathcal{A}(i,3(j-1)+1)\cup\mathcal{A}(i,3(j-1)+2)\cup\mathcal{C}(i,j),\
1\leq j\leq l.$

It is clear that $\mathcal{F}(i,j)$ is the block set of a simple
S${}_{3}(2,3,gn+s)$ and all $\mathcal{F}(i,j)$s are mutually disjoint. For
$1\leq q\leq l(n-1)$, construct $\mathcal{F}^{\prime}$ by taking the union of
$\mathcal{F}$ and $q$ block sets in $\\{\mathcal{F}(i,j):1\leq i\leq n-1,1\leq
j\leq l\\}$. Then we get an NWBTS$(v;b)$ by Lemma 3.1 for any
$b=q\binom{gn+s}{2}+b^{\prime}$ where $0\leq q\leq l(n-1)$ and
$3b^{\prime}=\lambda\binom{gn+s}{2}+\varepsilon$.∎

######  Construction 3.4.

Suppose that there exists a $g$-PCS${}_{2}((2ag)^{n}:2)$ where $g\equiv
0\pmod{3}$ and $g,n\geq 3$. Denote $v=2agn+2$ and let $\lambda=2,4$. Suppose
that there exists a simple S${}_{6}(2,3,2ag+2)$ and an NWBTS$(2ag+2;c)$ where
$(2ag+2,c)$ satisfies (C1) associating with $(\lambda,\varepsilon)$. Then
there exists an NWBTS$(v;b)$ for any $b=qv(v-1)+b^{\prime}$ where $0\leq q\leq
n-1$ and $3b^{\prime}=\lambda{v\choose 2}+\varepsilon$.

###### Proof.

Let $V=(Z_{n}\times Z_{g}\times Z_{2a})\cup S$, $|S|=2$, and
$\mathcal{G}=\\{G_{0},G_{1},\ldots,G_{n-1}\\}$ where $G_{i}=\\{i\\}\times
Z_{g}\times Z_{2a}$. Let $(V,S,\mathcal{G},\mathcal{A})$ be a
$g$-PCS${}_{2}((2ag)^{n}:2)$. Then the block set $\mathcal{A}$ contains a
subset $\mathcal{A}^{\prime}$ and $\mathcal{A}^{\prime}$ can be partitioned
into $gn$ pairwise disjoint parts $\mathcal{A}(i,j)$, $0\leq i\leq n-1$ and
$0\leq j\leq g-1$, such that each $\mathcal{A}(i,j)$ is the block set of a
GDD${}_{2}(2,3,v)$ of type $1^{2ag(n-1)}(2ag+2)^{1}$ with long group
$G_{i}\cup S$. By assumption, for $\lambda=2,4$, there exists an
NWBTS$(2ag+2;c)$ on $G_{0}\cup S$ with block set ${\cal C}_{\lambda}$ where
$3c=\lambda\binom{2ag+2}{2}+\varepsilon$. Construct a family $\mathcal{F}$
containing all blocks of ${\cal C}_{\lambda}$ and all blocks of $\lambda/2$
sets in $\\{\mathcal{A}(0,j):j=0,1\\}$. Then by Lemma 3.2, $\mathcal{F}$ is an
NWBTS$(v;b^{\prime})$ where $3b^{\prime}=\lambda{v\choose 2}+\varepsilon$.
This proves the case of $q=0$.

For each $1\leq i\leq n-1$, construct a simple S${}_{6}(2,3,2ag+2)$ on
$G_{i}\cup S$ with block set $\mathcal{B}(i)$, which exists by assumption. Let
$\mathcal{F}(i)=(\cup^{2}_{j=0}\mathcal{A}(i,j))\cup\mathcal{B}(i)$. Then each
$\mathcal{F}(i)$ is the block set of a simple S${}_{6}(2,3,v)$. Construct a
family $\mathcal{F}^{\prime}$ containing $q$ $\mathcal{F}(i)$s. Then
$\mathcal{F}\cup\mathcal{F}^{\prime}$ is an NWBTS$(v;b)$ by Lemma 3.1 for any
$b=qv(v-1)+b^{\prime}$ where $0\leq q\leq n-1$ and
$3b^{\prime}=\lambda{v\choose 2}+\varepsilon$. ∎

##  4 NWBTSs for (C1)

This section shows the existence of an NWBTS$(v;b)$ if $(v,b)$ satisfies (C1).

######  Lemma 4.1.

There exists an NWBTS$(5;b)$ where $b\equiv 3,4,6,7\pmod{10}$ and an
NWBTS$(11;b)$ where $b\equiv 18,19,36,37\pmod{55}$.

###### Proof.

From Lemma 2.5, we only need to consider the existence of an NWBTS$(v;b)$
where $b\in\\{3,4\\}$ if $v=5$ and $b\in\\{18,19,36,37,73,74\\}$ if $v=11$.
The case of $v=5$ is solved by [6] and the needed small examples of $v=11$
exist by Lemma A.2 in Appendix. ∎

######  Lemma 4.2.

There exists an NWBTS$(17;b)$ whenever $b\equiv 45,46,90,91\pmod{136}$.

###### Proof.

By Lemma 2.5, we can assume that $b\leq 680$. There exists a PCS$(3^{5}:2)$
from [15, Example 2.3]. An NWBTS$(5;c)$ exists for any $c$ if $(5,c)$
satisfies (C1) from Lemma 4.1. Let $(5,c)$ associate with
$(\lambda,\varepsilon)$ where $\lambda=1,2$. A simple S${}_{3}(2,3,5)$ exists
trivially. Apply Construction 3.3 to get an NWBTS$(17;b)$ for
$3b=136q+{136\lambda+\varepsilon\over 3}$, where $0\leq q\leq 4$,
$\varepsilon\in\\{-1,2\\}$ if $\lambda=1$ and $\varepsilon\in\\{-2,1\\}$ if
$\lambda=2$. This covers all admissible block numbers $b\equiv
45,46,90,91\pmod{136}$ and $b\leq 680$. ∎

######  Lemma 4.3.

Let $v\equiv 5\pmod{6}$ and $(v,b)$ satisfy (C1). An NWBTS$(v;b)$ exists.

###### Proof.

Lemmas 4.1 and 4.2 proved the cases $v=5,11,17$. Next we let $v\geq 23$. By
Lemma 2.5, we can assume that $b\leq\frac{v(v-1)(v-2)}{12}$. Write $v=6n+5$
and $b=q\frac{v(v-1)}{2}+b^{\prime}$ where $0\leq
q\leq\left\lfloor\frac{v-2}{6}\right\rfloor=n$ and
$b^{\prime}\in\\{\lfloor\frac{\lambda v(v-1)}{6}\rfloor,\lceil\frac{\lambda
v(v-1)}{6}\rceil:\lambda=1,2\\}$. Thus $(v,b^{\prime})$ associates with
$(\lambda,\varepsilon)$ where $\lambda=1,2$. Apply Construction 3.3 from a
PCS$(6^{n}:5)$, which exists by [18, Lemma 11]. An NWBTS$(11;c)$ exists for
any $c$ if $(11,c)$ satisfies (C1) from Lemma 4.1. There exist two disjoint
simple S${}_{3}(2,3,11)$s defined over a 11-set with each block not contained
in a fixed 5-subset from [20, Lemma 5.4]. Then there exists an NWBTS$(v;b)$,
where $b=q\frac{v(v-1)}{2}+b^{\prime}$ with $0\leq q\leq 2(n-1)$; this
completes the proof.∎

######  Lemma 4.4.

Let $v\equiv 2,8,20\pmod{24}$, $v\not\in\\{50,74\\}$, and $(v,b)$ satisfy
(C1). There exists an NWBTS$(v;b)$.

###### Proof.

For $v=8$, see Lemma A.3 in Appendix. For $v\geq 20$, we can assume that
$b\leq\frac{v(v-1)(v-2)}{6}$ by Lemma 2.5. Write $v=6n+2$ and
$b=qv(v-1)+b^{\prime}$ where $0\leq q\leq\frac{v-2}{6}-1=n-1$, $0\leq
b^{\prime}\leq v(v-1)$. According to Lemma 2.5, we can assume that $0\leq
b^{\prime}\leq\frac{v(v-1)}{2}$, i.e.,
$b^{\prime}\in\\{\lfloor\frac{v(v-1)}{3}\rfloor$,
$\lceil\frac{v(v-1)}{3}\rceil\\}$. Thus $(v,b^{\prime})$ associates with
$(2,\varepsilon)$. There exists from [20, Lemma 6.8] a
$3$-PCS${}_{2}(6^{n}:2)$ for every $n\equiv 0,1,3\pmod{4}$, $n\geq 3$, and
$n\not\in\\{8,12\\}$. An NWBTS$(8;c)$ exists for any $c$ if $(8,c)$ satisfies
(C1) from Lemma A.3 in Appendix; and a simple S${}_{6}(2,3,8)$ trivially
exists. Then there exists an NWBTS$(v;b)$ by Construction 3.4, where
$b=qv(v-1)+b^{\prime}$ and $0\leq q\leq n-1$. ∎

######  Lemma 4.5.

Let $v\equiv 14\pmod{24}$ and $(v,b)$ satisfy (C1). There exists an
NWBTS$(v;b)$.

###### Proof.

For $v=14$, see Lemma A.4 in Appendix. For $v\geq 38$, we only need to
consider $b\leq\frac{v(v-1)(v-2)}{12}$ by Lemma 2.5. Write $v=12n+2$ and
$b=qv(v-1)+b^{\prime}$ where $0\leq q\leq\frac{v-2}{12}-1=n-1$ and
$b^{\prime}\in\\{\lfloor\frac{\lambda v(v-1)}{6}\rfloor$, $\lceil\frac{\lambda
v(v-1)}{6}\rceil:\lambda=2,4\\}$. Thus $(v,b^{\prime})$ associates with
$(\lambda,\varepsilon)$ where $\lambda=2,4$. There exists a
$3$-PCS${}_{2}(12^{n}:2)$ for $n\equiv 1\pmod{2}$ and $n\geq 3$ from [20,
Lemma 6.14]. An NWBTS$(14;c)$ exists for any $c$ if $(14,c)$ satisfies (C1)
from Lemma A.4 in Appendix. A simple S${}_{6}(2,3,14)$ exists as a
WBTS$(14;182)$ exists by Theorem 1.4. Then there exists an NWBTS$(v;b)$ by
Construction 3.4, where $b=qv(v-1)+b^{\prime}$ and $0\leq q\leq n-1$.∎

######  Lemma 4.6.

Let $v\in\\{50,74\\}$ and $(v,b)$ satisfy (C1). There exists an NWBTS$(v;b)$.

###### Proof.

See Lemma A.1 in Appendix.

Combing Lemmas 4.3-4.6 we have established the main result of this section.

######  Theorem 4.7.

An NWBTS$(v;b)$ exists if $(v,b)$ satisfies (C1).

##  5 Case $v\equiv 0\pmod{6}$ for (C2)

In this section we define a variant of partitionable candelabra systems with
an empty stem, present a product-type construction and prove the existence of
an NWBTS$(v;b)$ for any $v\equiv 0$ (mod 6) and $(v,b)$ satisfying (C2).

Let $m,n$, and $u$ be nonnegative integers. Suppose that $V$ is a set of
$v=2(m+n)+u$ points and ${\cal G}=\\{G_{1},\ldots,G_{m+n},U\\}$ is a partition
of $V$ into groups where $|G_{i}|=2~{}(1\leq i\leq m+n)$ and $|U|=u$. Let
$\mathcal{B}$ be a set of triples (blocks) on $V$. Then $(V,{\cal
G},\mathcal{B})$ is a _nearly_ GDD$(2,3,v)$, simply by NGDD$(2,3,v)$, of type
$2^{(m,n)}u^{1}$ if

(1) any pair of $G_{i}$ with $1\leq i\leq m$ is contained in exactly two
blocks of $\mathcal{B}$,

(2) any pair in $\binom{U}{2}\cup\\{G_{i}:m+1\leq i\leq m+n\\}$ is contained
in no blocks, and

(3) any of the other pairs of $V$ is contained in exactly one block.

Let $0\leq r\leq 3g$ be an integer. A CS$((2g)^{3}:0)$
$(X,\emptyset,\mathcal{G},\mathcal{A})$ is partitionable and denoted by
PICS${}_{2}^{r}((2g)^{3}:0)$ if its block set $\mathcal{A}$ can be partitioned
into $4g-2$ subsets
$\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{4g-2}$ with the
properties: (1) for $1\leq i\leq 2g$, each $\mathcal{A}_{i}$ is the block set
of an S${}_{2}(2,3,6g)$, and at least one of $\mathcal{A}_{i}$’s has a subset
$\mathcal{A}^{\prime}$ such that $\mathcal{A}^{\prime}$ is the block set of an
NGDD$(2,3,6g)$ of type $2^{(r,3g-r)}$; (2) for $2g+1\leq i\leq 4g-2$, each
$\mathcal{A}_{i}$ is the block set of a GDD$(2,3,6g)$ of type $(2g)^{3}$ with
the group set $\mathcal{G}$. It is obvious that a PICS${}_{2}^{r}((2g)^{3}:0)$
is also a PICS${}_{2}^{3g-r}((2g)^{3}:0)$.

######  Lemma 5.1.

There exists a PICS${}_{2}^{r}(4^{3}:0)$ for any $r=0,3,6$.

###### Proof.

For $r=0,6$, there exists a PICS${}_{2}^{r}(4^{3}:0)$ by [20, Lemma 7.2].
Next, a PICS${}_{2}^{3}(4^{3}:0)$ can be constructed on $Z_{2}\times
Z_{2}\times Z_{3}$ with group set $\mathcal{G}=\\{Z_{2}\times
Z_{2}\times\\{j\\}:j=0,1,2\\}$. Here we write a point $(x,y,i)$ as
$(x,y)_{i}$.

We first construct two GDD$(2,3,12)$s of type $4^{3}$ with the group set
$\mathcal{G}$. Their blocks are generated by two sets of base blocks (each set
in a line) under the action of the group $Z_{2}\times Z_{2}$.

$\\{(0,0)_{0},(0,0)_{1},(0,0)_{2}\\}$ | $\\{(0,0)_{0},(0,1)_{1},(1,0)_{2}\\}$ | $\\{(0,0)_{0},(1,0)_{1},(1,1)_{2}\\}$ | $\\{(0,0)_{0},(1,1)_{1},(0,1)_{2}\\}$
---|---|---|---
$\\{(0,0)_{0},(0,0)_{1},(0,1)_{2}\\}$ | $\\{(0,0)_{0},(0,1)_{1},(1,1)_{2}\\}$ | $\\{(0,0)_{0},(1,0)_{1},(1,0)_{2}\\}$ | $\\{(0,0)_{0},(1,1)1,(0,0)_{2}\\}$

Then, we construct an initial S${}_{2}(2,3,12)$ on $Z_{2}\times Z_{2}\times
Z_{3}$ with the following blocks; the underlined blocks form the blocks of an
NGDD$(2,3,12)$ of type $2^{(3,3)}$, where $\lambda_{p}=2$ for
$p\in\\{\\{(0,0)_{0},(0,1)_{0}\\}$,
$\\{(0,0)_{1},(0,1)_{1}\\},\\{(0,0)_{2},(0,1)_{2}\\}\\}$ and $\lambda_{q}=0$
for $q\in\\{\\{(1,0)_{0},(1,1)_{0}\\}$, $\\{(1,0)_{1}$, $(1,1)_{1}\\}$,
$\\{(1,0)_{2}$, $(1,1)_{2}\\}\\}$.

$\underline{\\{(0,0)_{0},(0,1)_{0},(1,0)_{1}\\}}$ | $\underline{\\{(0,0)_{0},(0,1)_{0},(0,1)_{2}\\}}$ | $\underline{\\{(0,0)_{0},(1,0)_{0},(0,1)_{1}\\}}$ | $\underline{\\{(0,0)_{0},(1,1)_{0},(1,1)_{1}\\}}$
---|---|---|---
$\underline{\\{(0,0)_{0},(0,0)_{2},(1,1)_{2}\\}}$ | $\underline{\\{(0,0)_{0},(0,0)_{1},(1,0)_{2}\\}}$ | $\underline{\\{(0,1)_{0},(1,1)_{1},(0,0)_{2}\\}}$ | $\underline{\\{(0,1)_{0},(1,0)_{0},(1,0)_{2}\\}}$
$\underline{\\{(0,1)_{0},(0,0)_{1},(0,1)_{1}\\}}$ | $\underline{\\{(0,0)_{1},(0,1)_{1},(1,1)_{2}\\}}$ | $\underline{\\{(0,1)_{0},(1,1)_{0},(1,1)_{2}\\}}$ | $\underline{\\{(1,0)_{0},(0,0)_{1},(1,0)_{1}\\}}$
$\underline{\\{(1,0)_{0},(1,1)_{1},(1,1)_{2}\\}}$ | $\underline{\\{(1,0)_{0},(0,0)_{2},(0,1)_{2}\\}}$ | $\underline{\\{(0,1)_{1},(0,0)_{2},(0,1)_{2}\\}}$ | $\underline{\\{(1,1)_{0},(0,0)_{1},(0,0)_{2}\\}}$
$\underline{\\{(1,1)_{0},(0,1)_{1},(1,0)_{1}\\}}$ | $\underline{\\{(1,1)_{0},(0,1)_{2},(1,0)_{2}\\}}$ | $\underline{\\{(0,0)_{1},(1,1)_{1},(0,1)_{2}\\}}$ | $\underline{\\{(0,1)_{1},(1,1)_{1},(1,0)_{2}\\}}$
$\underline{\\{(1,0)_{1},(0,0)_{2},(1,0)_{2}\\}}$ | $\underline{\\{(1,0)_{1},(0,1)_{2},(1,1)_{2}\\}}$ | $\\{(0,0)_{0},(0,0)_{1},(1,1)_{2}\\}$ | $\\{(1,0)_{0},(1,1)_{0},(1,0)_{1}\\}$
$\\{(0,1)_{0},(1,1)_{0},(0,1)_{1}\\}$ | $\\{(1,1)_{0},(0,0)_{1},(1,1)_{1}\\}$ | $\\{(0,0)_{0},(0,1)_{1},(0,0)_{2}\\}$ | $\\{(0,1)_{0},(1,0)_{0},(0,0)_{1}\\}$
$\\{(1,0)_{0},(0,1)_{1},(1,1)_{1}\\}$ | $\\{(0,1)_{0},(1,1)_{1},(0,1)_{2}\\}$ | $\\{(0,0)_{0},(1,0)_{1},(1,1)_{1}\\}$ | $\\{(0,1)_{0},(1,0)_{1},(1,1)_{2}\\}$
$\\{(1,0)_{0},(1,1)_{0},(0,0)_{2}\\}$ | $\\{(0,0)_{0},(1,0)_{0},(0,1)_{2}\\}$ | $\\{(0,0)_{0},(1,1)_{0},(1,0)_{2}\\}$ | $\\{(0,1)_{0},(0,0)_{2},(1,0)_{2}\\}$
$\\{(1,0)_{0},(1,0)_{2},(1,1)_{2}\\}$ | $\\{(1,1)_{0},(0,1)_{2},(1,1)_{2}\\}$ | $\\{(1,0)_{1},(1,1)_{1},(1,0)_{2}\\}$ | $\\{(0,0)_{1},(1,0)_{1},(0,0)_{2}\\}$
$\\{(0,0)_{1},(0,1)_{2},(1,0)_{2}\\}$ | $\\{(1,1)_{1},(0,0)_{2},(1,1)_{2}\\}$ | $\\{(0,1)_{1},(1,0)_{1},(0,1)_{2}\\}$ | $\\{(0,1)_{1},(1,0)_{2},(1,1)_{2}\\}$

We obtain four S${}_{2}(2,3,13)$s under the action of the group $Z_{2}\times
Z_{2}$. All block sets of these GDD$(2,3,12)$s and S${}_{2}(2,3,12)$s are
pairwise disjoint and they form a CS$(4^{3}:0)$. ∎

Let $X$ be a set of cardinality $tu$, and $\mathcal{H}$ be a partition of $X$
into $u$ subsets of size $t$ (elements of $\mathcal{H}$ are _holes_). Let $L$
be a square array of size $tu$, indexed by $X$, which satisfies: (1) if
$x,y\in H\in\mathcal{H}$, then $L(x,y)$ is empty, otherwise $L(x,y)$ contains
a symbol of $X$; and (2) row or column $x$ of $L$ contains the symbols in
$X\backslash H$, where $x\in H\in\mathcal{H}$. $L$ is called a _partitioned
incomplete Latin square_ (or PILS) of type $t^{u}$. $L$ is _symmetric_ if
$L(x,y)=L(y,x)$ for all $x,y$ not in the same hole.

Now we give a recursive construction for PICS${}^{r}_{2}$, which is similar to
the constructions in [8, 20, 15].

######  Construction 5.2.

Suppose that there exists a PICS${}^{r}_{2}((2m)^{3}:0)$ for any $0\leq r\leq
3m$ with $r\equiv 0\pmod{3}$. Then for each integer $u>2$ and $u\neq 6$, there
exists a PICS${}^{e}_{2}((2mu)^{3}:0)$ for any $0\leq e\leq 3mu$ and $e\equiv
0\pmod{3}$.

###### Proof.

We construct a PICS${}^{e}_{2}((2mu)^{3}:0)$ on $X=Z_{2m}\times Z_{u}\times
Z_{3}$, with the group set $\mathcal{G}=\\{Z_{2m}\times
Z_{u}\times\\{i\\}:0\leq i\leq 2\\}$ for $0\leq e\leq 3mu$ and $e\equiv
0\pmod{3}$. The construction proceeds in three steps.

Step 1: Let $(Z_{u}\times Z_{3},\mathcal{G}^{\prime},\mathcal{A}_{0})$ be a
resolvable GDD$(2,3,3u)$ of type $u^{3}$, which exists from [1], where
$\mathcal{G}^{\prime}=\\{Z_{u}\times\\{i\\}:0\leq i\leq 2\\}$. For each $h\in
Z_{u}$, define
$\mathcal{A}_{h}=\\{\\{(a,0),(b,1),(c+h,2)\\}:\\{(a,0),(b,1),(c,2)\\}\in\mathcal{A}_{0}\\}$.
Then each $(Z_{u}\times Z_{3},\mathcal{G}^{\prime},\mathcal{A}_{h})$ is a
GDD$(2,3,3u)$, and $\\{\mathcal{A}_{h}:h\in Z_{u}\\}$ is a partition of all
triples from three distinct groups.

For $A=\\{(a,0),(b,1),(c,2)\\}\in\mathcal{A}_{h}$ with $1\leq h\leq u-1$,
construct $2m$ pairwise disjoint GDD$(2,3,6m)$s on $Z_{2m}\times A$ with
groups $Z_{2m}\times\\{x\\},~{}x\in A$. The required $2m$ GDD$(2,3,6m)$s have
block set $\mathcal{B}^{i}_{A},i\in Z_{2m}$, which contains the blocks
$\\{(i_{0},a,0),(i_{1},b,1),(i_{2},c,2)\\},$ $i_{0}+i_{1}+i_{2}\equiv i$ (mod
$2m),i_{0},i_{1},i_{2}\in Z_{2m}$.

For $1\leq h\leq u-1$ and $i\in Z_{2m}$, define
$\mathcal{F}_{(u-1)i+h}=\bigcup_{A\in\mathcal{A}_{h}}\mathcal{B}^{i}_{A}$.
Then for $1\leq l\leq 2m(u-1)$, each $(X,\mathcal{G},\mathcal{F}_{l})$ is a
GDD$(2,3,6mu)$ of type $(2mu)^{3}$.

Step 2: Because $\mathcal{A}_{0}$ is resolvable, we can partition it into $u$
parallel classes, $P_{0},P_{1},\ldots,P_{u-1}$, each of which partitions
$Z_{u}\times Z_{3}$. For each $A\in P_{h}$ with $h\in Z_{u}$, we may take an
$e_{A}$ with $0\leq e_{A}\leq 3m$ and $e_{A}\equiv 0\pmod{3}$ such that
$e=\sum_{A\in P_{h}}e_{A}$. We construct a PICS${}^{e_{A}}_{2}((2m)^{3}:0)$ on
$Z_{2m}\times A$ with groups $Z_{2m}\times\\{x\\}$, $x\in A$, which exists by
assumption. Denote its block set by $\mathcal{B}_{A}$. Then $\mathcal{B}_{A}$
can be partitioned into $4m-2$ parts $\mathcal{B}^{i}_{A}$, $1\leq i\leq
4m-2$, so that each $\mathcal{B}^{i}_{A}$ $(1\leq i\leq 2m)$ is the block set
of an S${}_{2}(2,3,6m)$, each $\mathcal{B}^{i}_{A}$ $(2m+1\leq i\leq 4m-2)$ is
the block set of a GDD$(2,3,6m)$ of type $(2m)^{3}$, and there is a subset
$\mathcal{B}^{\prime}_{A}\subset\mathcal{B}^{1}_{A}$ that is the block set of
an NGDD$(2,3,6m)$ of type $2^{(e_{A},3m-e_{A})}$.

For $1\leq i\leq 2m-2$, define
$\mathcal{F}_{2m(u-1)+i}=\bigcup_{A\in\mathcal{A}_{0}}\mathcal{B}^{2m+i}_{A}$.
Then for $2m(u-1)+1\leq l\leq 2mu-2$, each $(X,\mathcal{G},\mathcal{F}_{l})$
is a GDD$(2,3,6mu)$ of type $(2mu)^{3}$.

Step 3: For each $P_{h}\in\\{P_{0},P_{1},\ldots,P_{u-1}\\}$ and $i\in Z_{3}$,
define a permutation $\alpha^{i}_{h}$ of $Z_{u}$ by $\alpha^{i}_{h}(x)=y$ if
the pair $\\{(x,i),(y,i+1)\\}$ is contained in some block of $P_{h}$.

Next let $L$ be a symmetric PILS of type $(2m)^{u}$, having holes
$Z_{2m}\times\\{h\\},h\in Z_{u}$. Such a square exists by [9]. Now for $h\in
Z_{u},i\in Z_{2m}$, define two GDD$(2,3,6mu)$s of type $(6m)^{u}$ on $X$, with
group set $\\{Z_{2m}\times A:A\in P_{h}\\}$, having block sets

$\mathcal{C}^{0}_{i,h}=\big{\\{}\\{(a,b,i),(a^{\prime},b^{\prime},i),(a^{\prime\prime}+i,\alpha^{i}_{h}(b^{\prime\prime}),i+1)\\}:a,a^{\prime}\in
Z_{2m},~{}b,b^{\prime}\in Z_{u},~{}b<b^{\prime},~{}i\in Z_{3}$, and
$(a^{\prime\prime},b^{\prime\prime})=L((a,b),(a^{\prime},b^{\prime}))\big{\\}}$,
and

$\mathcal{C}^{1}_{i,h}=\big{\\{}\\{(a,b,i),(a^{\prime},b^{\prime},i),(a^{\prime\prime}+i,(\alpha^{i-1}_{h})^{-1}(b^{\prime\prime}),i-1)\\}:a,a^{\prime}\in
Z_{2m},~{}b,b^{\prime}\in Z_{u},~{}b<b^{\prime},~{}i\in Z_{3}$, and
$(a^{\prime\prime},b^{\prime\prime})=L((a,b),(a^{\prime},b^{\prime}))\big{\\}}$.

For $h\in Z_{u}$, $i\in Z_{2m}$, define $\mathcal{F}_{i,h}=(\bigcup_{A\in
P_{h}}\mathcal{B}^{i+1}_{A})\cup\mathcal{C}^{0}_{i,h}\cup\mathcal{C}^{1}_{i,h}$
and $\mathcal{F}^{\prime}=(\bigcup_{A\in
P_{0}}\mathcal{B}^{\prime}_{A})\cup\mathcal{C}^{0}_{00}$. Then each
$(X,\mathcal{F}_{i,h})$ is an S${}_{2}(2,3,6mu)$ and
$\mathcal{F}^{\prime}\subset\mathcal{F}_{00}$ is the block set of an
NGDD$(2,3,6mu)$ of type $2^{(e,3mu-e)}$.

All these block sets $\mathcal{F}_{l}$ and $\mathcal{F}_{i,h}$ are pairwise
disjoint, so we obtain a PICS${}^{e}_{2}((2mu)^{3}:0)$ for $0\leq e\leq mu$
and $e\equiv 0\pmod{3}$.∎

######  Lemma 5.3.

There exists a PICS${}^{e}_{2}((2n)^{3}:0)$ whenever $n$ is a positive
integer, $0\leq e\leq 3n$, and $e\equiv 0\pmod{3}$.

###### Proof.

There exists a PICS${}_{2}^{e}(2^{3}:0)$ by [20, Lemma 7.1] where $e=0,3$. For
$n=2$, see Lemma 5.1. For $n\geq 3$ and $n\neq 6$, apply Construction 5.2 with
$m=1$ and $u=n$. For $n=6$, apply Construction 5.2 with $m=2$ and $u=3$.∎

######  Theorem 5.4.

Let $v\equiv 0\pmod{6}$ and $(v,b)$ satisfy (C2). There exists an
NWBTS$(v;b)$.

###### Proof.

According to Lemma 2.5, we can assume that $b\leq\frac{v(v-1)(v-2)}{12}$.
Write $v=6n$ and $b=q\frac{v(v-1)}{3}+b^{\prime}$ where $0\leq
q\leq\left\lfloor\frac{v-2}{4}\right\rfloor\leq 2n-1$ and
$\frac{v(v-1)}{6}-\frac{v}{6}<b^{\prime}<\frac{v(v-1)}{6}+\frac{v}{6}$. We
first deal with the case $q=0$ by considering the parity of $b=b^{\prime}$.
For any even integer $b^{\prime}$ in the range
$\frac{v(v-1)}{6}-\frac{v}{6}<b^{\prime}<\frac{v(v-1)}{6}+\frac{v}{6}$, we may
write $b^{\prime}=\frac{v(v-1)}{6}-\frac{v}{6}+2e=6n^{2}-2n+2e$ where $1\leq
e\leq n-1$. For any odd integer $b^{\prime}$ in this range, we write
$b^{\prime}=\frac{v(v-1)}{6}-\frac{v}{6}+2e+1=6n^{2}-2n+2e+1$ where $0\leq
e\leq n-1$.

Start with a PICS${}^{3e}_{2}((2n)^{3}:0)$
$(V,\emptyset,\mathcal{G},\mathcal{A})$ where $0\leq e\leq n$, which exists
from Lemma 5.3. Its block set contains $2n$ pairwise disjoint subsets
$\mathcal{A}_{l}$, $1\leq l\leq 2n$, and each $\mathcal{A}_{l}$ is the block
set of an S${}_{2}(2,3,6n)$ and $\mathcal{A}_{1}$ has a subset
$\mathcal{B}_{1}$ which is the block set of an NGDD$(2,3,6n)$ of type
$2^{(3e,3n-3e)}$. Clearly $|\mathcal{B}_{1}|=6n^{2}-2n+2e$.

For the even case $b^{\prime}=6n^{2}-2n+2e$ where $1\leq e\leq n-1$, we now
show that $\mathcal{B}_{1}$ actually forms an NWBTS$(v;b^{\prime})$. Since
$\mathcal{B}_{1}$ is a simple NGDD$(2,3,6n)$ of type $2^{(3e,3n-3e)}$ and
$3b^{\prime}=\binom{v}{2}+\varepsilon$ where $\varepsilon=-3n+6e\equiv{v\over
2}\pmod{2}$, we only need to check that a nearly 2-balanced TS$(v;b^{\prime})$
should have a defect graph $D=D_{1}\cup D_{-1}$ isomorphic to
$H_{v,\varepsilon}^{0}$ with $|D_{1}|=3e$ and $|D_{-1}|=3n-3e$. Noticing
$({v+2\varepsilon\over 4},{v-2\varepsilon\over 4})=(3e,3n-3e)$ yields that
$\mathcal{B}_{1}$ is nearly 2-balanced and hence nearly well-balanced. Let
$\mathcal{F}=\mathcal{B}_{1}$ in this case.

We then treat the case of $b^{\prime}=6n^{2}-2n+2e+1$ where $0\leq e\leq n-1$
based on the previous block set $\mathcal{B}_{1}$, whose defect graph is
$D=D_{1}\cup D_{-1}$ (noting ${\cal B}_{1}$ is a GDD and $D_{1}=\emptyset$
when $e=0$). If there exists a block
$\\{x,y,z\\}\in\mathcal{A}_{1}\backslash\mathcal{B}_{1}$ where
$\\{x,y\\},\\{z,w\\}\in D_{-1}$ for some element $w\notin\\{x,y,z\\}$, then we
let $\mathcal{F}=\mathcal{B}_{1}\cup\\{\\{x,y,z\\}\\}$ and it is readily
checked that $\mathcal{F}$ is an NWBTS$(v;b^{\prime})$ by showing its defect
graph isomorphic to $H_{v,\varepsilon}^{1}$ where $\varepsilon=-3n+6e+3$ and
$({v+2+2\varepsilon\over 4},{v+2-2\varepsilon\over 4})=(3e+2,3n-3e-1)$. This
solves the case of odd $b^{\prime}$. Now we show such a block $\\{x,y,z\\}$
exists. We trace back to the proof of Construction 5.2. Observe that $e\leq
n-1$ and we need to input at least one PICS${}^{r}_{2}((2m)^{3}:0)$ with
$r\leq 3(m-1)$ and $r\equiv 0\pmod{3}$ for Construction 5.2. By the
construction method we only need to check that such an input design
PICS${}^{r}_{2}((2m)^{3}:0)$ with $r\leq 3(m-1)$ and $r\equiv 0\pmod{3}$ has
such a desirable property. Checking the proof of Lemma 5.3, we only need the
input designs for $m=1,2$. A PICS${}^{0}_{2}((2m)^{3}:0)$ certainly satisfies
this and it also holds for the example of PICS${}^{3}_{2}(4^{3}:0)$ in Lemma
5.1.

Finally when $1\leq q\leq 2n-1$, construct a family $\mathcal{F}^{\prime}$
consisting of $q$ $\mathcal{A}_{l}$’s for $2\leq l\leq 2n$. Then by Lemma 3.1,
$\mathcal{F}\cup\mathcal{F}^{\prime}$ is an NWBTS$(v;b)$, where
$b=q\frac{v(v-1)}{3}+b^{\prime}$.∎

##  6 Cases $v\equiv 2,4\pmod{6}$ for (C2)

In this section, we define candelabra systems with another three types of
partitions and give three corresponding recursive constructions via $s$-fan
designs. By constructing candelabra systems with special partitions, we
establish the existence of a large portion of NWBTSs for the parameters
satisfying (C2) and $v\equiv 2,4\pmod{6}$.

###  6.1 Recursive constructions

Let $(X,S,\mathcal{G},\mathcal{A})$ be a CS$((2ag)^{n}:s)$ with $a,g$ being
positive integers. Further let $r$ be an integer with $0\leq r\leq ag(n-1)$.
Then $(X,S,\mathcal{G},\mathcal{A})$ is called $(g,r)$-partitionable with
index two and denoted by $g$-PCS${}^{r}_{2}((2ag)^{n}:s)$ if its block set
$\mathcal{A}$ contains a subset $\mathcal{A}^{\prime}$ and
$\mathcal{A}^{\prime}$ can be partitioned into $gn$ pairwise disjoint parts
$\mathcal{A}_{i}$, $0\leq i\leq gn-1$, fulfilling that

(i) for each group $G\in\mathcal{G}$, there are exactly $g$
$\mathcal{A}_{i}$’s such that each $\mathcal{A}_{i}$ is the block set of a
GDD${}_{2}(2,3,2agn+s)$ of type $1^{2ag(n-1)}(2ag+s)^{1}$ with $G\cup S$ as
its long group, and

(ii) at least one of the $gn$ GDDs contains a sub-NGDD$(2,3,2agn+s)$ of type
$2^{(r,ag(n-1)-r)}(2ag+s)^{1}$.

We strengthen a little on the conditions of $g$-PCS${}^{r}_{2}((2ag)^{n}:s)$
to define $g^{+}$-PCS${}^{r}_{2}((2ag)^{n}:s)$. Let
$(X,S,\mathcal{G},\mathcal{A})$ be a CS$((2ag)^{n}:s)$ with $a,g$ being
positive integers. It is said to be $(g^{+},r)$-partitionable with index two
and denoted by $g^{+}$-PCS${}^{r}_{2}((2ag)^{n}:s)$ if its block set
$\mathcal{A}$ contains a subset $\mathcal{A}^{\prime}$ and
$\mathcal{A}^{\prime}$ can be partitioned into $gn+1$ pairwise disjoint parts
$\mathcal{A}_{i}$, $0\leq i\leq gn$, when it satisfies that

(i) for a fixed group $G_{0}\in\mathcal{G}$, there are exactly $g+1$
$\mathcal{A}_{i}$’s such that each $\mathcal{A}_{i}$ is the block set of a
GDD${}_{2}(2,3,2agn+s)$ of type $1^{2ag(n-1)}(2ag+s)^{1}$ with long group
$G_{0}\cup S$ and at least one of $\mathcal{A}_{i}$’s contains a subset
$\mathcal{A}^{\prime}$ such that $\mathcal{A}^{\prime}$ is the block set of an
NGDD$(2,3,2agn+s)$ of type $2^{(r,ag(n-1)-r)}(2ag+s)^{1}$, and

(ii) for each group $G\in\mathcal{G}\backslash\\{G_{0}\\}$, there are exactly
$g$ $\mathcal{A}_{i}$’s such that each $\mathcal{A}_{i}$ is the block set of a
GDD${}_{2}(2,3,2agn+s)$ of type $1^{2ag(n-1)}(2ag+s)^{1}$ with long group
$G\cup S$.

Let $(X,\mathcal{G},\mathcal{A})$ be a GDD$(3,3,2g(n+1))$ of type $(2g)^{n+1}$
and $G_{0}$ be one of its groups. Such a GDD is _partitionable with index two_
, denoted by PGDD${}_{2}((2g)^{n+1})$ (as in [20]), if the block set
$\mathcal{A}$ can be partitioned into $\mathcal{A}_{i}$, $1\leq i\leq gn+2g$,
so that

(i) for each $G\in\mathcal{G}$ with $G\neq G_{0}$, there are exactly $g$
$\mathcal{A}_{i}$’s such that each $\mathcal{A}_{i}$ is the block set of a
GDD${}_{2}(2,3,2gn)$ of type $(2g)^{n}$ with group set
$\mathcal{G}\backslash\\{G\\}$, and

(ii) each $(X\backslash
G_{0},\mathcal{G}\backslash\\{G_{0}\\},\mathcal{A}_{i})$, $gn+1\leq i\leq
gn+2g$, is a GDD$(2,3,2gn)$ of type $(2g)^{n}$.

######  Lemma 6.1.

A PGDD${}_{2}((2g)^{n+1})$ exists if and only if $n\geq 3$ and $gn(n-1)\equiv
0\pmod{3}$.

###### Proof.

The necessary conditions for the existence of a PGDD${}_{2}((2g)^{n+1})$ are
$n\geq 3$ and $gn(n-1)\equiv 0\pmod{3}$. For $n\geq 3$ and $gn(n-1)\equiv
0\pmod{3}$ with $(n,g)\neq(4,1)$, there exists a PGDD${}_{2}((2g)^{n+1})$,
which can be produced by the so-called generalized frame
F$(3,3,(n+1)\\{2g\\})$ in [14, Corollary 3.3]. A PGDD${}_{2}(2^{5})$ exists by
[20, Lemma 6.4]. Then we obtain the desired result. ∎

Let $(X,S,\mathcal{G},\mathcal{A})$ be a CS$(3,K,v)$ of type
$(g^{n_{1}}_{1}g^{n_{2}}_{2}\cdots g^{n_{u}}_{u}:s)$ with
$S=\\{\infty_{1},\ldots,\infty_{s}\\}$. For $1\leq i\leq s$, let
$\mathcal{A}_{i}=\\{A\backslash\\{\infty_{i}\\}:A\in\mathcal{A},~{}\infty_{i}\in
A\\}$ and $\mathcal{A}_{T}=\\{A\in\mathcal{A}:A\cap S=\emptyset\\}$. Then the
$(s+3)$-tuple $(X\backslash
S,\mathcal{G},\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{s},\mathcal{A}_{T})$
is called an $s$-_fan design_ (as in [12]). If the block sizes of
$\mathcal{A}_{i}$ and $\mathcal{A}_{T}$ are from $K_{i}~{}(1\leq i\leq s)$ and
$K_{T}$, respectively, then the $s$-fan design is denoted by
$s$-FG$(3,(K_{1},\ldots,K_{s},K_{T})$, $\sum^{u}_{i=1}n_{i}g_{i})$ of type
$g^{n_{1}}_{1}g^{n_{2}}_{2}\cdots g^{n_{u}}_{u}$.

######  Construction 6.2.

Suppose that there exists an $h$-FG$(3,(K_{1},\ldots$, $K_{h}$,$K_{T}),gn)$ of
type $g^{n}$, where $mg(n-1)\equiv 0\pmod{3}$ and $m(k-1)\equiv 0\pmod{3}$ for
each $k\in K_{1}$. Further suppose that there exists

  1. (i)

an $m$-PCS${}^{r}_{2}((2am)^{k}:l)$ for each $k\in K_{1}$ and each $0\leq
r\leq am(k-1)$ with $r\equiv 0\pmod{3a}$;

  2. (ii)

a PGDD${}_{2}((2am)^{k+1})$ for each $k\in K_{j}~{}(2\leq j\leq h)$; and

  3. (iii)

a PGDD${}_{2}((2am)^{k})$ for each $k\in K_{T}$.

Then there exists an $mg$-PCS${}^{e}_{2}((2amg)^{n}:2(h-1)am+l)$ for any
$0\leq e\leq amg(n-1)$ and $e\equiv 0\pmod{3a}$.

###### Proof.

Let
$(X,\mathcal{G},\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{h},\mathcal{T})$
be the given $h$-FG$(3,(K_{1},K_{2},\ldots,K_{h},K_{T}),gn)$ of type $g^{n}$.
Let $S=\\{\infty\\}\times Z_{s}$, where $s=2(h-1)am+l$. We shall construct the
desired design on $X^{\prime}=(X\times Z_{m}\times Z_{2a})\cup S$ with the
group set $\mathcal{G}^{\prime}=\\{G^{\prime}=G\times Z_{m}\times
Z_{2a}:G\in\mathcal{G}\\}$ and the stem $S$, where $(X\times Z_{m}\times
Z_{2a})\cap S=\emptyset$. We shall describe its block set $\mathcal{F}$ below.

Denote $G_{x}=\\{x\\}\times Z_{m}\times Z_{2a}$ for $x\in X$ and $S=S_{1}\cup
S_{2}\cup\cdots\cup S_{h}$, where $S_{1}=\\{\infty\\}\times
Z_{l},~{}S_{j}=\\{(\infty,l+2(j-2)am),\ldots,(\infty,l+2(j-1)am-1)\\}$ for
$2\leq j\leq h$. Fix a point $x_{0}\in X$ where $x_{0}\in
G_{0}\in\mathcal{G}$. Let $\mathcal{A}_{1,0}=\\{A\in\mathcal{A}_{1}:x_{0}\in
A\\}$. We know that $\\{A\backslash\\{x_{0}\\}:A\in\mathcal{A}_{1,0}\\}$ is a
partition of $X\backslash G_{0}$. For any $e$ with $0\leq e\leq amg(n-1)$ and
$e\equiv 0\pmod{3a}$, we may take $e_{A}$ with $0\leq e_{A}\leq am(|A|-1)$ and
$e_{A}\equiv 0\pmod{3a}$ for $A\in\mathcal{A}_{1}$ such that
$e=\sum_{A\in\mathcal{A}_{1,0}}e_{A}$.

For each block $A\in\mathcal{A}_{1}$, construct a
PCS${}^{e_{A}}_{2}((2am)^{|A|}:l)$ on $(A\times Z_{m}\times Z_{2a})\cup S_{1}$
having $\\{G_{x}:x\in A\\}$ as its group set, $S_{1}$ as its stem. Such a
design exists by assumption. Denote its block set by $\mathcal{D}^{*}_{A}$
which contains a subset $\mathcal{D}_{A}$ and $\mathcal{D}_{A}$ can be
partitioned into $m|A|$ pairwise disjoint block sets $\mathcal{D}_{A}(x,i)$,
$(x,i)\in A\times Z_{m}$, satisfying that

(D1) each $\mathcal{D}_{A}(x,i)$ is the block set of a GDD${}_{2}(2,3$,
$2am|A|+l)$ of type $1^{2am(|A|-1)}$ $(2am+l)^{1}$ and with long group
$G_{x}\cup S_{1}$;

(D2) if $x_{0}\in A$, then there is a subset
$\mathcal{D}^{\prime}_{A}(x_{0},0)$ of $\mathcal{D}_{A}(x_{0},0)$ such that
$\mathcal{D}^{\prime}_{A}(x_{0},0)$ is the block set of an
NGDD$(2,3,2am|A|+l)$ of type $2^{(e_{A},am(|A|-1)-e_{A})}(2am+l)^{1}$.

For any $2\leq j\leq h$ and any block $A\in\mathcal{A}_{j}$, construct a
PGDD${}_{2}((2am)^{|A|+1})$ on $(A\times Z_{m}\times Z_{2a})\cup S_{j}$ having
$\Gamma_{A}=\\{G_{x}:x\in A\\}\cup\\{S_{j}\\}$ as its group set. Such a design
exists by assumption. Denote its block set by $\mathcal{C}^{*j}_{A}$ which
contains a subset $\mathcal{C}^{j}_{A}$. Two disjoint GDD$(2,3,2am|A|)$s with
the same group set can be seen as a GDD${}_{2}(2,3,2am|A|)$ containing a sub-
GDD$(2,3,2am|A|)$. So $\mathcal{C}^{j}_{A}$ can be partitioned into $m|A|$
pairwise disjoint block sets $\mathcal{C}^{j}_{A}(x,i)$, $(x,i)\in A\times
Z_{m}$, satisfying that

(N1) for any $(x,i)\in A\times Z_{m}$, each $\mathcal{C}^{j}_{A}(x,i)$ is the
block set of a GDD${}_{2}(2,3,2am|A|)$ of type $(2am)^{|A|}$ with the group
set $\Gamma_{A}\backslash\\{G_{x}\\}$;

(N2) if $x_{0}\in A$, then there is a subset $\mathcal{C}^{\prime
j}_{A}(x_{0},0)$ of $\mathcal{C}^{j}_{A}(x_{0},0)$ so that
$\mathcal{C}^{\prime j}_{A}(x_{0},0)$ is the block set of a GDD$(2,3,2am|A|)$
with the group set $\Gamma_{A}\backslash\\{G_{x_{0}}\\}$.

For each $A\in\mathcal{T}$, construct a PGDD${}_{2}((2am)^{|A|})$ on $A\times
Z_{m}\times Z_{2a}$ having $\Gamma_{A}^{\prime}=\\{G_{x}:x\in A\\}$ as its
group set. Such a design exists by assumption. Denote its block set by
$\mathcal{B}^{*}_{A}$ which contains a subset $\mathcal{B}_{A}$; the block set
$\mathcal{B}_{A}$ can be partitioned into $m|A|$ pairwise disjoint block sets
$\mathcal{B}_{A}(x,i)$, $(x,i)\in A\times Z_{m}$, satisfying that

(B1) each $\mathcal{B}_{A}(x,i)$ is the block set of a
GDD${}_{2}(2,3,2am(|A|-1))$ with group set
$\Gamma_{A}^{\prime}\backslash\\{G_{x}\\}$;

(B2) if $x_{0}\in A$, then there is a subset
$\mathcal{B}^{\prime}_{A}(x_{0},0)$ of $\mathcal{B}_{A}(x_{0},0)$ so that
$\mathcal{B}^{\prime}_{A}(x_{0},0)$ is the block set of a
GDD$(2,3,2am(|A|-1))$ with the group set
$\Gamma_{A}^{\prime}\backslash\\{G_{x_{0}}\\}$.

For each $(x,i)\in X\times Z_{m}$, define

$\mathcal{F}(x,i)=\big{(}\bigcup_{x\in
A,A\in\mathcal{A}_{1}}\mathcal{D}_{A}(x,i)\big{)}\bigcup\big{(}\bigcup_{2\leq
j\leq h}~{}\bigcup_{x\in
A,A\in\mathcal{A}_{j}}\mathcal{C}^{j}_{A}(x,i)\big{)}\bigcup\big{(}\bigcup_{x\in
A,A\in\mathcal{T}}\mathcal{B}_{A}(x,i)\big{)}.$

Define

$\mathcal{F}^{\prime}(x_{0},0)=\big{(}\bigcup_{x_{0}\in
A,A\in\mathcal{A}_{1}}\mathcal{D}^{\prime}_{A}(x_{0},0)\big{)}\bigcup\big{(}\bigcup_{2\leq
j\leq h}~{}\bigcup_{x_{0}\in A,A\in\mathcal{A}_{j}}\mathcal{C}^{\prime
j}_{A}(x_{0},0)\big{)}\bigcup\big{(}\bigcup_{x_{0}\in
A,A\in\mathcal{T}}\mathcal{B}^{\prime}_{A}(x_{0},0)\big{)},$

Finally, define

$\mathcal{F}=\big{(}\bigcup_{A\in\mathcal{A}_{1}}\mathcal{D}^{*}_{A}\big{)}\bigcup\big{(}\bigcup_{2\leq
j\leq
h}~{}\bigcup_{A\in\mathcal{A}_{j}}\mathcal{C}^{*j}_{A}\big{)}\bigcup\big{(}\bigcup_{A\in\mathcal{T}}\mathcal{B}^{*}_{A}\big{)}.$

It is readily checked that $(X^{\prime},S,\mathcal{G}^{\prime},\mathcal{F})$
is a $mg$-PCS${}^{e}_{2}((2amg)^{n}:2(h-1)am+l)$; each $\mathcal{F}(x,i)$ is
the block set of a GDD${}_{2}(2,3,2amgn+s)$ of type
$1^{2amg(n-1)}(2amg+s)^{1}$ with long group $(G\times Z_{m}\times Z_{2a})\cup
S$; $\mathcal{F}^{\prime}(x_{0},0)\subset\mathcal{F}(x_{0},0)$ is the block
set of an NGDD$(2,3,2amgn+s)$ of type $2^{(e,amg(n-1)-e)}(2amg+s)^{1}$. ∎

We also employ a variant of Construction 6.2.

######  Construction 6.3.

Suppose that there exists a $1$-FG$(3,(K_{1},K_{T}),gn)$ of type $g^{n}$ where
$mg(n-1)\equiv 0\pmod{3}$ and $m(k-1)\equiv 0\pmod{3}$ for each $k\in K_{1}$.
Further suppose that there exists an $m^{+}$-PCS${}^{r}_{2}((4m)^{k}:l)$ for
each $k\in K_{1}$ and each $0\leq r\leq 2m(k-1)$ with $r\equiv 0\pmod{6}$, and
a PGDD${}_{2}((4m)^{k})$ for each $k\in K_{T}$, then there exists an
$(mg)^{+}$-PCS${}^{e}_{2}((4mg)^{n}:l)$ for any $0\leq e\leq 2mg(n-1)$ and
$e\equiv 0\pmod{6}$.

######  Theorem 6.4.

A $1$-FG$(3,(\\{3,5\\},\\{4,6\\}),n)$ of type $1^{n}$ exists for any $n\equiv
1\pmod{2}$.

###### Proof.

An S$(3,\\{4,6\\},n+1)$ exists if and only if $n\equiv 1$ (mod 2) [11].
Deleting a point yields the desired 1-fan design.∎

######  Lemma 6.5.

Let $m\equiv n\equiv 0\pmod{3}$, $u\equiv 2,4\pmod{6}$, and $v=2m+2n+u$.
Further let $c<{u\choose 3}$ and $(u,c)$ satisfy (C2) associating with
$(\lambda,\delta)$. Suppose that

  1. (i)

$(V,{\cal A})$ is a simple NGDD$(2,3,v)$ of type $2^{(m,n)}u^{1}$ with
$G_{i},1\leq i\leq m+n,$ being groups of size $2$ and $U$ the group of size
$u$;

  2. (ii)

$(V,{\cal B})$ is a simple GDD${}_{\lambda-1}(2,3,v)$ of type $1^{2m+2n}u^{1}$
with long group $U$; and

  3. (iii)

$(U,\mathcal{C})$ is an NWBTS$(u;c)$.

If the block sets ${\cal A},{\cal B}$ and ${\cal C}$ are mutually disjoint,
then ${\cal F}:={\cal A}\cup{\cal B}\cup{\cal C}$ forms the block set of an
NWBTS$(v;|{\cal F}|)$, where $3|{\cal F}|=\lambda{v\choose 2}+m-n+\delta$.

###### Proof.

It is obvious that $3|{\cal A}|={v\choose 2}-{u\choose 2}+m-n$ and $3|{\cal
B}|=\big{(}\lambda-1)({v\choose 2}-{u\choose 2}\big{)}$. And using
$3c=\lambda{u\choose 2}+\delta$ yields $3|{\cal F}|=3(|{\cal A}|+|{\cal
B}|+c)=\lambda{v\choose 2}+\varepsilon$ where $\varepsilon=m-n+\delta$.
Clearly $-\frac{v}{2}<\varepsilon<\frac{v}{2}$ as $v=2m+2n+u$ and
$-\frac{u}{2}<\delta<\frac{u}{2}$. Let $(U,\mathcal{C})$ have a defect graph
$D=D_{1}\cup D_{-1}$, isomorphic to $H_{u,\delta}^{0}$ or one of
$H_{u,\delta}^{1}$ and $H_{u,\delta}^{2}$, corresponding to the case ${u\over
2}-\delta$ being even or odd. By the construction we have that every pair of
$V$ is contained in $\lambda$ blocks of ${\cal F}$ except that each pair in
$D_{1}^{\cal F}:=P_{1}\cup D_{1}$ is contained in $\lambda+1$ blocks and each
pair in $D_{-1}^{\cal F}:=P_{-1}\cup D_{-1}$ is contained in $\lambda-1$
blocks, where $P_{1}=\\{G_{i}:1\leq i\leq m\\}$ and $P_{-1}=\\{G_{i}:m+1\leq
i\leq m+n\\}$ represent the groups contained in two blocks and no blocks of
${\cal A}$, respectively. Thus we determined the defect graph $D^{\cal
F}=D_{1}^{\cal F}\cup D_{-1}^{\cal F}$ of ${\cal F}$. In the case of ${u\over
2}\equiv\delta$ (mod 2), we have ${v\over 2}\equiv\varepsilon$ (mod 2),
$({v+2\varepsilon\over 4},{v-2\varepsilon\over 4})=(m+|D_{1}|,n+|D_{-1}|)$,
$D^{\cal F}$ is isomorphic to $H_{v,\varepsilon}^{0}$, and hence ${\cal F}$ is
nearly 2-balanced. Similar arguments yield that $D^{\cal F}$ is isomorphic to
$H_{v,\varepsilon}^{i}$ $(i=1,2)$ if $D$ is isomorphic to $H_{u,\delta}^{i}$
in the case ${u\over 2}\not\equiv\delta$ (mod 2). Finally noticing ${\cal F}$
is a simple family yields that it is nearly well-balanced. ∎

######  Construction 6.6.

Let $g\equiv 0\pmod{3}$, $s\equiv 2,4\pmod{6}$, and $\lambda\in\\{1,3,5\\}$.
Suppose that there exists a $g$-PCS${}^{e}_{2}((2ag)^{n}:s)$ for any $0\leq
e\leq ag(n-1)$ and $e\equiv 0\pmod{3a}$. Further let $u=2ag+s$ and suppose
that there exists

  1. (i)

a simple S${}_{6}(2,3,u)$ defined over a $u$-set $X$, each of whose blocks is
not strictly contained in an $s$-subset $Y\subset X$; and

  2. (ii)

an NWBTS$(u;c)$ for any $c=(\lambda{u\choose 2}+\delta)/3$ where $-{u\over
2}<\delta<{u\over 2}$.

Let $v=2agn+s$. Then there exists an NWBTS$(v;b)$ for any
$b=qv(v-1)+b^{\prime}$, where $0\leq q\leq n-1$ and
$3b^{\prime}=\lambda\binom{v}{2}+\varepsilon$ with $-{v\over
2}<\varepsilon<{v\over 2}$.

###### Proof.

Let $V=(Z_{n}\times Z_{g}\times Z_{2a})\cup S$, $|S|=s$, and
$\mathcal{G}=\\{G_{0},G_{1},\ldots,G_{n-1}\\}$ where $G_{i}=\\{i\\}\times
Z_{g}\times Z_{2a}$. For any fixed $e$ with $0\leq e\leq ag(n-1)$ and $e\equiv
0\pmod{3a}$, let $(V,S,\mathcal{G},\mathcal{A})$ be an assumed
$g$-PCS${}_{2}^{e}((2ag)^{n}:s)$. Then the block set $\mathcal{A}$ contains a
subset $\mathcal{A}^{\prime}$ and $\mathcal{A}^{\prime}$ can be partitioned
into $gn$ pairwise disjoint parts $\mathcal{A}(i,j)$, $0\leq i\leq n-1$,
$0\leq j\leq g-1$, so that each $\mathcal{A}(i,j)$ is the block set of a
GDD${}_{2}(2,3,v)$ of type $1^{2ag(n-1)}u^{1}$ with long group $G_{i}\cup S$,
and $\mathcal{A}(0,0)$ contains a subset $\mathcal{A}^{\prime}(0,0)$ that is
the block set of an NGDD$(2,3,v)$ of type $2^{(e,ag(n-1)-e)}u^{1}$. By
assumption, there exists an NWBTS$(u;c)$ on $G_{0}\cup S$ with block set
$\mathcal{C}_{\lambda}$ where $3c=\lambda\binom{u}{2}+\delta$. Construct a
family $\mathcal{F}$ containing all blocks of $\mathcal{A}^{\prime}(0,0)$,
$(\lambda-1)/2$ sets in $\\{\mathcal{A}(0,j):j=1,2\\}$, and all blocks of
$\mathcal{C}_{\lambda}$. Then by Lemma 6.5, $\mathcal{F}$ is an
NWBTS$(v;b^{\prime})$, where
$3b^{\prime}=\lambda\binom{v}{2}-ag(n-1)+2e+\delta$.

For each $1\leq i\leq n-1$, construct a simple S${}_{6}(2,3,u)$ on $G_{i}\cup
S$ with block set $\mathcal{B}(i)\subset\binom{G_{i}\cup
S}{3}\backslash\binom{S}{3}$ which exists by assumption. Let
$\mathcal{F}(i)=(\cup^{2}_{j=0}\mathcal{A}(i,j))\cup\mathcal{B}(i)$. Then each
$\mathcal{F}(i)$ is the block set of a simple S${}_{6}(2,3,v)$ and all
$\mathcal{F}(i)$s are pairwise disjoint. Construct a family
$\mathcal{F}^{\prime}$ containing $q$ $\mathcal{F}(i)$s where $0\leq q\leq
n-1$. Thus $\mathcal{F}\cup\mathcal{F}^{\prime}$ is an
NWBTS$(v;qv(v-1)+b^{\prime})$ by Lemma 3.1, where
$3b^{\prime}=\lambda\binom{v}{2}+\varepsilon$ with $\varepsilon=-{v-u\over
2}+2e+\delta$.

Now let $s\equiv 2\pmod{6}$. We know that $(u,c)$ associates with
$(\lambda,\delta)$. When $e,\delta$ take all admissible integers, it is
readily checked that the value of $\varepsilon$ covers all admissible integers
in the intervals $I_{e}:=[-{v\over 2}+2e+\tau_{1},-{v\over 2}+u+2e+\tau_{2}]$
with $0\leq e\leq ag(n-1)$ and $e\equiv 0\pmod{3a}$, where

$(\tau_{1},\tau_{2})=\left\\{\begin{array}[]{ll}{(3,-2)},&{\rm if\
}\lambda=1,\\\ {(1,-1)},&{\rm if\ }\lambda=3,\\\ {(2,-3)},&{\rm if\
}\lambda=5.\\\ \end{array}\right.$

It is not difficult to compare the right endpoint $y_{j}$ of $I_{j}$ with the
left endpoint $x_{j+1}$ of $I_{j+1}$ to have that $y_{j}+3\geq x_{j+1}$. Thus
these intervals cover all admissible integers $\varepsilon$ with $-{v\over
2}<\varepsilon<{v\over 2}$. As a consequence, an NWBTS$(v;qv(v-1)+b^{\prime})$
exists for any $0\leq q\leq n-1$ and
$3b^{\prime}=\lambda\binom{v}{2}+\varepsilon$ with $-{v\over
2}<\varepsilon<{v\over 2}$. Similar arguments also apply for the case $s\equiv
4\pmod{6}$; in this case we always have $(\tau_{1},\tau_{2})=(2,-3)$. The
details are omitted. ∎

###  6.2 $v\equiv 2\pmod{6}$

In this subsection, we construct $3$-PCS${}^{e}_{2}((6a)^{n}:2)$s for $a=1,2$
and $0\leq e\leq 3a(n-1)$ with $e\equiv 0\pmod{3a}$, and then show the
existence of an NWBTS$(v;b)$ if $v\equiv 2\pmod{6}$, $v\neq 50,74$, and
$(v,b)$ satisfies (C2).

######  Lemma 6.7.

There is a $3$-PCS${}^{r}_{2}(6^{3}:2)$ for any $r=0,3,6$.

###### Proof.

For $r=0,6$, a $3$-PCS${}^{r}_{2}(6^{3}:2)$ exists by [20, Lemma 6.6]. Next,
we construct a $3$-PCS${}^{3}_{2}(6^{3}:2)$ on $X=Z_{18}\cup\\{a,b\\}$ with
stem $S=\\{a,b\\}$ and groups $G_{i}=\\{i,i+3,i+6,\ldots,i+15\\},i=0,1,2$. We
first construct a GDD${}_{2}(2,3,20)$ of type $1^{12}8^{1}$ with long group
$G_{0}\cup S$, whose blocks are generated from the following blocks by adding
9 (mod 18); the underlined base blocks and their developments form the blocks
of an NGDD$(2,3,20)$ of type $2^{(3,3)}8^{1}$, where
$\lambda_{1,10}=\lambda_{4,13}=\lambda_{7,16}=2$ and
$\lambda_{2,11}=\lambda_{5,14}=\lambda_{8,17}=0$.

$\underline{\\{a,1,2\\}}$ | $\underline{\\{a,4,8\\}}$ | $\underline{\\{a,7,14\\}}$ | $\underline{\\{b,4,5\\}}$ | $\underline{\\{b,7,11\\}}$ | $\underline{\\{b,1,8\\}}$ | $\underline{\\{2,6,17\\}}$ | $\underline{\\{5,6,11\\}}$
---|---|---|---|---|---|---|---
$\underline{\\{1,3,10\\}}$ | $\underline{\\{3,4,13\\}}$ | $\underline{\\{3,7,16\\}}$ | $\underline{\\{0,1,7\\}}$ | $\underline{\\{6,7,8\\}}$ | $\underline{\\{5,7,10\\}}$ | $\underline{\\{8,13,16\\}}$ | $\underline{\\{3,11,17\\}}$
$\underline{\\{2,10,13\\}}$ | $\underline{\\{1,6,13\\}}$ | $\underline{\\{6,10,14\\}}$ | $\underline{\\{0,8,10\\}}$ | $\underline{\\{0,2,4\\}}$ | $\underline{\\{4,6,16\\}}$ | $\underline{\\{2,3,5\\}}$ | $\underline{\\{3,8,14\\}}$
$\underline{\\{4,9,14\\}}$ | $\underline{\\{2,7,9\\}}$ | $\underline{\\{0,14,17\\}}$ | $\\{2,3,11\\}$ | $\\{3,5,14\\}$ | $\\{3,8,17\\}$ | $\\{b,2,4\\}$ | $\\{b,7,17\\}$
$\\{3,10,13\\}$ | $\\{1,3,7\\}$ | $\\{3,4,16\\}$ | $\\{0,2,14\\}$ | $\\{0,7,8\\}$ | $\\{1,6,16\\}$ | $\\{a,4,17\\}$ | $\\{b,1,14\\}$
$\\{0,1,5\\}$ | $\\{0,13,16\\}$ | $\\{0,11,17\\}$ | $\\{0,4,10\\}$ | $\\{6,10,17\\}$ | $\\{4,6,11\\}$ | $\\{a,1,11\\}$ | $\\{a,5,7\\}$
$\\{1,2,17\\}$ | $\\{7,11,14\\}$ | $\\{2,6,7\\}$ | $\\{4,5,8\\}$ | $\\{5,6,13\\}$ | $\\{6,8,14\\}$ |  |

Nine GDD${}_{2}(2,3,20)$s are obtained by developing modulo 18, forming a
$3$-PCS${}^{3}_{2}(6^{3}:2)$.∎

######  Lemma 6.8.

Let $n\equiv 1\pmod{2}$ and $n\geq 3$. Then there is a
$3$-PCS${}_{2}^{e}(6^{n}:2)$ for any $0\leq e\leq 3(n-1)$ and $e\equiv
0\pmod{3}$.

###### Proof.

For $n=3$, see Lemma 6.7. For $n=5$, there exists a 1-FG$(3,(4,\\{4,5\\}),15)$
of type $3^{5}$, which is obtained by deleting two points from the known
S$(3,5,17)$ (see [10]). Apply Construction 6.2 with $h=a=m=1$ and $l=2$. The
input designs are $1$-PCS${}^{i}_{2}(2^{4}:2)$s $(i=0,3)$ in [20, Lemma 6.1]
and PGDD${}_{2}(2^{j})$s $(j=4,5)$ in Lemma 6.1. Then we get a
$3$-PCS${}^{r}_{2}(6^{5}:2)$ for any $r\in\\{0,3,6,9,12\\}$.

For $n\equiv 1$ (mod 2) and $n\geq 7$, start with a
1-FG$(3,(\\{3,5\\},\\{4,6\\}),n)$ of type $1^{n}$ which exists by Lemma 6.4. A
$3$-PCS${}^{r}_{2}(6^{k}:2)$ exists by the above proofs for $k\in\\{3,5\\}$
and $r\leq 3(k-1)$ with $r\equiv 0\pmod{3}$. A PGDD${}_{2}(6^{k})$ for
$k\in\\{4,6\\}$ exists by Lemma 6.1. Then we obtain the desired design by
applying Construction 6.2 with $h=a=1$, $l=2$, and $m=3$.∎

######  Lemma 6.9.

Let $v\equiv 8\pmod{12}$ and $(v,b)$ satisfy (C2). There exists an
NWBTS$(v;b)$.

###### Proof.

For $v=8$, see Lemma A.3 in Appendix. For $v\geq 20$, we can assume that
$b\leq\frac{v(v-1)(v-2)}{6}$ by Lemma 2.5. Write $v=6n+2$ where $n$ is odd;
write $b=qv(v-1)+b^{\prime}$ where $0\leq q\leq\frac{v-2}{6}-1=n-1$ and $0\leq
b^{\prime}\leq v(v-1)$. According to Lemma 2.5, we can assume that $0\leq
b^{\prime}\leq\frac{v(v-1)}{2}$ and $\frac{\lambda
v(v-1)}{6}-\frac{v}{6}<b^{\prime}<\frac{\lambda v(v-1)}{6}+\frac{v}{6}$ with
$\lambda=1,3$. Thus $(v,b^{\prime})$ associates with $(\lambda,\varepsilon)$
where $\lambda=1,3$ and $-\frac{v}{2}<\varepsilon<\frac{v}{2}$. Apply
Construction 6.6 with $a=1$ and $(g,s)=(3,2)$. There exists a
$3$-PCS${}_{2}^{e}(6^{n}:2)$ for any $0\leq e\leq 3(n-1)$ and $e\equiv
0\pmod{3}$ from Lemma 6.8. An NWBTS$(8;c)$ exists for all $c$ with $(8,c)$
satisfying (C2) from Lemma A.3 in Appendix. A simple S${}_{6}(2,3,8)$ exists
trivially. Then there exists an NWBTS$(v;b)$. ∎

######  Lemma 6.10.

There is a $3$-PCS${}^{r}_{2}(12^{3}:2)$ for $r=0,6,12$.

###### Proof.

For $r=0,12$, a $3$-PCS${}^{r}_{2}(12^{3}:2)$ exists by [20, Lemma 6.12]. For
$r=6$, we construct a $3$-PCS${}^{6}_{2}(12^{3}:2)$ on $X=(Z_{12}\times
Z_{3})\cup\\{a,b\\}$ with stem $S=\\{a,b\\}$ and groups
$G_{j}=\\{(0,j),(1,j),\ldots,(11,j)\\},~{}j=0,1,2$. We first construct an
initial GDD${}_{2}(2,3,38)$ of type $1^{24}14^{1}$ with long group $G_{0}\cup
S$. The blocks are generated from the base blocks in the following by adding
$3$ (mod 12) to the first coordinate; the underlined base blocks and their
developments form the blocks of an NGDD$(2,3,38)$ of type $2^{(6,6)}14^{1}$,
where
$\lambda_{(0,1),(6,1)}=\lambda_{(1,1),(7,1)}=\lambda_{(2,1),(8,1)}=\lambda_{(3,1),(9,1)}=\lambda_{(4,1),(10,1)}=\lambda_{(5,1),(11,1)}=2$
and
$\lambda_{(0,2),(6,2)}=\lambda_{(1,2),(7,2)}=\lambda_{(2,2),(8,2)}=\lambda_{(3,2),(9,2)}=\lambda_{(4,2),(10,2)}=\lambda_{(5,2),(11,2)}=0$.

$\underline{\\{a,(0,1),(1,2)\\}}$ | $\underline{\\{a,(1,1),(3,2)\\}}$ | $\underline{\\{a,(2,1),(2,2)\\}}$ | $\underline{\\{b,(0,1),(2,2)\\}}$
---|---|---|---
$\underline{\\{b,(1,1),(1,2)\\}}$ | $\underline{\\{b,(2,1),(9,2)\\}}$ | $\underline{\\{(0,1),(6,1),(0,2)\\}}$ | $\underline{\\{(5,0),(1,1),(7,1)\\}}$
$\underline{\\{(5,0),(2,1),(8,1)\\}}$ | $\underline{\\{(0,0),(0,1),(2,1)\\}}$ | $\underline{\\{(1,0),(1,1),(4,1)\\}}$ | $\underline{\\{(10,0),(0,1),(5,1)\\}}$
$\underline{\\{(2,0),(1,1),(2,1)\\}}$ | $\underline{\\{(2,0),(9,1),(1,2)\\}}$ | $\underline{\\{(11,0),(0,1),(3,1)\\}}$ | $\underline{\\{(0,0),(1,1),(0,2)\\}}$
$\underline{\\{(4,1),(0,2),(5,2)\\}}$ | $\underline{\\{(1,0),(2,1),(7,1)\\}}$ | $\underline{\\{(4,0),(0,1),(1,1)\\}}$ | $\underline{\\{(6,0),(1,1),(3,1)\\}}$
$\underline{\\{(3,0),(2,1),(6,1)\\}}$ | $\underline{\\{(0,0),(6,1),(5,2)\\}}$ | $\underline{\\{(6,0),(1,2),(2,2)\\}}$ | $\underline{\\{(3,0],(0,2),(2,2)\\}}$
$\underline{\\{(5,0),(2,2),(7,2)\\}}$ | $\underline{\\{(1,0),(1,2),(3,2)\\}}$ | $\underline{\\{(2,0),(7,1),(2,2)\\}}$ | $\underline{\\{(1,1),(6,1),(11,2)\\}}$
$\underline{\\{(3,0),(7,1),(1,2)\\}}$ | $\underline{\\{(3,0),(1,1),(5,2)\\}}$ | $\underline{\\{(6,0),(2,1),(0,2)\\}}$ | $\underline{\\{(0,0),(5,1),(1,2)\\}}$
$\underline{\\{(9,0),(0,2),(1,2)\\}}$ | $\underline{\\{(8,0),(1,2),(4,2)\\}}$ | $\underline{\\{(2,0),(8,1),(0,2)\\}}$ | $\underline{\\{(11,0),(0,2),(3,2)\\}}$
$\underline{\\{(1,1),(5,1),(6,2)\\}}$ | $\underline{\\{(2,1),(4,1),(1,2)\\}}$ | $\underline{\\{(0,1),(4,1),(7,2)\\}}$ | $\underline{\\{(10,0),(2,2),(3,2)\\}}$
$\underline{\\{(7,0),(1,2),(5,2)\\}}$ | $\underline{\\{(1,0),(0,2),(4,2)\\}}$ | $\underline{\\{(4,0),(3,1),(0,2)\\}}$ | $\underline{\\{(7,0),(2,2),(4,2)\\}}$
$\underline{\\{(10,0),(2,1),(3,1)\\}}$ | $\underline{\\{(4,0),(2,1),(5,2)\\}}$ | $\underline{\\{(8,0),(6,1),(2,2)\\}}$ | $\underline{\\{(8,1),(2,2),(5,2)\\}}$
$\underline{\\{(11,0),(2,2),(6,2)\\}}$ | $\underline{\\{(3,1),(1,2),(6,2)\\}}$ | $\underline{\\{(2,1),(5,1),(7,2)\\}}$ | $\\{a,(0,1),(4,2)\\}$
$\\{a,(2,1),(9,2)\\}$ | $\\{a,(1,1),(11,2)\\}$ | $\\{(2,1),(0,2),(6,2)\\}$ | $\\{b,(0,1),(10,2)\\}$
$\\{b,(1,1),(9,2)\\}$ | $\\{b,(2,1),(5,2)\\}$ | $\\{(3,0),(1,2),(7,2)\\}$ | $\\{(5,0),(2,2),(8,2)\\}$
$\\{(4,0),(0,2),(1,2)\\}$ | $\\{(2,1),(7,1),(10,2)\\}$ | $\\{(9,0),(0,2),(5,2)\\}$ | $\\{(0,0),(1,2),(2,2)\\}$
$\\{(1,1),(3,1),(5,2)\\}$ | $\\{(5,0),(1,1),(1,2)\\}$ | $\\{(8,0),(1,2),(3,2)\\}$ | $\\{(7,0),(3,1),(0,2)\\}$
$\\{(9,0),(0,1),(5,1)\\}$ | $\\{(4,0),(2,2),(5,2)\\}$ | $\\{(6,1),(2,2),(6,2)\\}$ | $\\{(3,0),(1,1),(8,2)\\}$
$\\{(1,0),(11,1),(1,2)\\}$ | $\\{(6,0),(0,2),(3,2)\\}$ | $\\{(2,0),(4,1),(3,2)\\}$ | $\\{(8,0),(0,2),(2,2)\\}$
$\\{(1,0),(0,1),(3,2)\\}$ | $\\{(5,0),(2,1),(3,2)\\}$ | $\\{(2,0),(1,2),(4,2)\\}$ | $\\{(2,0),(2,1),(6,1)\\}$
$\\{(9,0),(1,1),(6,1)\\}$ | $\\{(4,0),(1,1),(4,1)\\}$ | $\\{(0,0),(0,1),(7,2)\\}$ | $\\{(7,0),(1,2),(6,2)\\}$
$\\{(3,0),(2,2),(3,2)\\}$ | $\\{(0,1),(1,1),(6,2)\\}$ | $\\{(9,0),(2,1),(3,1)\\}$ | $\\{(6,0),(1,1),(5,1)\\}$
$\\{(0,0),(1,1),(2,1)\\}$ | $\\{(1,0),(2,1),(4,1)\\}$ | $\\{(2,0),(1,1),(2,2)\\}$ | $\\{(1,0),(3,1),(8,2)\\}$
$\\{(10,0),(2,1),(2,2)\\}$ | $\\{(3,1),(2,2),(4,2)\\}$ | $\\{(7,0),(0,1),(2,1)\\}$ | $\\{(1,0),(7,1),(4,2)\\}$
$\\{(11,0),(2,1),(5,1)\\}$ | $\\{(8,1),(2,2),(7,2)\\}$ | $\\{(11,0),(0,1),(4,1)\\}$ | $\\{(5,0),(0,1),(3,1)\\}$
$\\{(10,1),(0,2),(4,2)\\}$ | $\\{(8,1),(1,2),(5,2)\\}$ |  |

Nine pairwise disjoint GDD${}_{2}(2,3,38)$s result by developing over both
coordinates. The remaining blocks are forced by the definition of a
$3$-PCS${}_{2}^{6}(12^{3}:2)$.∎

######  Lemma 6.11.

Let $n\equiv 1\pmod{2}$ and $n\geq 3$. Then there exists a
$3$-PCS${}^{e}_{2}(12^{n}:2)$ for any $0\leq e\leq 6(n-1)$ and $e\equiv
0\pmod{6}$.

###### Proof.

For $n=3$, see Lemma 6.10. For $n=5$, there exists a 2-FG$(3,(4,4,4),30)$ of
type $6^{5}$ from [16, Lemma 3.5]. It forms a 1-FG$(3,(4,4),30)$ of the same
type. Apply Construction 6.2 with $h=a=m=1$ and $l=2$; the input designs are
$1$-PCS${}^{i}_{2}(2^{4}:2)$s $(i=0,3)$ in [20, Lemma 6.1] and a
PGDD${}_{2}(2^{4})$ in Lemma 6.1. Then we obtain a
$6$-PCS${}^{r}_{2}(12^{5}:2)$, also a $3$-PCS${}^{r}_{2}(12^{5}:2)$, where
$r\in\\{0,6,12,18,24\\}$.

For $n\equiv 1\pmod{2}$ and $n\geq 7$, start with a
1-FG$(3,(\\{3,5\\},\\{4,6\\}),n)$ of type $1^{n}$ which exists by Lemma 6.4
and use the above $3$-PCS${}^{r}_{2}(12^{k}:2)$s for $k\in\\{3,5\\}$ and
$r\leq 6(k-1)$ with $r\equiv 0\pmod{6}$. A PGDD${}_{2}(12^{k})$ for
$k\in\\{4,6\\}$ exists by Lemma 6.1. Apply Construction 6.2 with $h=1$,
$a=l=2$, and $m=3$ to obtain the required $3$-PCS${}^{e}_{2}(12^{n}:2)$ for
any $0\leq e\leq 6(n-1)$ and $e\equiv 0\pmod{6}$.∎

######  Lemma 6.12.

Let $v\equiv 14\pmod{24}$ and $(v,b)$ satisfy (C2). There exists an
NWBTS$(v;b)$.

###### Proof.

For $v=14$, see Lemma A.4 in Appendix. For $v\geq 38$, we only need to
consider $b\leq\frac{v(v-1)(v-2)}{12}$ by Lemma 2.5. Let $v=12n+2$ where $n$
is odd. Write $b=qv(v-1)+b^{\prime}$ where $0\leq q\leq\frac{v-2}{12}-1=n-1$
and $\frac{\lambda v(v-1)}{6}-\frac{v}{6}<b^{\prime}<\frac{\lambda
v(v-1)}{6}+\frac{v}{6}$ with $\lambda=1,3,5$. Thus $(v,b^{\prime})$ associates
with $(\lambda,\varepsilon)$ where $\lambda=1,3,5$ and
$-\frac{v}{2}<\varepsilon<\frac{v}{2}$. There exists a
$3$-PCS${}^{e}_{2}(12^{n}:2)$ where $0\leq e\leq 6(n-1)$ and $e\equiv
0\pmod{6}$ from Lemma 6.11. An NWBTS$(14;c)$ exists for all $c$ with $(14,c)$
satisfying (C2) by Lemma A.4 in Appendix. A simple S${}_{6}(2,3,14)$ exists by
Theorem 1.4. Then there exists an NWBTS$(v;b)$ by Construction 6.6.∎

It remains to treat the case when $v\equiv 2\pmod{24}$. We define a
generalization of $g$-PCS${}^{r}_{2}((2g)^{n}:s)$.

Let $(X,S,\mathcal{G},\mathcal{A})$ be a CS$((6g)^{n}:s)$ with $g$ being a
positive integer. It is denoted by $g$-PCS${}^{r}_{6}((6g)^{n}:s)$ if its
block set $\mathcal{A}$ contains a subset $\mathcal{A}^{\prime}$ and
$\mathcal{A}^{\prime}$ can be partitioned into $gn+2$ pairwise disjoint parts
$\mathcal{A}_{i}$, $1\leq i\leq gn+2$, when it satisfies that

(i) for a fixed group $G_{0}\in\mathcal{G}$, there are exactly $g+2$
$\mathcal{A}_{i}$’s satisfying that (a) each $\mathcal{A}_{i}~{}(1\leq i\leq
3)$ is the block set of a GDD${}_{2}(2,3,6gn+s)$ of type
$1^{6g(n-1)}(6g+s)^{1}$ with long group $G_{0}\cup S$ and at least one of
$\mathcal{A}_{i}$’s contains a subset $\mathcal{A}^{\prime}$ such that
$\mathcal{A}^{\prime}$ is the block set of an NGDD$(2,3,6gn+s)$ of type
$2^{(r,3g(n-1)-r)}(6g+s)^{1}$, and (b) each $\mathcal{A}_{i}~{}(4\leq i\leq
g+2)$ is the block set of a GDD${}_{6}(2,3,6gn+s)$ of type
$1^{6g(n-1)}(6g+s)^{1}$ with long group $G_{0}\cup S$, and

(ii) for each group $G\in\mathcal{G}\backslash\\{G_{0}\\}$, there are exactly
$g$ $\mathcal{A}_{i}$’s such that each $\mathcal{A}_{i}$ is the block set of a
GDD${}_{6}(2,3,6gn+s)$ of type $1^{6g(n-1)}(6g+s)^{1}$ with long group $G\cup
S$.

Clearly, a $3g$-PCS${}^{r}_{2}((6g)^{n}:s)$ gives a
$g$-PCS${}^{r}_{6}((6g)^{n}:s)$. We will make use of the following two
immediate variants of Constructions 6.2 and 6.6.

######  Construction 6.13.

Suppose that there exists a $1$-FG$(3,(K_{1},K_{T}),gn)$ of type $g^{n}$. If
there exists an $m$-PCS${}^{r}_{6}((6m)^{k}:l)$ for each $k\in K_{1}$ and each
$0\leq r\leq 3m(k-1)$ with $r\equiv 0\pmod{3}$, and a PGDD${}_{2}((6m)^{k})$
for each $k\in K_{T}$, then there exists an $mg$-PCS${}^{e}_{6}((6mg)^{n}:l)$
for any $0\leq e\leq 3mg(n-1)$ and $e\equiv 0\pmod{3}$.

######  Construction 6.14.

Let $s\equiv 2,4\pmod{6}$ and $\lambda\in\\{1,3,5\\}$. Suppose that there
exists a $g$-PCS${}^{e}_{6}((6g)^{n}:s)$ for any $0\leq e\leq 3g(n-1)$ and
$e\equiv 0\pmod{3}$. Further let $u=6g+s$ and suppose that there exists

  1. (i)

a simple S${}_{6}(2,3,u)$ defined over a $u$-set $X$, each of whose blocks is
not strictly contained in an $s$-subset $Y\subset X$; and

  2. (ii)

an NWBTS$(u;c)$ for any $c=(\lambda{u\choose 2}+\delta)/3$ where $-{u\over
2}<\delta<{u\over 2}$.

Let $v=6gn+s$. Then there exists an NWBTS$(v;b)$ for any
$b=qv(v-1)+b^{\prime}$, where $0\leq q\leq n-1$ and
$3b^{\prime}=\lambda\binom{v}{2}+\varepsilon$ with $-{v\over
2}<\varepsilon<{v\over 2}$.

######  Lemma 6.15.

Let $n\equiv 0\pmod{4}$ and $n\neq 8,12$. Then there exists a
$1$-PCS${}^{e}_{6}(6^{n}:2)$ for any $0\leq e\leq 3(n-1)$ and $e\equiv
0\pmod{3}$.

###### Proof.

For $n\equiv 0\pmod{4}$ and $n\neq 8,12$, there exists a
1-FG$(3,(\\{3,4,5\\},\\{4,5,6\\}),n)$ of type $1^{n}$ which can be obtained by
deleting a point from an S$(3,\\{4,5,6\\},n+1)$ in [17]. A
$3$-PCS${}^{r}_{2}(6^{4}:2)$ for any $r\in\\{0,9\\}$ exists by Lemma B.2 in
Appendix. There exists a $1$-PCS${}^{r}_{6}(6^{4}:2)$ for any $r\in\\{3,6\\}$
by Lemma B.3 in Appendix. Since a $3$-PCS${}^{r}_{2}(6^{k}:2)$ exists for
$k\in\\{3,5\\}$ and $r\leq 3(k-1)$ with $r\equiv 0\pmod{3}$ by Lemma 6.8, and
a PGDD${}_{2}(6^{k})~{}(k\in\\{4,5,6\\})$ also exists by Lemma 6.1, we can
obtain a $1$-PCS${}_{6}^{e}(6^{n}:2)$ for any $0\leq e\leq 3(n-1)$ and
$e\equiv 0\pmod{3}$ by using Construction 6.13. ∎

######  Lemma 6.16.

Let $v\equiv 2\pmod{24}$, $v\neq 50,74$, and $(v,b)$ satisfy (C2). There
exists an NWBTS$(v;b)$.

###### Proof.

We can assume that $b\leq\frac{v(v-1)(v-2)}{6}$ by Lemma 2.5. Write $v=6n+2$
where $n\equiv 0\pmod{4}$ and $n\neq 8,12$; write $b=qv(v-1)+b^{\prime}$ where
$0\leq q\leq\frac{v-2}{6}-1=n-1$ and $0\leq b^{\prime}\leq v(v-1)$. According
to Lemma 2.5, we can assume that $0\leq b^{\prime}\leq\frac{v(v-1)}{2}$ and
$\frac{\lambda v(v-1)}{6}-\frac{v}{6}<b^{\prime}<\frac{\lambda
v(v-1)}{6}+\frac{v}{6}$ with $\lambda=1,3$. Thus $(v,b^{\prime})$ associates
with $(\lambda,\varepsilon)$ where $\lambda=1,3$ and
$-\frac{v}{2}<\varepsilon<\frac{v}{2}$. Apply Construction 6.14 with $s=2$.
There exists a $1$-PCS${}_{6}^{e}(6^{n}:2)$ for any $0\leq e\leq 3(n-1)$ and
$e\equiv 0\pmod{3}$ by Lemma 6.15. An NWBTS$(8;c)$ exists for all $c$ with
$(8,c)$ satisfying (C2) from Lemma A.3 in Appendix. A simple S${}_{6}(2,3,8)$
exists trivially. Then there exists an NWBTS$(v;b)$ for $v\equiv 2\pmod{24}$
and $v\neq 50,74$. ∎

Combining Lemmas 6.9, 6.12, and 6.16 yields the main result of this
subsection.

######  Theorem 6.17.

Let $v\equiv 2\pmod{6}$, $v\neq 50,74$, and $(v,b)$ satisfy (C2). There exists
an NWBTS$(v;b)$.

Analogous to the proofs of Lemmas 6.15 and 6.16, if there exists a
$3$-PCS${}^{3}_{2}(6^{4}:2)$, then we could also prove Theorem 6.17 by
applying Constructions 6.2 and 6.6. However, we did not find out a
$3$-PCS${}^{3}_{2}(6^{4}:2)$ and had to turn to its generalization
$1$-PCS${}^{3}_{6}(6^{4}:2)$. This design we constructed (Lemma B.3 in
Appendix) does not impose nice automorphism groups for all the component
designs, because otherwise no positive result appeared after a long time of
computer search.

###  6.3 $v\equiv 4\pmod{6}$

In this subsection, we show the existence of an NWBTS$(v;b)$ if $v\equiv
10,16,22\pmod{24}$ and $(v,b)$ satisfies (C2).

######  Lemma 6.18.

There is a $3$-PCS${}^{r}_{2}(6^{3}:4)$ for any $r=0,3,6$.

###### Proof.

We construct the desired design on $X=Z_{18}\cup\\{a,b,c,d\\}$ with stem
$S=\\{a,b,c,d\\}$ and groups $G_{i}=\\{i,i+3,i+6,\ldots,i+15\\},i=0,1,2$. Let
$\mathcal{H}$ be an automorphism group generated by the following two
permutations:

$\alpha=(0,6,12)(1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(a)(b)(c)(d)(e);$
$\beta=(0,2,4)(1,15,17)(6,8,10)(3,5,7)(12,14,16)(9,11,13)(a)(b)(c)(d)(e).$

For $r=0,6$, we first construct a GDD${}_{2}(2,3,22)$ of type $1^{12}10^{1}$
with long group $G_{0}\cup S$ in the following; the underlined blocks form the
blocks of an NGDD$(2,3,22)$ of type $2^{(0,6)}10^{1}$, where
$\lambda_{2,11}=\lambda_{4,13}=\lambda_{8,17}=\lambda_{1,10}=\lambda_{5,14}=\lambda_{7,16}=0$.

$\\{\underline{a,1,2}\\}$ | $\\{\underline{a,7,11}\\}$ | $\\{\underline{a,5,13}\\}$ | $\\{\underline{a,4,14}\\}$ | $\\{\underline{a,8,10}\\}$ | $\\{\underline{a,16,17}\\}$ | $\\{\underline{b,1,5}\\}$
---|---|---|---|---|---|---
$\\{\underline{b,10,17}\\}$ | $\\{\underline{b,4,8}\\}$ | $\\{\underline{b,2,7}\\}$ | $\\{\underline{b,11,13}\\}$ | $\\{\underline{b,14,16}\\}$ | $\\{\underline{c,1,8}\\}$ | $\\{\underline{c,7,17}\\}$
$\\{\underline{c,13,14}\\}$ | $\\{\underline{c,2,16}\\}$ | $\\{\underline{c,4,11}\\}$ | $\\{\underline{c,5,10}\\}$ | $\\{\underline{d,1,11}\\}$ | $\\{\underline{d,10,14}\\}$ | $\\{\underline{d,2,4}\\}$
$\\{\underline{d,13,17}\\}$ | $\\{\underline{d,5,16}\\}$ | $\\{\underline{d,7,8}\\}$ | $\\{\underline{0,2,5}\\}$ | $\\{\underline{1,12,14}\\}$ | $\\{\underline{6,8,16}\\}$ | $\\{\underline{6,11,17}\\}$
$\\{\underline{5,11,12}\\}$ | $\\{\underline{0,8,11}\\}$ | $\\{\underline{0,14,17}\\}$ | $\\{\underline{4,12,17}\\}$ | $\\{\underline{5,15,17}\\}$ | $\\{\underline{2,8,12}\\}$ | $\\{\underline{2,6,10}\\}$
$\\{\underline{2,13,15}\\}$ | $\\{\underline{3,11,14}\\}$ | $\\{\underline{2,3,17}\\}$ | $\\{\underline{1,9,17}\\}$ | $\\{\underline{2,9,14}\\}$ | $\\{\underline{3,5,8}\\}$ | $\\{\underline{6,7,14}\\}$
$\\{\underline{4,5,6}\\}$ | $\\{\underline{1,6,13}\\}$ | $\\{\underline{5,7,9}\\}$ | $\\{\underline{8,9,13}\\}$ | $\\{\underline{8,14,15}\\}$ | $\\{\underline{9,11,16}\\}$ | $\\{\underline{4,9,10}\\}$
$\\{\underline{10,11,15}\\}$ | $\\{\underline{10,12,16}\\}$ | $\\{\underline{4,15,16}\\}$ | $\\{\underline{1,7,15}\\}$ | $\\{\underline{7,12,13}\\}$ | $\\{\underline{0,1,4}\\}$ | $\\{\underline{0,7,10}\\}$
$\\{\underline{0,13,16}\\}$ | $\\{\underline{1,3,16}\\}$ | $\\{\underline{3,4,7}\\}$ | $\\{\underline{3,10,13}\\}$ | $\\{a,1,8\\}$ | $\\{a,11,13\\}$ | $\\{10,12,13\\}$
$\\{a,4,17\\}$ | $\\{a,5,16\\}$ | $\\{a,2,7\\}$ | $\\{a,10,14\\}$ | $\\{0,1,10\\}$ | $\\{0,5,14\\}$ | $\\{1,10,15\\}$
$\\{8,15,17\\}$ | $\\{5,14,15\\}$ | $\\{2,11,15\\}$ | $\\{4,13,15\\}$ | $\\{7,15,16\\}$ | $\\{0,2,11\\}$ | $\\{0,4,13\\}$
$\\{0,7,16\\}$ | $\\{0,8,17\\}$ | $\\{b,1,11\\}$ | $\\{b,16,17\\}$ | $\\{b,4,14\\}$ | $\\{b,2,13\\}$ | $\\{b,5,10\\}$
$\\{b,7,8\\}$ | $\\{c,5,7\\}$ | $\\{c,1,14\\}$ | $\\{c,13,17\\}$ | $\\{c,2,4\\}$ | $\\{c,8,16\\}$ | $\\{c,10,11\\}$
$\\{d,4,5\\}$ | $\\{d,7,14\\}$ | $\\{d,8,13\\}$ | $\\{d,11,16\\}$ | $\\{d,1,17\\}$ | $\\{d,2,10\\}$ | $\\{4,8,10\\}$
$\\{2,3,8\\}$ | $\\{6,8,14\\}$ | $\\{2,14,16\\}$ | $\\{5,8,12\\}$ | $\\{8,9,11\\}$ | $\\{1,2,6\\}$ | $\\{2,5,9\\}$
$\\{2,12,17\\}$ | $\\{3,4,11\\}$ | $\\{3,13,14\\}$ | $\\{5,6,11\\}$ | $\\{1,5,13\\}$ | $\\{3,5,17\\}$ | $\\{7,11,17\\}$
$\\{6,10,17\\}$ | $\\{9,14,17\\}$ | $\\{11,12,14\\}$ | $\\{1,3,7\\}$ | $\\{3,10,16\\}$ | $\\{4,6,16\\}$ | $\\{6,7,13\\}$
$\\{1,4,9\\}$ | $\\{1,12,16\\}$ | $\\{4,7,12\\}$ | $\\{7,9,10\\}$ | $\\{9,13,16\\}$ |  |

Nine GDD${}_{2}(2,3,20)$s are obtained by developing the above 124 base blocks
under the action of the automorphism group $\mathcal{H}$ and the remaining
blocks are forced by the definition of the $3$-PCS${}^{0}_{2}(6^{3}:4)$.

For a $3$-PCS${}^{3}_{2}(6^{3}:4)$, we first construct a GDD${}_{2}(2,3,22)$
of type $1^{12}10^{1}$ with long group $G_{0}\cup S$ in the following; the
underlined blocks form the blocks of an NGDD$(2,3,22)$ of type
$2^{(3,3)}10^{1}$, where $\lambda_{2,11}=\lambda_{4,13}=\lambda_{8,17}=2$ and
$\lambda_{1,10}=\lambda_{5,14}=\lambda_{7,16}=0$.

$\\{\underline{a,1,2}\\}$ | $\\{\underline{a,7,14}\\}$ | $\\{\underline{a,4,8}\\}$ | $\\{\underline{a,5,13}\\}$ | $\\{\underline{a,10,17}\\}$ | $\\{\underline{a,11,16}\\}$ | $\\{\underline{7,15,16}\\}$
---|---|---|---|---|---|---
$\\{\underline{1,10,15}\\}$ | $\\{\underline{4,11,15}\\}$ | $\\{\underline{8,13,15}\\}$ | $\\{\underline{2,15,17}\\}$ | $\\{\underline{5,14,15}\\}$ | $\\{\underline{b,2,4}\\}$ | $\\{\underline{b,7,8}\\}$
$\\{\underline{b,13,17}\\}$ | $\\{\underline{b,5,16}\\}$ | $\\{\underline{b,1,14}\\}$ | $\\{\underline{b,10,11}\\}$ | $\\{\underline{c,4,5}\\}$ | $\\{\underline{c,1,17}\\}$ | $\\{\underline{c,7,11}\\}$
$\\{\underline{c,8,10}\\}$ | $\\{\underline{c,13,14}\\}$ | $\\{\underline{c,2,16}\\}$ | $\\{\underline{d,4,14}\\}$ | $\\{\underline{d,1,8}\\}$ | $\\{\underline{d,2,7}\\}$ | $\\{\underline{d,5,10}\\}$
$\\{\underline{d,11,13}\\}$ | $\\{\underline{d,16,17}\\}$ | $\\{\underline{0,14,17}\\}$ | $\\{\underline{0,8,11}\\}$ | $\\{\underline{0,2,5}\\}$ | $\\{\underline{1,9,11}\\}$ | $\\{\underline{2,8,9}\\}$
$\\{\underline{2,3,13}\\}$ | $\\{\underline{2,12,14}\\}$ | $\\{\underline{2,6,10}\\}$ | $\\{\underline{5,8,12}\\}$ | $\\{\underline{6,8,16}\\}$ | $\\{\underline{3,8,14}\\}$ | $\\{\underline{11,12,17}\\}$
$\\{\underline{3,5,11}\\}$ | $\\{\underline{1,5,7}\\}$ | $\\{\underline{3,7,17}\\}$ | $\\{\underline{6,11,14}\\}$ | $\\{\underline{5,9,14}\\}$ | $\\{\underline{5,6,17}\\}$ | $\\{\underline{4,9,17}\\}$
$\\{\underline{10,14,16}\\}$ | $\\{\underline{1,4,6}\\}$ | $\\{\underline{6,7,13}\\}$ | $\\{\underline{0,1,13}\\}$ | $\\{\underline{1,10,12}\\}$ | $\\{\underline{0,4,16}\\}$ | $\\{\underline{0,7,10}\\}$
$\\{\underline{1,3,16}\\}$ | $\\{\underline{3,4,10}\\}$ | $\\{\underline{4,7,12}\\}$ | $\\{\underline{7,9,16}\\}$ | $\\{\underline{9,10,13}\\}$ | $\\{\underline{12,13,16}\\}$ | $\\{a,1,17\\}$
$\\{a,4,5\\}$ | $\\{a,7,11\\}$ | $\\{a,2,10\\}$ | $\\{a,8,13\\}$ | $\\{a,14,16\\}$ | $\\{b,1,11\\}$ | $\\{b,2,13\\}$
$\\{b,5,7\\}$ | $\\{b,10,14\\}$ | $\\{b,4,17\\}$ | $\\{b,8,16\\}$ | $\\{c,2,7\\}$ | $\\{c,1,8\\}$ | $\\{c,5,13\\}$
$\\{c,10,17\\}$ | $\\{c,4,14\\}$ | $\\{c,11,16\\}$ | $\\{d,8,10\\}$ | $\\{d,4,11\\}$ | $\\{d,1,5\\}$ | $\\{d,2,16\\}$
$\\{d,7,17\\}$ | $\\{d,13,14\\}$ | $\\{8,12,17\\}$ | $\\{4,12,13\\}$ | $\\{0,4,13\\}$ | $\\{0,7,8\\}$ | $\\{0,1,14\\}$
$\\{0,5,11\\}$ | $\\{0,2,17\\}$ | $\\{0,10,16\\}$ | $\\{2,11,12\\}$ | $\\{2,9,11\\}$ | $\\{6,8,17\\}$ | $\\{4,6,7\\}$
$\\{5,6,16\\}$ | $\\{1,2,15\\}$ | $\\{1,6,13\\}$ | $\\{1,12,16\\}$ | $\\{2,3,5\\}$ | $\\{2,4,8\\}$ | $\\{2,6,14\\}$
$\\{3,8,11\\}$ | $\\{3,14,17\\}$ | $\\{4,10,15\\}$ | $\\{1,4,9\\}$ | $\\{1,3,7\\}$ | $\\{3,4,16\\}$ | $\\{3,10,13\\}$
$\\{5,10,12\\}$ | $\\{5,8,15\\}$ | $\\{5,9,17\\}$ | $\\{6,10,11\\}$ | $\\{7,9,10\\}$ | $\\{7,12,14\\}$ | $\\{7,13,15\\}$
$\\{8,9,14\\}$ | $\\{9,13,16\\}$ | $\\{11,13,17\\}$ | $\\{11,14,15\\}$ | $\\{15,16,17\\}$ |  |

Nine GDD${}_{2}(2,3,20)$s are obtained by developing the above 124 base blocks
under the action of the automorphism group $\mathcal{H}$; the remaining blocks
are forced by the definition of the $3$-PCS${}^{3}_{2}(6^{3}:4)$.∎

######  Lemma 6.19.

For $n\equiv 1\pmod{2}$ and $n\geq 3$. There exists a
$3$-PCS${}_{2}^{e}(6^{n}:4)$ for any $0\leq e\leq 3(n-1)$ and $e\equiv
0\pmod{3}$.

###### Proof.

For $n=3$, see Lemma 6.18. For $n=5$, there exists a 2-FG$(3,(4,3,4),15)$ of
type $3^{5}$, which can be obtained by deleting one point from an S$(3,4,16)$
containing a subdesign S$(2,4,16)$ (see [5]). Apply Construction 6.2 with
$a=m=1$ and $h=l=2$. The input designs are $1$-PCS${}^{i}_{2}(2^{4}:2)$s
$(i=0,3)$ in [20, Lemma 6.1] and a PGDD${}_{2}(2^{4})$ in Lemma 6.1. Then we
get a $3$-PCS${}^{r}_{2}(6^{5}:4)$ for $r\in\\{0,3,6,9,12\\}$.

For $n\equiv 1$ (mod 2) and $n\geq 7$, start with a
1-FG$(3,(\\{3,5\\},\\{4,6\\}),n)$ of type $1^{n}$ which exists by Lemma 6.4. A
PGDD${}_{2}(6^{k})$ for $k\in\\{4,6\\}$ exists by Lemma 6.1 and use the above
$3$-PCS${}^{r}_{2}(6^{k}:4)$s for $k\in\\{3,5\\}$ and $r\leq 3(k-1)$ with
$r\equiv 0\pmod{3}$. Then we obtain the desired design by applying
Construction 6.2 with $h=a=1$, $m=3$, and $l=4$.∎

######  Lemma 6.20.

Let $v\equiv 10\pmod{12}$ and $(v,b)$ satisfy (C2). There exists an
NWBTS$(v;b)$.

###### Proof.

For $v=10$, see Lemma A.5 in Appendix. For $v\geq 22$, we only need to
consider $b\leq\frac{v(v-1)(v-2)}{12}$ by Lemma 2.5. Write $v=6n+4$ where $n$
is odd, $b=qv(v-1)+b^{\prime}$ where $0\leq q\leq\frac{v-2}{12}-1<n-1$ and
$\frac{\lambda v(v-1)}{6}-\frac{v}{6}<b^{\prime}<\frac{\lambda
v(v-1)}{6}+\frac{v}{6}$ with $\lambda=1,3,5$. Thus $(v,b^{\prime})$ associates
with $(\lambda,\varepsilon)$ where $\lambda=1,3,5$ and
$-\frac{v}{2}<\varepsilon<\frac{v}{2}$. There exists a
$3$-PCS${}^{e}_{2}(6^{n}:4)$ for any $0\leq e\leq 3(n-1)$ and $e\equiv
0\pmod{3}$ from Lemma 6.19. An NWBTS$(10;c)$ exists for all $c$ with $(10,c)$
satisfying (C2) by Lemma A.5 in Appendix. There exists a simple
S${}_{6}(2,3,10)$ defined over a 10-set with each block not contained in a
fixed 4-subset from Lemma C.1 in Appendix. Then there exists an
NWBTS$(v;b=qv(v-1)+b^{\prime})$ by applying Construction 6.6.∎

######  Lemma 6.21.

Let $n\equiv 1\pmod{2}$ and $n\geq 3$. Then there is a
$3^{+}$-PCS${}^{e}_{2}(12^{n}:4)$ for any $0\leq e\leq 6(n-1)$ and $e\equiv
0\pmod{6}$.

###### Proof.

For $n=3$, see Lemma B.1 in Appendix. For $n=5$, there exists a
2-FG$(3,(4,4,4),30)$ of type $6^{5}$ from [16, Lemma 3.5]. Apply Construction
6.2 with $a=m=1$ and $h=l=2$; the input designs are
$1$-PCS${}^{i}_{2}(2^{4}:2)$s $(i=0,3)$ in [20, Lemma 6.1] and
PGDD${}_{2}(2^{j})$s $(j=4,5)$ in Lemma 6.1. Then we obtain a
$6$-PCS${}^{r}_{2}(12^{5}:4)$, also a $3^{+}$-PCS${}^{r}_{2}(12^{5}:4)$ for
$r\in\\{0,6,12,18,24\\}$.

For $n\equiv 1\pmod{2}$ and $n\geq 7$, start with a
1-FG$(3,(\\{3,5\\},\\{4,6\\}),n)$ of type $1^{n}$ which exists by Lemma 6.4. A
PGDD${}_{2}(12^{k})$ for $k\in\\{4,6\\}$ exists by Lemma 6.1 and use the above
$3^{+}$-PCS${}^{r}_{2}(12^{k}:4)$ for $k\in\\{3,5\\}$ and $r\leq 6(k-1)$ with
$r\equiv 0\pmod{6}$. Apply Construction 6.3 with $m=3$ and $l=4$. So we obtain
a $3^{+}$-PCS${}^{e}_{2}(12^{n}:4)$ where $0\leq e\leq 6(n-1)$ and $e\equiv
0\pmod{6}$.∎

######  Lemma 6.22.

Let $X$ be a set of $16$ points, $Y$ be a $4$-subset of $X$, and let
$b\in[38,42]$. Then there exist two disjoint designs defined over $X$, an
S${}_{6}(2,3,16)$ and an NWBTS$(16,b)$, whose blocks all belong to
$\binom{X}{3}\backslash\binom{Y}{3}$.

###### Proof.

See Lemma A.6 in Appendix.∎

######  Lemma 6.23.

Let $v\equiv 16\pmod{24}$ and $(v,b)$ satisfy (C2). There exists an
NWBTS$(v;b)$.

###### Proof.

For $v=16$, see Lemma A.7 in Appendix. For $v\geq 40$, we only need to
consider $b\leq\frac{v(v-1)(v-2)}{12}$ from Lemma 2.5. Write $v=12n+4$ where
$n$ is odd. Further write $b=qv(v-1)+b^{\prime}$ where $(v,b^{\prime})$
associates with $(\lambda,\varepsilon)$,
$-\frac{v}{2}<\varepsilon<\frac{v}{2}$, and if $0\leq
q\leq\left\lfloor\frac{v-2}{12}\right\rfloor-1=n-1$, then $\lambda=1,3,5$, and
that if $q=n$, then $\lambda=1$.

Let $V=(Z_{n}\times Z_{3}\times Z_{4})\cup S$, $|S|=4$, and
$\mathcal{G}=\\{G_{0},G_{1},\ldots,G_{n-1}\\}$ where $G_{i}=\\{i\\}\times
Z_{3}\times Z_{4}$. Let $(V,S,\mathcal{G},\mathcal{A})$ be a
$3^{+}$-PCS${}^{e}_{2}(12^{n}:4)$ where $0\leq e\leq 6(n-1)$ and $e\equiv
0\pmod{6}$ from Lemma 6.21. Then the block set $\mathcal{A}$ contains a subset
$\mathcal{A}^{\prime}$ and $\mathcal{A}^{\prime}$ can be partitioned into
$(3n+1)$ pairwise disjoint parts $\mathcal{A}(i,j)$, $0\leq i\leq n-1$, $0\leq
j\leq 2$, plus ${\cal A}(0,3)$, so that each $\mathcal{A}(i,j)$ is the block
set of a GDD${}_{2}(2,3,12n+4)$ of type $1^{12(n-1)}16^{1}$ with long group
$G_{i}\cup S$, and $\mathcal{A}(0,3)$ contains a subset
$\mathcal{A}^{\prime}(0,3)$ which is the block set of an NGDD$(2,3,12n+6)$ of
type $2^{(e,6(n-1)-e)}16^{1}$. Applying Construction 6.6 with $\lambda=1,3,5$
yields the existence of an NWBTS$(v;qv(v-1)+b^{\prime})$ for any $q\leq n-1$,
because a simple S${}_{6}(2,3,16)$ exists by Lemma 6.22 with all blocks in
$\binom{G_{i}\cup S}{3}\backslash\binom{S}{3}$ and an NWBTS$(16;c)$ exists for
all $c$ with $(16,c)$ satisfying (C2).

Now we consider that $q=n$. Thus $b=nv(v-1)+b^{\prime}$ and $(v,b^{\prime})$
associates with $(1,\varepsilon)$. By Lemma 6.22, we may construct on
$G_{i}\cup S$ ($0\leq i\leq n-1$) two disjoint designs, an S${}_{6}(2,3,16)$
with block set $\mathcal{B}(i)$ and an NWBTS$(16,c)$ ($c\in[38,42]$) with
block set ${\cal C}(i)$, whose blocks all belong to $\binom{G_{i}\cup
S}{3}\backslash\binom{S}{3}$. Let
$\mathcal{F}(i)=(\bigcup^{2}_{j=0}\mathcal{A}(i,j))\cup\mathcal{B}(i)$. Each
$\mathcal{F}(i)$ is the block set of a simple S${}_{6}(2,3,12n+4)$ and all
$\mathcal{F}(i)$s are pairwise disjoint. Construct a family
$\mathcal{F}=\mathcal{A}^{\prime}(0,3)\cup{\cal
C}(0)\cup(\bigcup_{i=0}^{n-1}\mathcal{F}(i))$. Then by Lemmas 6.5 and 3.1,
$\mathcal{F}$ is an NWBTS$(v;nv(v-1)+b^{\prime})$ where $3b^{\prime}={v\choose
2}+\varepsilon$.∎

Combining Lemmas 6.20 and 6.23 yields the main result of this subsection.

######  Theorem 6.24.

Let $v\equiv 10,16,22\pmod{24}$ and $(v,b)$ satisfy (C2). There exists an
NWBTS$(v;b)$.

##  7 Conclusion

In this section we summarize our main results and propose a promising approach
to resolving entirely the existence problem of NWBTSs and optimal data
placements for triple replication.

######  Theorem 7.1.

Let $v\geq 3$ and $b\geq 0$ be integers. An NWBTS$(v;b)$ exists for any $v,b$
satisfying (C1) or (C2), possibly except that $v\equiv 4\pmod{24}$ or
$v=50,74$, and $\frac{\lambda v(v-1)}{6}-\frac{v}{6}<b<\frac{\lambda
v(v-1)}{6}+\frac{v}{6}$ for an odd integer $\lambda$.

###### Proof.

Combine Theorems 4.7, 5.4, 6.17 and 6.24 to prove that an NWBTS$(v;b)$ exists
if $(v,b)$ satisfies (C1) or (C2) possibly except that $v\equiv 4\pmod{24}$ or
$v=50,74$, and $(v,b)$ satisfies (C2). ∎

######  Theorem 7.2.

An optimal data placement TS$(v;b)$ exists for any positive integers $v,b$
with $v\geq 3$, possibly except that $v\equiv 4\pmod{24}$ or $v=50,74$, and
$\frac{\lambda v(v-1)}{6}-\frac{v}{6}<b<\frac{\lambda v(v-1)}{6}+\frac{v}{6}$
for an odd integer $\lambda$.

###### Proof.

The conclusion follows by combining Corollary 1.5 with Theorems 2.3 and 7.1.∎

For the possible exceptions in Theorems 7.1 and 7.2, we put forward a
promising working approach. Let $v\equiv 4\pmod{24}$ and $(v,b)$ satisfy (C2).
For $n\equiv 0$ (mod 4) and $n\neq 8,12$, there exists a
1-FG$(3,(\\{3,4,5\\},\\{4,5,6\\}),n)$ of type $1^{n}$. A
$3$-PCS${}^{r}_{2}(6^{4}:4)$ (also a $1$-PCS${}^{r}_{6}(6^{4}:4)$) for any
$r\in\\{0,9\\}$ can be found by computer research. Suppose that there exists a
$1$-PCS${}^{r}_{6}(6^{4}:4)$ for any $r\in\\{3,6\\}$. Since a
$3$-PCS${}^{r}_{2}(6^{k}:4)$ exists for $k\in\\{3,5\\}$ and $r\leq 3(k-1)$
with $r\equiv 0\pmod{3}$ by Lemma 6.19, and a
PGDD${}_{2}(6^{k})~{}(k\in\\{4,5,6\\})$ also exists by Lemma 6.1, we can
obtain a $1$-PCS${}_{6}^{e}(6^{n}:4)$ for any $0\leq e\leq 3(n-1)$ and
$e\equiv 0\pmod{3}$ by using Construction 6.13. Then applying Construction
6.14 we get an NWBTS$(v;b)$ ($v\not\in\\{50,52,74,76\\}$), also an optimal
data placement for triple replication.

To sum up, we could reduce an almost solution of the existence of NWBTSs (and
optimal data placements) to one small design $1$-PCS${}^{3}_{6}(6^{4}:4)$.
Certainly the final complete solution also depends on the small cases of
$v\in\\{50,52,74,76\\}$. We conjecture that NWBTS$(v;b)$s always exist without
no exceptions, although techniques of direct constructions and effective
computational methods are two important concerns.

## References

  * [1] R. J. R. Abel, C. J. Colbourn, and J. H. Dinitz, Mutually orthogonal Latin squares (MOLS), in: C. J. Colbourn, J. H. Dinitz (eds.) The CRC Handbook of Combinatorial Designs, 2nd edn. CRC Press, Boca Raton, FL, (2007) 160-193.
  * [2] A.-E. Baert, V. Boudet, and A. Jean-Marie, Performance analysis of data replication in grid delivery networks, in International Conference on Complex Intelligent and Software Intensive Systems, Barcelona, (2008) 369-374.
  * [3] A.-E. Baert, V. Boudet, A. Jean-Marie, and X. Roche, Minimization of download times in a distributed VOD system, in ICPP08: The International Conference on Parallel Processing, IEEE, Los Alamitos, CA, (2008) 173-180.
  * [4] A.-E. Baert, V. Boudet, A. Jean-Marie, and X. Roche, Minimization of download time variance in a distributed VOD system, Scalable Comput. Pract. Exp., West University of Timisoara, 10 (2009) 75-86.
  * [5] R. D. Baker, Partitioning the planes of AG2m(2) into $2$-designs, Discrete Math. 15 (1976) 205-211.
  * [6] J.-C. Bermond, A. Jean-Marie, D. Mazauric, and J. Yu, Well-balanced designs for data placement, J. Combin. Des. 24 (2016) 55-76.
  * [7] H. Cao, L. Ji, and L. Zhu, Large sets of disjoint packings on $6k+5$ points, J. Combin. Theory Ser. A 108 (2004) 169-183.
  * [8] D. Chen, C. C. Lindner, and D. R. Stinson, Further results on large sets of disjoint group-divisible designs, Discrete Math. 110 (1992) 35-42.
  * [9] C.-M. K. Fu, The intersection problem for pentagon systems, Ph. D. Thesis, Auburn University, 1987.
  * [10] H. Hanani, A class of three-designs, J. Combin. Theory Ser. A 26 (1979) 1-19.
  * [11] H. Hanani, On some tactical configurations, Can. J. Math. 15 (1963) 702-722.
  * [12] A. Hartman, The fundamental construction for $3$-designs, Discrete Math. 124 (1994) 107-132.
  * [13] A. Jean-Marie, X. Roche, V. Boudet, and A.-E. Baert, Combinatorial designs and availability, Research Report RR-7119 INRIA (2009).
  * [14] L. Ji, A complete solution to existence of $H$ designs, J. Combin. Des. 27 (2019) 75-81.
  * [15] L. Ji, A new existence proof for large sets of disjoint Steiner triple systems, J. Combin. Theory Ser. A 112 (2005) 308-327.
  * [16] L. Ji, On the 3BD-closed set B${}_{3}(\\{4,5\\})$, Discrete Math. 287 (2004) 55-67.
  * [17] L. Ji, On the 3BD-closed set B${}_{3}(\\{4,5,6\\})$, J. Combin. Des. 12 (2004) 92-102.
  * [18] L. Ji, Partition of triples of order $6k+5$ into $6k+3$ optimal packings and one packing of size $8k+4$, Graph Combin. 22 (2006) 251-260.
  * [19] H. Moh csy and D. K. Ray-Chaudhuri, Candelabra systems and designs, J. Statist. Plann. Inference 106 (2002) 419-448.
  * [20] H. Wei, G. Ge, and C. J. Colbourn, The existence of well-balanced triple systems, J. Combin. Des. 24 (2016) 77-100.

Appendix

##  A A NWBTSs

######  Lemma A.1.

Let $v\in\\{50,74\\}$ and $(v,b)$ satisfy (C1). There exists an NWBTS$(v;b)$.

###### Proof.

By Lemma 2.5, we assume that $b\leq\frac{v(v-1)(v-2)}{12}$. We first produce a
nearly 2-balanced simple triple system. Write $w=v/2$. For $w\in\\{25,37\\}$,
we partition $\frac{(w-1)^{2}}{24}$ inequivalent simple difference triples
modulo $w$ into $\frac{w-1}{4}$ classes (the rows) each having $\frac{w-1}{6}$
difference triples $(a,b,c)$ with $a+b+c\equiv 0\pmod{w}$; in each class,
every nonzero element $(\pm)$ appears exactly once.

$D_{00}$ | $d_{0}=(1,2,-3)$ | $f_{0}=(5,6,-11)$ | $(4,9,12)$ | $(7,8,10)$
---|---|---|---|---
$D_{01}$ | $(1,5,-6)$ | $(2,8,-10)$ | $(3,9,-12)$ | $(4,7,-11)$
$D_{02}$ | $(1,8,-9)$ | $(2,3,-5)$ | $(4,10,11)$ | $(6,7,12)$
$D_{10}$ | $d_{1}=(4,8,-12)$ | $(3,6,-9)$ | $(1,10,-11)$ | $(2,5,-7)$
---|---|---|---|---
$D_{11}$ | $(1,4,-5)$ | $(2,7,-9)$ | $(3,10,12)$ | $(6,8,11)$
$D_{12}$ | $(1,11,-12)$ | $(4,5,-9)$ | $(2,6,-8)$ | $(3,7,-10)$
$D_{00}$ | $d_{0}=(2,8,-10)$ | $f_{0}=(7,11,-18)$ | $(1,12,-13)$ | $(3,14,-17)$ | $(4,5,-9)$ | $(6,15,16)$
---|---|---|---|---|---|---
$D_{01}$ | $(1,7,-8)$ | $(2,9,-11)$ | $(3,13,-16)$ | $(4,10,-14)$ | $(5,15,17)$ | $(6,12,-18)$
$D_{02}$ | $(1,8,-9)$ | $(2,13,-15)$ | $(3,7,-10)$ | $(4,12,-16)$ | $(5,14,18)$ | $(6,11,-17)$
$D_{10}$ | $d_{1}=(4,13,-17)$ | $(1,6,-7)$ | $(2,14,-16)$ | $(3,9,-12)$ | $(5,10,-15)$ | $(8,11,18)$
$D_{11}$ | $(1,2,-3)$ | $(4,15,18)$ | $(5,6,-11)$ | $(7,9,-16)$ | $(8,12,17)$ | $(10,13,14)$
$D_{12}$ | $(1,3,-4)$ | $(2,7,-9)$ | $(5,12,-17)$ | $(6,13,18)$ | $(8,14,15)$ | $(10,11,16)$
$D_{20}$ | $d_{2}=(3,6,-9)$ | $(1,10,-11)$ | $(2,12,-14)$ | $(4,16,17)$ | $(5,13,-18)$ | $(7,8,-15)$
$D_{21}$ | $(1,4,-5)$ | $(2,15,-17)$ | $(3,11,-14)$ | $(6,7,-13)$ | $(8,10,-18)$ | $(9,12,16)$
$D_{22}$ | $(1,5,-6)$ | $(2,17,18)$ | $(3,8,-11)$ | $(4,9,-13)$ | $(7,14,16)$ | $(10,12,15)$

We treat difference triples as _ordered_ , using the ordering given. For each
$0\leq\alpha<\frac{w-1}{12}$, $0\leq\beta<3$, let
$S_{\alpha,\beta}=\bigcup_{(a,b,c)\in D_{\alpha,\beta}}\bigcup_{i\in
Z_{w}}\\{\\{i,a+i,a+b+i\\}\\}$, and $T_{\alpha,\beta}=\bigcup_{(a,b,c)\in
D_{\alpha,\beta}}\bigcup_{i\in Z_{w}}\\{\\{i,b+i,a+b+i\\}\\}$, arithmetic in
$Z_{w}$. Then $|S_{\alpha,\beta}|=|T_{\alpha,\beta}|=\frac{w(w-1)}{6}$. For
each $\alpha$, let $R_{\alpha,0}=\bigcup_{(a,b,c)\in
D_{\alpha,0}\setminus\\{d_{\alpha}\\}}$ $\bigcup_{i\in
Z_{w}}\\{\\{i,a+i,a+b+i\\}\\}$. Then $S_{\alpha,\beta}$ and $T_{\alpha,\beta}$
are S$(2,3,w)$s on $Z_{w}$. They share no triples with
$S_{\alpha^{\prime},\beta^{\prime}}$ or $T_{\alpha^{\prime},\beta^{\prime}}$
unless $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$, because no
(unordered) difference triple of $D_{\alpha,\beta}$ is equivalent to one in
$D_{\alpha^{\prime},\beta^{\prime}}$. Moreover, $S_{\alpha,\beta}$ and
$T_{\alpha,\beta}$ are disjoint, because $\\{0,a,a+b\\}\neq\\{j,b+j,a+b+j\\}$
when the difference triple is simple.

For $w=25$, let $E=\\{(0,p,q)\\}=\\{(0,6,20)\\}$ and $C$ be a parallel class
on $Z_{w}\backslash\\{0\\}$: $\\{(1,7,21)$, $(2,8,22)$,
$(3,4,6),(5,19,24),(9,14,20),(10,11,13),(12,17,23),(15,16,18)\\}$.

For $w=37$, let $E=\\{(0,p,q)\\}=\\{(0,11,30)\\}$ and $C$ be a parallel class
on $Z_{w}\backslash\\{0\\}$: $\\{(1,3,11)$, $(2,13,32)$,
$(4,23,30),(5,24,31),(6,17,36),(7,14,25),(8,27,34),(9,28,35),(10,12,20),(15,22,33)$,
$(16,18,26)$, $(19,21,29)\\}$.

(All triples of $C$ are translates of $d_{0}$ or $f_{0}$ i.e., for given
difference triple $(a,b,c)\in\\{d_{0},f_{0}\\}$, we have $\\{i,a+i,a+b+i\\}\in
C$ for some $i\in Z_{w}$, and all triples in $E\cup C$ are ordered in the same
way.) Let $C^{\prime}$ be obtained from $C$ by adding $p$ to each entry
(arithmetic in $Z_{w}$). Because $C$ contains all elements of $Z_{w}$ except
for 0, $C^{\prime}$ contains all elements except for $p$. Each triple in
$C\cup C^{\prime}$ is in $S_{0,0}$, and $C$ and $C^{\prime}$ share no triples.

We form a nearly 2-balanced simple triple system on $Z_{w}\times\\{0,1\\}$,
representing $(i,j)$ as $i_{j}$. For a 3-subset $B=\\{a,b,c\\}$ of $Z_{w}$ and
$\sigma\in\\{0,1\\}$, we set
$l_{\sigma}(B)=\\{\\{a_{l},b_{j},c_{k}\\}:l+j+k\equiv\sigma$ (mod 2)}. When
$d_{\alpha}=(a,b,c)$, let

$M_{\alpha}=\\{\\{i_{0},i_{1},(i+x)_{j}\\}:i\in Z_{w}$,
$x\in\\{a,b,-(a+b)\\},~{}j\in\\{0,1\\}\\}$,

$N_{\alpha}=\\{\\{i_{0},i_{1},(i+x)_{j}\\}:i\in Z_{w}$,
$x\in\\{-a,-b,a+b\\},~{}j\in\\{0,1\\}\\}$,

$P_{\alpha}=M_{\alpha}\cup l_{0}(R_{\alpha,0})\cup l_{1}(T_{\alpha,0})\cup
l_{0}(S_{\alpha,1})\cup l_{1}(T_{\alpha,1})\cup l_{0}(S_{\alpha,2})\cup
l_{1}(T_{\alpha,2})$,

$Q_{\alpha}=N_{\alpha}\cup l_{1}(R_{\alpha,0})\cup l_{0}(T_{\alpha,0})\cup
l_{1}(S_{\alpha,1})\cup l_{0}(T_{\alpha,1})\cup l_{1}(S_{\alpha,2})\cup
l_{0}(T_{\alpha,2})$.

For each $\alpha$, each of $P_{\alpha}$ and $Q_{\alpha}$ is an
S${}_{6}(2,3,v)$, the triple systems so produced are disjoint.

Write $b=(2q+q^{\prime})v(v-1)+b^{\prime}$ with $0\leq b^{\prime}<v(v-1)$,
$q^{\prime}\in\\{0,1\\}$, and $0\leq q<\frac{w-1}{12}$. Place in $\mathcal{F}$
all triples in $P_{\alpha}$ and $Q_{\alpha}$ for $1\leq\alpha\leq q$, and also
the triples of $Q_{0}$ when $q^{\prime}=1$. When $2q+q^{\prime}=0$,
$b=b^{\prime}$. We treat cases for $b^{\prime}$. Let $(v,b^{\prime})$
associate with $(\lambda^{\prime},\varepsilon)$ where $\lambda^{\prime}=2,4$.

For $b^{\prime}\in\\{\frac{1}{6}(8w^{2}-4w-4),\frac{1}{6}(8w^{2}-4w+2)\\}$. We
have that $\lambda^{\prime}=2$ and $\varepsilon=-2,1$. $\mathcal{F}^{\prime}$
include (1) all triples in $l_{0}(S_{0,0}\backslash C)$; (2)
$\\{\\{x_{0},x_{1},y_{j}\\},\\{y_{0},y_{1},z_{j}\\},\\{z_{0},z_{1},x_{j}\\}:j\in\\{0,1\\},(x,y,z)\in
C\\}$ (none is in $N_{0}$); (3) $\\{0_{0},0_{1},p_{0}\\}$; (4) all triples in
$l_{1}(T_{0,0})$ except for those that contain $\\{0_{0},p_{0}\\}$ or
$\\{0_{1},p_{0}\\}$, denoted by
$U=\\{\\{0_{0},p_{0},f_{1}\\},\\{0_{1},p_{0},f_{0}\\}\\}$; and (5) any
$b^{\prime}-\frac{1}{6}(8w^{2}-4w-10)$ triples from $U$. The defect graph of
$(Z_{w}\times\\{0,1\\},\mathcal{F}^{\prime})$ is isomorphic to the graph
$G_{-2}$ and $G_{1}$ in Figure 1, respectively. So $\mathcal{F}^{\prime}$ is a
nearly 2-balanced simple triple system, also an NWBTS$(v;b^{\prime})$ by Lemma
2.4.

For $b^{\prime}\in\\{\frac{1}{6}(16w^{2}-8w-2),\frac{1}{6}(16w^{2}-8w+4)\\}$.
We have that $\lambda^{\prime}=4$ and $\varepsilon=-1,2$.
$\mathcal{F}^{\prime}$ include (1) all triples in
$l_{0}(S_{0,0}\backslash(C\cup C^{\prime}))$ except for two triples
$\\{0_{0},p_{0},q_{0}\\}$ and $\\{0_{1},p_{1},q_{0}\\}$ in $l_{0}(E)$; (2) all
triples in $l_{1}(T_{0,0})$, $l_{0}(S_{0,1})$, and $l_{1}(T_{0,1})$; (3)
$\\{\\{x_{0},x_{1},y_{j}\\},\\{y_{0},y_{1},z_{j}\\},\\{z_{0},z_{1},x_{j}\\}:j\in\\{0,1\\},(x,y,z)\in
C\cup C^{\prime}\\}$; (4) $\\{0_{0},0_{1},q_{0}\\}$ and
$\\{p_{0},p_{{}_{1}},q_{0}\\}$; and (5) any
$b^{\prime}-\frac{1}{6}(16w^{2}-8w-8)$ triples from
$\\{\\{0_{0},0_{1},p_{0}\\},\\{0_{1},p_{1},f_{0}\\}\\}$, where
$\\{0_{1},p_{1},f_{0}\\}$ which is contained in $l_{0}(S_{0,2})$,
respectively. The defect graph of
$(Z_{w}\times\\{0,1\\},\mathcal{F}^{\prime})$ is isomorphic to the graph
$G_{-1}$ and $G_{2}$ in Figure 1, respectively. So $\mathcal{F}^{\prime}$ is a
nearly 2-balanced simple triple system, also an NWBTS$(v;b^{\prime})$ by Lemma
2.4.

When $2q+q^{\prime}\geq 1$, $\mathcal{F}\cup\mathcal{F}^{\prime}$ is an
NWBTS$(v;b)$ by Lemma 3.1. ∎

######  Lemma A.2.

There exists an NWBTS$(11;b)$, where $b\equiv 18,19,36,37\pmod{55}$.

###### Proof.

We only need to consider that $b\in\\{18,19,36,37,73,74\\}$ by Lemma 2.5. An
NWBTS$(11;18)$ can be constructed on $Z_{11}$, where $\lambda_{0,1}=2$ and
$\lambda_{0,10}=\lambda_{1,10}=0$.

$\\{2,6,10\\}$ | $\\{3,4,10\\}$ | $\\{7,9,10\\}$ | $\\{5,8,10\\}$ | $\\{0,2,4\\}$ | $\\{0,1,7\\}$ | $\\{1,3,5\\}$ | $\\{1,4,9\\}$ | $\\{3,6,7\\}$
---|---|---|---|---|---|---|---|---
$\\{0,3,8\\}$ | $\\{0,1,6\\}$ | $\\{0,5,9\\}$ | $\\{2,5,7\\}$ | $\\{1,2,8\\}$ | $\\{2,3,9\\}$ | $\\{4,5,6\\}$ | $\\{4,7,8\\}$ | $\\{6,8,9\\}$

We can get an NWBTS$(11;19)$ by adding a block $\\{1,2,10\\}$ on the above
NWBTS$(11;18)$, where $\lambda_{0,10}=0$ and
$\lambda_{0,1}=\lambda_{2,10}=\lambda_{1,2}=2$.

We get a simple S${}_{3}(2,3,11)$, whose blocks are generated by the base
blocks under the action of the group $Z_{11}$, which are different from all
blocks of the above NWBTS$(11;19)$.

{0, 1, 3} | {0, 1, 4} | {0, 1, 5} | {0, 2, 5} | {0, 2, 6}
---|---|---|---|---

Then we can obtain an NWBTS$(11;l)$ $(l=73,74)$ by combining all blocks of the
S${}_{3}(2,3,11)$ and the NWBTS$(11;r)$ $(r=18,19)$, respectively.

An NWBTS$(11;36)$ can be constructed on $Z_{11}$, where $\lambda_{0,1}=3$ and
$\lambda_{0,9}=\lambda_{1,10}=\lambda_{9,10}=1$.

$\\{0,1,2\\}$ | $\\{0,1,3\\}$ | $\\{0,1,4\\}$ | $\\{0,2,9\\}$ | $\\{0,3,4\\}$ | $\\{0,5,6\\}$ | $\\{0,5,7\\}$ | $\\{0,6,8\\}$ | $\\{2,6,7\\}$
---|---|---|---|---|---|---|---|---
$\\{0,7,10\\}$ | $\\{0,8,10\\}$ | $\\{1,2,10\\}$ | $\\{1,3,4\\}$ | $\\{1,5,6\\}$ | $\\{1,5,7\\}$ | $\\{1,6,8\\}$ | $\\{1,7,9\\}$ | $\\{2,6,9\\}$
$\\{1,8,9\\}$ | $\\{2,3,5\\}$ | $\\{3,5,8\\}$ | $\\{2,4,5\\}$ | $\\{4,5,9\\}$ | $\\{5,8,10\\}$ | $\\{5,9,10\\}$ | $\\{3,8,9\\}$ | $\\{3,6,7\\}$
$\\{2,4,8\\}$ | $\\{2,7,8\\}$ | $\\{4,7,8\\}$ | $\\{4,6,9\\}$ | $\\{4,6,10\\}$ | $\\{4,7,10\\}$ | $\\{2,3,10\\}$ | $\\{3,6,10\\}$ | $\\{3,7,9\\}$

We can get an NWBTS$(11;37)$ by adding a block $\\{0,9,10\\}$ on the
NWBTS$(11;36)$, where $\lambda_{1,10}=1$ and $\lambda_{0,1}=\lambda_{0,10}=3$.
∎

######  Lemma A.3.

There exists an NWBTS$(8;b)$, where $b\equiv
9,10,18,19,27,28,29,37,38,46,47\pmod{56}$.

###### Proof.

We only need to consider that $b\in\\{9,10,18,19,27,28\\}$ by Lemma 2.5. We
first construct an NWBTS$(8;9)$ on $Z_{8}$, where
$\lambda_{0,1}=\lambda_{2,3}=2$ and
$\lambda_{3,5}=\lambda_{3,6}=\lambda_{4,7}=0$.

$\\{0,1,2\\}$ | $\\{0,1,3\\}$ | $\\{0,4,5\\}$ | $\\{0,6,7\\}$ | $\\{1,4,6\\}$ | $\\{2,3,4\\}$ | $\\{2,3,7\\}$ | $\\{2,5,6\\}$ | $\\{1,5,7\\}$
---|---|---|---|---|---|---|---|---

We can get an NWBTS$(8;10)$ by adding a block $\\{3,5,6\\}$ on the above
NWBTS$(8;9)$, where $\lambda_{4,7}=0$ and
$\lambda_{0,1}=\lambda_{2,3}=\lambda_{5,6}=2$.

An NWBTS$(8;18)$ can be constructed on $Z_{8}$, where $\lambda_{0,1}=3$ and
$\lambda_{0,7}=\lambda_{1,6}=\lambda_{6,7}=1$.

$\\{0,1,2\\}$ | $\\{0,1,3\\}$ | $\\{0,1,4\\}$ | $\\{0,2,6\\}$ | $\\{0,3,7\\}$ | $\\{0,4,5\\}$ | $\\{2,3,7\\}$ | $\\{2,4,6\\}$ | $\\{2,4,7\\}$
---|---|---|---|---|---|---|---|---
$\\{0,5,6\\}$ | $\\{1,2,5\\}$ | $\\{1,3,6\\}$ | $\\{1,4,7\\}$ | $\\{1,5,7\\}$ | $\\{2,3,5\\}$ | $\\{3,4,5\\}$ | $\\{3,4,6\\}$ | $\\{5,6,7\\}$

We can get an NWBTS$(8;19)$ by adding a block $\\{0,6,7\\}$ on the above
NWBTS$(8;18)$, where $\lambda_{1,6}=1$ and $\lambda_{0,1}=\lambda_{0,6}=3$.

Next we construct an NWBTS$(8;27)$ on $Z_{8}$, where $\lambda_{2,3}=4$ and
$\lambda_{4,5}=\lambda_{6,7}=\lambda_{1,2}=\lambda_{0,2}=2$.

$\\{3,5,7\\}$ | $\\{0,1,3\\}$ | $\\{0,1,4\\}$ | $\\{0,1,5\\}$ | $\\{0,2,3\\}$ | $\\{0,2,4\\}$ | $\\{0,3,6\\}$ | $\\{1,4,6\\}$ | $\\{3,4,6\\}$
---|---|---|---|---|---|---|---|---
$\\{0,4,7\\}$ | $\\{0,5,6\\}$ | $\\{0,5,7\\}$ | $\\{0,6,7\\}$ | $\\{1,2,3\\}$ | $\\{2,3,5\\}$ | $\\{2,3,6\\}$ | $\\{3,4,7\\}$ | $\\{1,6,7\\}$
$\\{1,2,7\\}$ | $\\{2,4,6\\}$ | $\\{2,4,7\\}$ | $\\{2,5,6\\}$ | $\\{2,5,7\\}$ | $\\{1,4,5\\}$ | $\\{3,4,5\\}$ | $\\{1,5,6\\}$ | $\\{1,3,7\\}$

We can get an NWBTS$(8;28)$ by adding a block $\\{0,1,2\\}$ on the above
NWBTS$(8;27)$, where $\lambda_{0,1}=\lambda_{2,3}=4$ and
$\lambda_{4,5}=\lambda_{6,7}=2$. ∎

######  Lemma A.4.

There exists an NWBTS$(14;b)$, where $b\equiv 60,61,121,122\pmod{182}$ for
(C1) and $b\equiv 29,30,31,32,89,90,91,92,93,150,151,152,153\pmod{182}$ for
(C2).

###### Proof.

We only need to consider that $b\in[60,61]\cup[121,122]$ for (C1) and
$b\in[29,32]\cup[89,93]\cup[150,153]$ for (C2) by Lemma 2.5. We first
construct an NWBTS$(14;60)$ on $Z_{14}$, where $\lambda_{0,1}=3$ and
$\lambda_{0,12}=\lambda_{1,13}=\lambda_{12,13}=1$.

$\\{0,1,2\\}$ | $\\{0,1,3\\}$ | $\\{0,1,4\\}$ | $\\{0,2,12\\}$ | $\\{0,3,4\\}$ | $\\{0,5,6\\}$ | $\\{6,8,12\\}$ | $\\{4,6,9\\}$
---|---|---|---|---|---|---|---
$\\{0,5,7\\}$ | $\\{0,6,7\\}$ | $\\{0,8,9\\}$ | $\\{0,8,10\\}$ | $\\{0,9,11\\}$ | $\\{0,10,13\\}$ | $\\{4,6,13\\}$ | $\\{6,9,13\\}$
$\\{0,11,13\\}$ | $\\{1,2,13\\}$ | $\\{1,3,4\\}$ | $\\{1,5,6\\}$ | $\\{1,5,7\\}$ | $\\{1,6,7\\}$ | $\\{2,7,9\\}$ | $\\{7,9,12\\}$
$\\{1,8,9\\}$ | $\\{1,8,10\\}$ | $\\{1,9,11\\}$ | $\\{1,10,12\\}$ | $\\{1,11,12\\}$ | $\\{2,5,8\\}$ |  $\\{2,4,12\\}$, | $\\{4,8,12\\}$
$\\{3,5,8\\}$ | $\\{2,3,5\\}$ | $\\{4,5,9\\}$ | $\\{5,9,10\\}$ | $\\{4,5,10\\}$ | $\\{5,11,12\\}$ | $\\{2,4,7\\}$ | $\\{2,8,13\\}$
$\\{5,11,13\\}$ | $\\{5,12,13\\}$ | $\\{2,3,11\\}$ | $\\{3,6,11\\}$ | $\\{3,6,8\\}$ | $\\{3,7,12\\}$ | $\\{4,7,13\\}$ | $\\{7,8,13\\}$
$\\{3,9,12\\}$ | $\\{3,7,10\\}$ | $\\{3,9,13\\}$ | $\\{3,10,13\\}$ | $\\{2,6,11\\}$ | $\\{4,8,11\\}$ | $\\{2,9,10\\}$ | $\\{6,10,12\\}$
$\\{7,8,11\\}$ | $\\{4,10,11\\}$ | $\\{7,10,11\\}$ | $\\{2,6,10\\}$ |  |  |  |

We can get an NWBTS$(14;61)$ by adding a block $\\{1,12,13\\}$ on the above
NWBTS$(14;60)$, where $\lambda_{0,12}=1$ and $\lambda_{0,1}=\lambda_{1,12}=3$.

We construct an NWBTS$(14;121)$ on $Z_{14}$, where $\lambda_{0,1}=5$ and
$\lambda_{0,13}=\lambda_{1,13}=3$.

$\\{0,1,2\\}$ | $\\{0,1,3\\}$ | $\\{0,1,4\\}$ | $\\{0,1,5\\}$ | $\\{0,1,6\\}$ | $\\{0,2,13\\}$ | $\\{0,3,13\\}$ | $\\{0,4,13\\}$
---|---|---|---|---|---|---|---
$\\{0,2,3\\}$ | $\\{0,2,4\\}$ | $\\{0,3,4\\}$ | $\\{0,5,6\\}$ | $\\{0,5,7\\}$ | $\\{0,5,8\\}$ | $\\{0,6,7\\}$ | $\\{0,6,8\\}$
$\\{0,7,9\\}$ | $\\{0,7,10\\}$ | $\\{0,8,11\\}$ | $\\{0,8,12\\}$ | $\\{0,9,10\\}$ | $\\{0,9,11\\}$ | $\\{0,9,12\\}$ | $\\{0,10,11\\}$
$\\{0,10,12\\}$ | $\\{0,11,12\\}$ | $\\{1,2,13\\}$ | $\\{1,3,13\\}$ | $\\{1,4,13\\}$ | $\\{1,2,3\\}$ | $\\{1,2,4\\}$ | $\\{1,3,4\\}$
$\\{1,5,6\\}$ | $\\{1,5,7\\}$ | $\\{1,5,8\\}$ | $\\{1,6,7\\}$ | $\\{1,6,8\\}$ | $\\{1,7,9\\}$ | $\\{1,7,10\\}$ | $\\{1,8,11\\}$
$\\{1,8,12\\}$ | $\\{1,9,10\\}$ | $\\{1,9,11\\}$ | $\\{1,9,12\\}$ | $\\{1,10,11\\}$ | $\\{1,10,12\\}$ | $\\{1,11,12\\}$ | $\\{2,3,4\\}$
$\\{2,3,5\\}$ | $\\{2,4,5\\}$ | $\\{2,5,6\\}$ | $\\{2,5,7\\}$ | $\\{2,6,7\\}$ | $\\{2,6,8\\}$ | $\\{2,6,9\\}$ | $\\{2,7,8\\}$
$\\{2,7,9\\}$ | $\\{2,8,10\\}$ | $\\{2,8,11\\}$ | $\\{2,9,10\\}$ | $\\{2,9,12\\}$ | $\\{2,10,11\\}$ | $\\{2,10,12\\}$ | $\\{2,11,12\\}$
$\\{2,11,13\\}$ | $\\{2,12,13\\}$ | $\\{3,4,5\\}$ | $\\{3,5,7\\}$ | $\\{3,5,9\\}$ | $\\{3,6,9\\}$ | $\\{3,6,10\\}$ | $\\{3,6,11\\}$
$\\{3,6,12\\}$ | $\\{3,7,10\\}$ | $\\{3,7,11\\}$ | $\\{3,7,12\\}$ | $\\{3,8,9\\}$ | $\\{3,8,10\\}$ | $\\{3,8,11\\}$ | $\\{3,8,12\\}$
$\\{3,9,11\\}$ | $\\{3,10,13\\}$ | $\\{3,12,13\\}$ | $\\{5,8,9\\}$ | $\\{5,8,10\\}$ | $\\{4,5,11\\}$ | $\\{4,5,12\\}$ | $\\{5,6,12\\}$
$\\{5,9,10\\}$ | $\\{5,9,13\\}$ | $\\{5,10,11\\}$ | $\\{5,10,13\\}$ | $\\{5,11,12\\}$ | $\\{5,11,13\\}$ | $\\{5,12,13\\}$ | $\\{4,6,11\\}$
$\\{4,7,11\\}$ | $\\{4,9,11\\}$ | $\\{6,7,11\\}$ | $\\{6,11,13\\}$ | $\\{7,11,13\\}$ | $\\{4,6,9\\}$ | $\\{6,9,13\\}$ | $\\{4,6,10\\}$
$\\{4,6,12\\}$ | $\\{6,8,13\\}$ | $\\{6,10,12\\}$ | $\\{6,10,13\\}$ | $\\{4,7,10\\}$ | $\\{4,8,10\\}$ | $\\{4,10,13\\}$ | $\\{4,7,8\\}$
$\\{4,7,12\\}$ | $\\{4,8,9\\}$ | $\\{4,8,13\\}$ | $\\{4,9,12\\}$ | $\\{7,9,13\\}$ | $\\{8,9,13\\}$ | $\\{7,8,12\\}$ | $\\{7,8,13\\}$
$\\{7,12,13\\}$ |  |  |  |  |  |  |

We can obtain an NWBTS$(14;122)$ by adding a block $\\{0,5,13\\}$ on the above
NWBTS$(14;121)$, where $\lambda_{1,13}=3$ and
$\lambda_{0,1}=\lambda_{5,13}=\lambda_{0,5}=5$.

Next we construct an NWBTS$(14;29)$ on $Z_{14}$, where
$\lambda_{2,3}=\lambda_{4,5}=2$ and
$\lambda_{0,5}=\lambda_{1,5}=\lambda_{6,7}=\lambda_{8,9}=\lambda_{10,11}=\lambda_{12,13}=0$.

$\\{0,6,8\\}$ | $\\{1,6,10\\}$ | $\\{2,6,12\\}$ | $\\{3,4,6\\}$ | $\\{5,6,9\\}$ | $\\{6,11,13\\}$ | $\\{1,7,9\\}$ | $\\{5,7,11\\}$
---|---|---|---|---|---|---|---
$\\{2,8,10\\}$ | $\\{1,8,13\\}$ | $\\{3,5,8\\}$ | $\\{4,7,8\\}$ | $\\{8,11,12\\}$ | $\\{0,3,10\\}$ | $\\{4,9,11\\}$ | $\\{9,10,12\\}$
$\\{1,3,12\\}$ | $\\{2,3,11\\}$ | $\\{2,3,7\\}$ | $\\{3,9,13\\}$ | $\\{0,2,9\\}$ | $\\{1,2,4\\}$ | $\\{4,5,10\\}$ | $\\{0,1,11\\}$
$\\{2,5,13\\}$ | $\\{0,4,13\\}$ | $\\{7,10,13\\}$ | $\\{0,7,12\\}$ | $\\{4,5,12\\}$ |  |  |

We can get an NWBTS$(14;30)$ by adding a block $\\{0,1,5\\}$ on the above
NWBTS$(14;29)$, where $\lambda_{0,1}=\lambda_{2,3}=\lambda_{4,5}=2$ and
$\lambda_{6,7}=\lambda_{8,9}=\lambda_{10,11}=\lambda_{12,13}=0$. We can obtain
an NWBTS$(14;31)$ by adding two blocks $\\{0,1,5\\}$ and $\\{6,7,8\\}$ on the
above NWBTS$(14;29)$, where
$\lambda_{0,1}=\lambda_{2,3}=\lambda_{4,5}=\lambda_{6,8}=\lambda_{7,8}=2$ and
$\lambda_{8,9}=\lambda_{10,11}=\lambda_{12,13}=0$.

We construct an NWBTS$(14;32)$ on $Z_{14}$, where $\lambda_{0,1}=0$ and
$\lambda_{2,3}=\lambda_{4,5}=\lambda_{6,7}=\lambda_{8,9}=\lambda_{10,11}=\lambda_{12,13}=2$.

$\\{0,2,3\\}$ | $\\{0,4,5\\}$ | $\\{0,6,7\\}$ | $\\{0,8,9\\}$ | $\\{0,10,11\\}$ | $\\{0,12,13\\}$ | $\\{7,8,11\\}$ | $\\{3,7,12\\}$
---|---|---|---|---|---|---|---
$\\{1,2,3\\}$ | $\\{1,4,5\\}$ | $\\{1,6,7\\}$ | $\\{1,8,9\\}$ | $\\{1,10,11\\}$ | $\\{1,12,13\\}$ | $\\{6,8,12\\}$ | $\\{3,8,13\\}$
$\\{2,4,6\\}$ | $\\{2,5,8\\}$ | $\\{2,7,10\\}$ | $\\{2,9,12\\}$ | $\\{2,11,13\\}$ | $\\{3,4,9\\}$ | $\\{3,6,10\\}$ | $\\{5,6,13\\}$
$\\{4,8,10\\}$ | $\\{4,7,13\\}$ | $\\{4,11,12\\}$ | $\\{5,7,9\\}$ | $\\{6,9,11\\}$ | $\\{3,5,11\\}$ | $\\{5,10,12\\}$ | $\\{9,10,13\\}$

An NWBTS$(14;89)$ can be constructed on $Z_{14}$, where $\lambda_{2,3}=4$ and
$\lambda_{0,2}=\lambda_{1,2}=\lambda_{4,5}=\lambda_{6,7}=\lambda_{8,9}=\lambda_{10,11}=\lambda_{12,13}=2$.

$\\{4,5,11\\}$ | $\\{0,1,3\\}$ | $\\{0,1,4\\}$ | $\\{0,1,5\\}$ | $\\{0,2,3\\}$ | $\\{0,2,4\\}$ | $\\{0,3,4\\}$ | $\\{5,8,10\\}$
---|---|---|---|---|---|---|---
$\\{0,5,6\\}$ | $\\{0,5,7\\}$ | $\\{0,6,7\\}$ | $\\{0,6,8\\}$ | $\\{0,7,8\\}$ | $\\{0,8,9\\}$ | $\\{0,9,10\\}$ | $\\{4,5,10\\}$
$\\{0,9,11\\}$ | $\\{0,10,12\\}$ | $\\{0,10,13\\}$ | $\\{0,11,12\\}$ | $\\{0,11,13\\}$ | $\\{0,12,13\\}$ | $\\{4,8,11\\}$ | $\\{3,8,11\\}$
$\\{1,2,3\\}$ | $\\{2,3,4\\}$ | $\\{2,3,5\\}$ | $\\{1,2,4\\}$ | $\\{2,5,6\\}$ | $\\{2,5,7\\}$ | $\\{2,6,7\\}$ | $\\{3,5,11\\}$
$\\{2,6,8\\}$ | $\\{2,7,8\\}$ | $\\{2,8,9\\}$ | $\\{2,9,10\\}$ | $\\{2,9,11\\}$ | $\\{2,10,12\\}$ | $\\{2,10,13\\}$ | $\\{3,8,10\\}$
$\\{2,11,12\\}$ | $\\{2,11,13\\}$ | $\\{2,12,13\\}$ | $\\{1,6,8\\}$ | $\\{1,6,12\\}$ | $\\{3,6,12\\}$ | $\\{4,6,12\\}$ | $\\{3,5,10\\}$
$\\{1,3,6\\}$ | $\\{3,6,9\\}$ | $\\{4,6,9\\}$ | $\\{4,6,10\\}$ | $\\{5,6,11\\}$ | $\\{6,9,13\\}$ | $\\{6,10,11\\}$ | $\\{7,10,11\\}$
$\\{6,10,13\\}$ | $\\{6,11,13\\}$ | $\\{1,4,13\\}$ | $\\{1,5,13\\}$ | $\\{1,7,13\\}$ | $\\{3,8,13\\}$ | $\\{4,8,13\\}$ | $\\{4,7,11\\}$
$\\{5,8,13\\}$ | $\\{3,4,13\\}$ | $\\{3,7,13\\}$ | $\\{5,9,13\\}$ | $\\{7,9,13\\}$ | $\\{1,8,10\\}$ | $\\{1,8,11\\}$ | $\\{4,7,10\\}$
$\\{1,5,9\\}$ | $\\{1,7,9\\}$ | $\\{1,7,10\\}$ | $\\{1,9,11\\}$ | $\\{1,10,12\\}$ | $\\{1,11,12\\}$ | $\\{3,9,10\\}$ | $\\{3,7,11\\}$
$\\{3,9,12\\}$ | $\\{3,7,12\\}$ | $\\{4,7,9\\}$ | $\\{4,9,12\\}$ | $\\{5,9,12\\}$ | $\\{4,8,12\\}$ | $\\{5,7,12\\}$ | $\\{7,8,12\\}$
$\\{5,8,12\\}$ |  |  |  |  |  |  |

We can obtain an NWBTS$(14;90)$ by adding a block $\\{0,1,2\\}$ on the above
NWBTS$(14;89)$, where $\lambda_{0,1}=\lambda_{2,3}=4$ and
$\lambda_{4,5}=\lambda_{6,7}=\lambda_{8,9}=\lambda_{10,11}=\lambda_{12,13}=2$.

We construct an NWBTS$(14;91)$ on $Z_{14}$, where
$\lambda_{0,1}=\lambda_{4,5}=\lambda_{6,7}=\lambda_{8,9}=4$ and
$\lambda_{0,2}=\lambda_{0,3}=\lambda_{10,11}=\lambda_{12,13}=2$.

$\\{0,1,2\\}$ | $\\{0,2,4\\}$ | $\\{0,1,3\\}$ | $\\{0,1,4\\}$ | $\\{0,1,5\\}$ | $\\{0,3,4\\}$ | $\\{0,5,6\\}$ | $\\{3,5,9\\}$
---|---|---|---|---|---|---|---
$\\{0,5,7\\}$ | $\\{0,6,7\\}$ | $\\{0,6,8\\}$ | $\\{0,7,8\\}$ | $\\{0,8,9\\}$ | $\\{0,9,10\\}$ | $\\{0,9,11\\}$ | $\\{4,5,9\\}$
$\\{0,10,12\\}$ | $\\{0,10,13\\}$ | $\\{0,11,12\\}$ | $\\{0,11,13\\}$ | $\\{0,12,13\\}$ | $\\{1,10,11\\}$ | $\\{7,9,12\\}$ | $\\{4,7,9\\}$
$\\{2,10,11\\}$ | $\\{1,2,10\\}$ | $\\{1,3,10\\}$ | $\\{2,3,10\\}$ | $\\{3,4,10\\}$ | $\\{4,5,10\\}$ | $\\{4,6,10\\}$ | $\\{4,9,12\\}$
$\\{5,6,10\\}$ | $\\{5,7,10\\}$ | $\\{6,7,10\\}$ | $\\{7,8,10\\}$ | $\\{8,9,10\\}$ | $\\{8,10,12\\}$ | $\\{9,10,13\\}$ | $\\{4,7,11\\}$
$\\{10,12,13\\}$ | $\\{1,2,12\\}$ | $\\{1,3,12\\}$ | $\\{1,4,12\\}$ | $\\{1,4,5\\}$ | $\\{1,5,6\\}$ | $\\{1,6,7\\}$ | $\\{5,7,12\\}$
$\\{1,6,8\\}$ | $\\{1,7,8\\}$ | $\\{1,7,9\\}$ | $\\{1,8,13\\}$ | $\\{1,9,11\\}$ | $\\{1,9,13\\}$ | $\\{1,11,13\\}$ | $\\{5,11,12\\}$
$\\{2,6,7\\}$ | $\\{2,6,12\\}$ | $\\{3,6,12\\}$ | $\\{4,6,12\\}$ | $\\{2,3,6\\}$ | $\\{3,6,9\\}$ | $\\{4,6,11\\}$ | $\\{5,8,12\\}$
$\\{6,8,13\\}$ | $\\{6,9,11\\}$ | $\\{6,9,13\\}$ | $\\{6,11,13\\}$ | $\\{2,7,13\\}$ | $\\{3,7,13\\}$ | $\\{4,7,13\\}$ | $\\{4,8,11\\}$
$\\{2,3,13\\}$ | $\\{2,4,13\\}$ | $\\{3,5,13\\}$ | $\\{4,5,13\\}$ | $\\{5,8,13\\}$ | $\\{2,8,9\\}$ | $\\{3,8,9\\}$ | $\\{3,7,12\\}$
$\\{2,4,8\\}$ | $\\{2,5,8\\}$ | $\\{2,5,9\\}$ | $\\{2,9,12\\}$ | $\\{2,5,11\\}$ | $\\{2,7,11\\}$ | $\\{3,4,8\\}$ | $\\{3,7,11\\}$
$\\{3,8,11\\}$ | $\\{8,11,12\\}$ | $\\{3,5,11\\}$ |  |  |  |  |

We can get an NWBTS$(14;92)$ by adding a block $\\{0,2,3\\}$ on the above
NWBTS$(14;91)$, where $\lambda_{10,11}=\lambda_{12,13}=2$ and
$\lambda_{0,1}=\lambda_{2,3}=\lambda_{4,5}=\lambda_{6,7}=\lambda_{8,9}=4$. We
can obtain an NWBTS$(14;93)$ by adding two blocks $\\{0,2,3\\}$ and
$\\{10,11,12\\}$ on the above NWBTS$(14;91)$, where
$\lambda_{0,1}=\lambda_{2,3}=\lambda_{4,5}=\lambda_{6,7}=\lambda_{8,9}=\lambda_{10,12}=\lambda_{11,12}=4$
and $\lambda_{12,13}=2$.

We construct an NWBTS$(14;150)$ on $Z_{14}$, whose blocks are generated by the
following 75 base blocks modulo 7, where $\lambda_{0,7}=6$ and
$\lambda_{1,8}=\lambda_{2,9}=\lambda_{3,10}=\lambda_{4,11}=\lambda_{5,12}=\lambda_{6,13}=4$.

$\\{0,1,7\\}$ | $\\{0,2,7\\}$ | $\\{0,3,7\\}$ | $\\{0,1,8\\}$ | $\\{1,2,8\\}$ | $\\{0,2,9\\}$ | $\\{1,2,9\\}$ | $\\{4,12,13\\}$
---|---|---|---|---|---|---|---
$\\{0,3,10\\}$ | $\\{1,3,10\\}$ | $\\{0,4,11\\}$ | $\\{1,4,11\\}$ | $\\{0,5,12\\}$ | $\\{1,5,12\\}$ | $\\{0,6,13\\}$ | $\\{4,6,12\\}$
$\\{1,6,13\\}$ | $\\{0,1,2\\}$ | $\\{0,1,3\\}$ | $\\{0,1,4\\}$ | $\\{0,2,3\\}$ | $\\{0,2,4\\}$ | $\\{0,3,4\\}$ | $\\{3,12,13\\}$
$\\{0,4,5\\}$ | $\\{0,5,6\\}$ | $\\{0,5,8\\}$ | $\\{0,5,9\\}$ | $\\{0,6,8\\}$ | $\\{0,6,9\\}$ | $\\{0,6,11\\}$ | $\\{3,11,13\\}$
$\\{0,8,12\\}$ | $\\{0,9,13\\}$ | $\\{0,10,11\\}$ | $\\{0,10,12\\}$ | $\\{0,10,13\\}$ | $\\{0,11,12\\}$ | $\\{0,11,13\\}$ | $\\{3,11,12\\}$
$\\{0,12,13\\}$ | $\\{1,2,3\\}$ | $\\{1,2,4\\}$ | $\\{1,3,4\\}$ | $\\{1,3,5\\}$ | $\\{1,4,5\\}$ | $\\{1,5,6\\}$ | $\\{3,6,12\\}$
$\\{1,6,10\\}$ | $\\{1,6,11\\}$ | $\\{1,6,12\\}$ | $\\{1,9,11\\}$ | $\\{1,9,12\\}$ | $\\{1,9,13\\}$ | $\\{1,10,11\\}$ | $\\{3,6,11\\}$
$\\{1,10,12\\}$ | $\\{1,10,13\\}$ | $\\{1,11,13\\}$ | $\\{2,3,5\\}$ | $\\{2,3,11\\}$ | $\\{2,3,13\\}$ | $\\{2,4,10\\}$ | $\\{3,4,12\\}$
$\\{2,4,12\\}$ | $\\{2,5,10\\}$ | $\\{2,5,11\\}$ | $\\{2,5,13\\}$ | $\\{2,6,10\\}$ | $\\{2,6,11\\}$ | $\\{2,6,12\\}$ | $\\{2,11,13\\}$