# Super-simple $(v,4,2)$ directed designs and a lower bound for the minimum
size of their defining set

M. Boostan, S. Golalizadeh and  N. Soltankhah
Department of Mathematical Sciences
Alzahra University
P.O. Box 19834
Tehran, I.R. Iran. Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we show that for all $v\equiv 1$ (mod 3), there exists a super-
simple $(v,4,2)$ directed design. Also, we show that for these parameters
there exists a super-simple $(v,4,2)$ directed design whose each defining set
has at least a half of the blocks.

Keywords: Super-simple directed design, Smallest defining set, Trade, Pairwise
balanced design, Group divisible design, Directed group divisible design

##  1 Introduction and preliminaries

A $t-(v,k,\lambda)$ design with parameters $v,k,\lambda$ is a pair
$(X,\mathcal{B})$ where $X$ is a set of $v$ elements, and $\mathcal{B}$ is a
family of $k-$subsets of distinct elements of $X$, called blocks, with the
property that every t-tuple of distinct elements occurs in exactly $\lambda$
blocks. The problem of evaluation $v$ such that there exists a
$t-(v,k,\lambda)$ design for a specific $k,\lambda$ is one of the most
important problems in combinatorics.

Some generalizations has been introduced for the concept of designs. Gronau
and Mullin [12] for the first time, introduced a new definition of block
designs called super-simple block designs. A super-simple $t-(v,k,\lambda)$
design is a block design such that any two blocks of the design intersect in
at most $t$ points. A simple block design is a block design such that it has
no repeated blocks. The existence of super-simple $(v,4,\lambda)$ designs have
been characterized for $2\leq\lambda\leq 9$ except $\lambda=7$, see [1, 2, 3,
4, 6, 12, 17]. Also, the existence of super-simple $(v,5,\lambda)$ designs
have been characterized for $2\leq\lambda\leq 5$, see [5, 7, 8, 9].

A $t-(v,k,\lambda)$ directed design is a pair $(X,\mathcal{B})$, where $X$ is
a $v$-set, $\mathcal{B}$ is a collection of ordered k-tuples of distinct
elements of $X$ (called blocks) such that each ordered t-tuples of distinct
elements of $X$ appears in exactly $\lambda$ blocks. By $t-(v,k,\lambda)$DD,
we mean a $t-(v,k,\lambda)$ directed design. A $t-(v,k,\lambda)$DD is super-
simple if its underlying $t-(v,k,(t!)\lambda)$ design is super-simple.

In the rest of this paper, we use various types of combinatorial objects such
as trade, defining set, group divisible design(GDD), directed group divisible
design(DGDD) and pairwise balanced design(PBD), that we review them here.

A set of blocks which is a subset of a unique $t-(v,k,\lambda)$DD is said to
be a defining set of the directed design. A minimal defining set is a defining
set, no proper subset of which is a defining set. A smallest defining set, is
a defining set with the smallest cardinality.

A $(v,k,t)$ directed trade of volume $s$ consists of two disjoint collections
$T_{1}$ and $T_{2}$, each of $s$ blocks, such that every ordered t-tuple of
distinct elements of $V$ is covered by precisely the same number of blocks of
$T_{1}$ as of $T_{2}$. Such a directed trade is usually denoted by
$T=T_{1}-T_{2}$. Blocks in $T_{1}(T_{2})$ are called the positive
(respectively, negative) blocks of $T$. In a $(v,k,t)$ directed trade, both
collections of blocks cover the same set of elements. This set of elements is
called the foundation of the trade. In [14], it has been shown that the
minimum volume of a $(v,k,t)$ directed trade is $2^{\lfloor t/2\rfloor}$ and
that directed trades of minimum volume and minimum foundation exist. In some
parts of this paper, we handle with a special type of directed trade, called a
cyclical trade, defined as follows. Let $T=T_{1}-T_{2}$ be a $(v,k,t)$
directed trade of volume s with blocks $b_{1},\cdots,b_{s}$ such that each
pair of consecutive blocks of $T_{1}$ $(b_{i},b_{i+1},i=1,\cdots,s$ (mod s))
is a trade of volume 2.

If $\mathcal{D}=(V,\mathcal{B})$ is a directed design, and if
$T_{1}\subseteq\mathcal{B}$, we say that $\mathcal{D}$ contains the directed
trade $T$. For example the following super-simple $2-(13,4,2)DD$

$\\{(6,5,0,2),(0,1,6,4),(0,1,11,5),(1,0,3,9)\\}$

by (+1 mod 13) contains the following directed trade:

$T_{1}$ | $T_{2}$
---|---
(0,1,11,5) | (1,0,11,5)
(1,0,3,9) | (0,1,3,9)

Defining sets for directed designs are strongly related to trades. This
relation is illustrated by the following result.

######  Proposition 1.

[13] Let $\mathcal{D}=(V,\mathcal{B})$ be a $t-(v,k,\lambda)DD$ and let
$S\subseteq\mathcal{B}$, then $S$ is a defining set of $\mathcal{D}$ if and
only if $S$ contains a block of every $(v,k,t)$ directed trade $T=T_{1}-T_{2}$
such that $T$ is contained in $\mathcal{D}$.

Each defining set of a $t-(v,k,\lambda)DD$, $\mathcal{D}$ contains at least
one block in every trade in $\mathcal{D}$. In particular, if $\mathcal{D}$
contains $m$ mutually disjoint directed trades then the smallest defining set
of $\mathcal{D}$ must contain at least $m$ blocks. If a directed design
$\mathcal{D}$ contains a cyclical trade of volume $s$, then each defining set
for $\mathcal{D}$ must contain at least $\lfloor\frac{s+1}{2}\rfloor$ blocks
of $T_{1}$.

The concept of directed trades and defining sets for directed designs were
investigated in articles [13, 14].

A pairwise balanced design of order $v$ with block sizes $k\in$K or
PBD$(v,K,\lambda)$ is a pair $(V,\mathcal{B})$, where $V$ is a $v-$set, and
$\mathcal{B}$ is a collection of subsets (called blocks) of $V$ such that if
$B\in\mathcal{B}$ then $|B|\in$K and every pair of distinct elements of $V$
appears in precisely $\lambda$ blocks. A PBD$(v,K,1)$ is denoted by
PBD$(v,K)$.

A group divisible design of order $v$ with block sizes $k\in$K or
$(K,\lambda)-$GDD of type $g_{1}^{u_{1}}g_{2}^{u_{2}}...g_{N}^{u_{N}}$, where
$u_{1},u_{2},...,u_{N}$ are non-negative integers, is a triple
$(V,\mathcal{G},\mathcal{B})$, where $V$ is a $v-$set that is partitioned into
parts (called groups) of sizes $g_{1},g_{2},...,g_{N}$, and $\mathcal{B}$ is a
collection of subsets (called blocks) of $V$ such that if $B\in\mathcal{B}$
then $|B|\in$K and every pair of distinct elements of $V$ appears in precisely
$\lambda$ blocks or one group but not in both. A $(K,1)-$ GDD and
$(\\{k\\},\lambda)-$GDD are denoted by $K-$GDD and $(k,\lambda)-$GDD,
respectively.

One can see deleting one point from a PBD$(v,K)$ gives a $K$-GDD of type
$g_{1}^{u_{1}}g_{2}^{u_{2}}...g_{N}^{u_{N}}$, where $u_{1},u_{2},...,u_{N}$
are non-negative integers and for all $i=1,2,...,N$, $g_{i}=k_{i}-1;k_{i}\in
K$.

A transversal design $TD(k,\lambda;n)$ is a $(k,\lambda)-$GDD of type $n^{k}$.
When $\lambda=1$, we simply write $TD(k,n)$.

A directed group divisible design $(K,\lambda)-$DGDD is a group divisible
design in which every block is ordered and each ordered pair formed from
distinct elements of different groups occurs is exactly $\lambda$ blocks. A
$(K,\lambda)-$DGDD of type $(g_{1}^{u_{1}},g_{2}^{u_{2}},...,g_{N}^{u_{N}})$
is super-simple if its underlying $(K,2\lambda)-$GDD of type
$(g_{1}^{u_{1}},g_{2}^{u_{2}},...,g_{N}^{u_{N}})$ is super-simple.

Some results have been obtained on $2-(v,k,\lambda)$DDs for special $k$ and
$\lambda$ and their defining set. For example, in [13], has been proved that
if $\mathcal{D}$ be a $2-(v,3,1)$DD, then a defining set of $\mathcal{D}$ has
at least $\frac{v}{2}$ blocks. In [11], it has been shown that for each
admissible value of $v$, there exists a simple $2-(v,3,1)$DD whose smallest
defining sets have at least a half of the blocks. In [15], it has been shown
that the necessary and sufficient condition for the existence of a super-
simple $2-(v,4,1)$DD is $v\equiv 1$ (mod 3) and for these values of $v$ except
$v=7$, there exists a super-simple $2-(v,4,1)$DD whose smallest defining sets
have at least a half of the blocks. Also, in [16], it has been shown that for
all $v\equiv 1,5$ (mod 10) except $v=5,15$, there exists a super-simple
$2-(v,5,1)$DD such that their smallest defining sets have at least a half of
the blocks. In this paper, we prove that the necessary and sufficient
condition for the existence of a super-simple $2-(v,4,2)$DD is $v\equiv 1$
(mod 3) $(v\geq 10)$ and for these values of $v$, there exists a super-simple
$2-(v,4,2)$DD whose their smallest defining sets have at least a half of the
blocks. For this goal, we introduce the following quantity

$d=\frac{the\ total\ number\ of\ blocks\ in\ a\ smallest\ defining\ set\ in\
\mathcal{D}}{the\ total\ number\ of\ blocks\ in\ \mathcal{D}}$

and we show for all admissible values of $v$, $d\geq\frac{1}{2}$.

##  2 Recursive Constructions

For some values of $v$, the existence of super-simple $(v,4,2)$DD will be
proved by recursive constructions that we present them in this section for
later use.

######  Construction 1.

(Weighting). Let $(X,\mathcal{G},\mathcal{B})$ be a super-simple GDD with
index $\lambda_{1}$, and let $w:X\rightarrow Z^{+}\bigcup\\{0\\}$ be a weight
function on $X$, where $Z^{+}$ is the set of positive integers. Suppose that
for each block $B\in\mathcal{B}$, there exists a super-simple
$(k,\lambda_{2})-GDD$ of type $\\{w(x):x\in B\\}$. Then there exists a super-
simple $(k,\lambda_{1}\lambda_{2})-GDD$ of type $\\{\sum_{x\in
G_{i}}w(x):G_{i}\in\mathcal{G}\\}$.

######  Remark 1.

In the above construction, if the master GDD and input designs are directed
and for all of them, we have $d\geq\frac{1}{2}$ then the resulted GDD is a
super-simple DGDD and for it, we have $d\geq\frac{1}{2}$.

######  Construction 2.

If there exists a super-simple $(k,\lambda)$-DGDD of type $g_{1}^{u_{1}}\cdots
g_{t}^{u_{t}}$ with $d\geq\frac{1}{2}$ and a super-simple
$(g_{i}+\eta,k,\lambda)$DD for each $i(1\leq i\leq t)$ with
$d\geq\frac{1}{2}$, then there exists a super-simple
$(\sum_{i=1}^{t}g_{i}u_{i}+\eta,k,\lambda)$DD with $d\geq\frac{1}{2}$, where
$\eta=0\ \ or\ \ 1$.

###### Proof.

Replacing each $G_{i}\in\mathcal{G}$, $|G_{i}|=g_{i}$ with a super-simple
$(g_{i}+\eta,k,\lambda)$DD gives a super-simple
$(\sum_{i=1}^{t}g_{i}u_{i}+\eta,k,\lambda)$DD. Obviously, if for master DGDD
and for all input designs $d\geq\frac{1}{2}$, then for the resulted design,
the inequality $d\geq\frac{1}{2}$ should be hold. ∎

######  Construction 3.

If there exists a $K-GDD$ of type
$g_{1}^{u_{1}}g_{2}^{u_{2}}...g_{n}^{u_{n}}$, a super-simple $(\alpha
g_{i}+1,4,2)$DD for each i, $i=1,2,...,n$ and a super-simple $4$-DGDD of type
$\alpha^{k}$ for each $k\in K$, then there exists a super-simple
$(\alpha\sum_{i=1}^{n}g_{i}u_{i}+1,4,2)$DD.

###### Proof.

Let $(X,\mathcal{G},\mathcal{B})$ be a group divisible design with blocks of
size $k\in K$ and groups of size $g_{i}$ for $i=1,2,\cdots,n$. Let $Y=X\times
Z_{\alpha}\cup\\{\infty\\}$. Replacing each element $x\in X$ with $\alpha$ new
points $\\{x_{1},x_{2},\cdots,x_{\alpha}\\}$ and each block $B\in\mathcal{B}$
of size $k\in K$ with a super-simple $4$-DGDD of type $\alpha^{k}$ such that
its groups are $\\{x_{1},x_{2},\cdots,x_{\alpha}\ :x\in X\\}$ gives us a
super-simple $4$-DGDD of type $(\alpha g_{1})^{u_{1}}(\alpha
g_{2})^{u_{2}}\cdots(\alpha g_{n})^{u_{n}}$. Finally, filling in the holes
with a new point $\infty$ and using a super-simple $2-(\alpha
g_{i}+1,4,2)$DDs, we obtain a super-simple
$2-(\alpha\sum_{i=1}^{n}g_{i}u_{i}+1,4,2)$DD. ∎

######  Remark 2.

In the above construction, if for master GDD and for all input designs, we
have $d\geq\frac{1}{2}$, then for the resulting design, the inequality
$d\geq\frac{1}{2}$ should be hold.

##  3 Direct Construction

In this section, we construct some super-simple $2-(v,4,2)$DDs for some small
admissible values of $v$ by direct construction and for these values of $v$,
we show that the parameter $d$ for constructed designs is at least
$\frac{1}{2}$. In the rest of paper, by $(v,4,2)$DD, we mean $2-(v,4,2)$DD.

######  Lemma 1.

There exists a super-simple $(v,4,2)$DD for all $v\in\\{10,13,16,19,22\\\
$,$25,28,31,34,40,43,58,67,79,94,103\\}$, whose their smallest defining sets
have at least a half of the blocks.

###### Proof.

$v=10$: The following base blocks by (+1 mod 10) form a super-simple
$(10,4,2)$DD.

(0,1,2,6) | (2,0,5,8)
---|---
(1,0,4,3) |

This design has 30 blocks, the first column has 10 disjoint directed trades of
volume 2 and the last column is a cyclical trade of volume 10. Since each
defining set for this super-simple directed design must contain at least one
4-tuple of each directed trades in first column and five 4-tuples of cyclical
trade in second column, then each defining set must contain at least 10+5=15
blocks. So for this super-simple $(10,4,2)$DD, we have $d\geq\frac{1}{2}$.

$v=13$: The following base blocks by (+1 mod 13) form a super-simple
$(13,4,2)$DD.

(0,1,11,5) | (0,1,6,4)
---|---
(1,0,3,9) | (6,5,0,2)

This design has 52 blocks, each of two columns has $13$ disjoint directed
trades of volume 2. Since each defining set for this super-simple directed
design must contain at least one 4-tuple of each directed trades, then each
defining set must contain at least $13+13=26$ blocks. So for this super-simple
$(13,4,2)$DD, we have $d\geq\frac{1}{2}$.

$v=16$: The following base blocks by (+1 mod 16) form a super-simple
$(16,4,2)$DD.

(0,1,6,8) | (4,1,0,10) | (0,1,9,5)
---|---|---
(1,0,11,14) | (12,0,3,14) |

There are 80 blocks in a super-simple $(16,4,2)$DD. The first two columns have
32 disjoint directed trades of volume 2, and the last column is a cyclical
trade of volume 16. Since each defining set for this super-simple directed
design must contain at least one 4-tuple of each directed trade in the first
two columns and eight 4-tuples of cyclical trade in the last column, then each
defining set must contain at least $16+16+8=40$ blocks. Therefore for this
super-simple$(16,4,2)$DD the inequality $d\geq\frac{1}{2}$ is satisfied.

$v=19$: The following base blocks by (+1 mod 19) form a super-simple
$(19,4,2)$DD.

(0,6,1,3) | (4,1,0,12) | (0,1,8,10)
---|---|---
(15,7,1,0) | (5,0,3,10) | (2,0,6,15)

This design has $114$ blocks, each of three columns of this design has 19
disjoint directed trades of volume 2. Since each defining set for this super-
simple $(19,4,2)$DD must contain at least one 4-tuple of each directed trades,
then each defining set must contain at least $19+19+19=57$ blocks. Therefore
for this super-simple $(19,4,2)$DD the inequality $d\geq\frac{1}{2}$ is
satisfied.

$v=22$: The following base blocks by (+1 mod 22) form a super-simple
$(22,4,2)$DD.

(0,3,1,6) | (1,5,7,0) | (15,1,10,0) | (0,14,2,10)
---|---|---|---
(0,4,1,16) | (2,0,13,9) | (8,0,19,3) |

This design contains 154 blocks, each of first three columns has 22 disjoint
directed trades of volume 2 and the last column is a cyclical trade of volume
22. Since each defining set for this design must contain one 4-tuple of each
directed trade in the first three columns and 11 4-tuples of cyclical trade in
the last column, then each defining set contains at least $22\times 3+11=77$
blocks. So $d\geq\frac{1}{2}$.

$v=25$: The following base blocks by (+1 mod 25) form a super-simple
$(25,4,2)$DD.

(0,1,18,3) | (0,11,1,7) | (13,6,1,0) | (2,0,16,12)
---|---|---|---
(16,0,8,23) | (14,9,3,0) | (16,21,0,4) | (19,0,22,24)

This design contains 200 blocks, each of four columns has 25 disjoint directed
trades of volume 2. Since each defining set for this design must contain one
4-tuple of each directed trade, then each defining set contains at least
$25\times 4=100$ blocks. So the desired inequality follows.

$v=28$: The following base blocks by (+1 mod 28) form a super-simple
$(28,4,2)$DD.

(4,0,2,1) | (1,20,0,26) | (0,3,20,7) | (0,23,3,14) | (0,18,2,12)
---|---|---|---|---
(15,0,1,5) | (19,0,6,13) | (17,0,10,5) | (19,11,7,0) |

This design has 252 blocks, each of the first four columns has 28 disjoint
directed trades of volume 2 and the last column is a cyclical trade of volume
28. Since each defining set for this design must contain one 4-tuple of each
directed trade in the first four columns and 14 4-tuples of cyclical trade in
the last column, then each defining set contains at least $28\times 4+14=126$
blocks. So the desired inequality follows.

$v=31$: The following base blocks by (+1 mod 31) form a super-simple
$(31,4,2)$DD.

(0,3,8,1) | (11,5,0,1) | (7,1,19,0) | (0,2,23,9) | (15,0,4,3)
---|---|---|---|---
(3,0,18,13) | (3,0,11,17) | (0,2,15,6) | (26,2,0,11) | (0,27,19,10)

This design contains 310 blocks, each of the five columns has 31 disjoint
directed trades of volume 2. Since each defining set for this design must
contain one 4-tuple of each directed trade, then each defining set contains at
least $31\times 5=155$ blocks. So $d\geq\frac{1}{2}$.

$v=34$: The following base blocks by (+1 mod 34) form a super-simple
$(34,4,2)$DD.

(25,0,4,11) | (2,6,0,21) | (0,1,9,3)
---|---|---
(2,12,9,0) | (7,3,29,0) | (1,7,19,0)
(20,0,2,5) |  |
(14,0,1,24) | (21,0,11,29) |
(5,1,0,17) | (28,20,11,0) |

This design has 374 blocks, the first column contains 34 cyclical trades of
volume 5 and the last two columns contain 102 disjoint directed trades of
volume 2. Since each defining set for this design must contain
$\frac{5+1}{2}\times 34$ 4-tuples of cyclical trade and one 4-tuple of each
directed trade of volume 2, then each defining set contains at least
$102+102=204$ blocks. So the desired inequality follows.

$v=40$: The following base blocks by (+1 mod 40) form a super-simple
$(40,4,2)$DD.

(4,1,0,2) | (0,4,11,32) | (0,16,7,29) | (26,20,0,5)
---|---|---|---
(0,3,18,23) | (5,0,33,17) | (32,1,0,38) |
(0,12,3,33) | (17,4,0,30) | (0,6,1,15) |
(23,0,10,2) | (18,0,4,29) | (22,30,6,0) |

This design has 520 blocks, contains 240 disjoint directed trades of volume 2
in the first three columns and a cyclical trade in the last column. Since each
defining set for this design must contain one 4-tuple of each directed trades
in the first three columns and 20 4-tuples of the cyclical trade, then each
defining set contains at least $240+20=260$ blocks. So $d\geq\frac{1}{2}$.

$v=43$: The following base blocks by (+1 mod 43) form a super-simple
$(43,4,2)$DD.

(0,1,8,3) | (0,1,4,10) | (1,19,0,12) | (12,0,42,29)
---|---|---|---
(0,5,20,27) | (11,22,3,0) | (0,10,35,26) | (4,16,0,29)
(0,4,2,23) | (2,17,0,8) | (4,20,0,9) |
(10,24,3,0) | (5,0,33,18) | (12,6,26,0) |

This design has 602 blocks, contains 301 disjoint directed trades of volume 2.
Since each defining set for this design must contain one 4-tuple of each
directed trades, then each defining set contains at least $301$ blocks. So
$d\geq\frac{1}{2}$.

$v=58$: The following base blocks by (+1 mod 58) form a super-simple
$(58,4,2)$DD.

(0,1,56,3) | (1,31,0,27) | (0,35,2,24) | (0,12,3,16)
---|---|---|---
(0,1,14,9) | (1,32,0,52) | (5,0,23,15) | (40,15,3,0)
(0,4,37,19) | (13,22,6,0) | (0,17,34,5) | (15,26,0,7)
(30,13,4,0) | (20,34,0,6) | (27,17,0,7) | (24,36,0,47)
(22,38,0,8) | (0,25,6,35) |  |
(0,56,40,19) |  |  |

This design has 1102 blocks, contains 522 disjoint directed trades of volume 2
and a cyclical trade of volume 58. Then each defining set contains at least
$522+29=551$ blocks. So the desired inequality follows.

$v=67$: The following base blocks by (+1 mod 67) form a super-simple
$(67,4,2)$DD.

(1,10,0,16) | (7,0,33,30) | (42,0,3,7) | (14,38,0,2)
---|---|---|---
(33,24,0,8) | (0,10,17,28) | (6,2,0,19) | (45,2,0,21)
(5,13,0,50) | (0,14,6,47) | (23,7,0,49) | (9,19,31,0)
(5,8,0,2) | (44,29,14,0) | (0,38,54,5) | (12,0,25,39)
(1,0,30,12) | (20,0,9,40) | (21,0,4,36) |
(11,0,1,46) | (0,20,41,1) | (27,32,4,0) |

This design has 1474 blocks, contains 737 disjoint directed trades of volume
2. Then each defining set for this design contains at least $737$ blocks. So
$d\geq\frac{1}{2}$.

$v=79$: The following base blocks by (+1 mod 79) form a super-simple
$(79,4,2)$DD.

(17,0,37,1) | (0,1,22,15) | (0,31,2,23) | (1,33,0,9) | (35,0,19,2)
---|---|---|---|---
(13,30,0,4) | (1,0,30,12) | (51,30,0,10) | (5,0,12,45) | (0,25,3,38)
(3,0,30,45) | (39,0,4,29) | (19,27,4,0) | (15,0,5,39) | (37,0,6,26)
(43,0,3,31) | (4,0,51,37) | (27,0,13,5) | (11,6,28,0) | (6,0,54,16)
(0,18,27,6) | (3,26,44,0) | (0,50,7,26) |  |
(20,2,0,34) | (9,25,2,0) | (0,43,24,11) |  |

This design has 2054 blocks, contains 1027 disjoint directed trades of volume
2. Then each defining set for this design contains at least $1027$ blocks. So
$d\geq\frac{1}{2}$.

$v=94$: The following base blocks by (+1 mod 94) form a super-simple
$(94,4,2)$DD.

(1,23,0,40) | (2,0,20,43) | (20,9,0,82) | (0,79,88,49) | (8,0,61,29)
---|---|---|---|---
(0,5,28,42) | (13,87,6,0) | (0,12,26,71) | (0,15,4,45) | (0,32,8,44)
(47,19,0,3) | (0,13,32,57) | (26,34,0,7) | (17,44,4,0) | (5,43,0,25)
(18,46,0,4) | (0,69,91,52) | (16,1,0,34) | (3,56,29,0) | (0,56,2,31)
(29,0,5,59) | (61,0,10,1) | (59,0,11,1) | (73,0,16,6) | (0,6,17,48)
(36,5,0,51) | (2,70,37,0) | (12,4,84,0) | (0,58,24,3) | (38,7,0,16)
(30,0,49,2) |  |  |  |

This design has 2914 blocks, contains 1410 disjoint directed trades of volume
2 and a cyclical trade of volume 94. So each defining set for this design
contains at least $1410+47=1457$ blocks. So the desired inequality follows.

$v=103$: The following base blocks by (+1 mod 103) form a super-simple
$(103,4,2)$DD.

(0,1,15,31) | (21,1,0,50) | (51,0,2,25) | (34,53,2,0) | (12,39,0,3)
---|---|---|---|---
(0,89,97,73) | (6,50,0,17) | (5,0,51,15) | (2,65,19,0) | (20,42,3,0)
(45,22,0,4) | (19,4,0,47) | (0,3,51,24) | (29,0,5,62) | (0,2,37,73)
(0,6,29,38) | (0,4,31,65) | (49,1,0,46) | (5,0,53,40) | (70,43,8,0)
(69,0,5,26) | (19,12,59,31) | (11,37,23,0) | (0,1,93,8) | (0,16,72,44)
(40,0,82,9) | (45,0,7,20) | (8,0,18,32) | (92,0,77,59) | (10,47,0,69)
(64,4,29,0) | (12,48,61,0) |  |  |
(0,66,41,86) | (22,13,0,6) |  |  |

This design has 3502 blocks, contains 1751 disjoint directed trades of volume
2. So each defining set of this design contains at least 1751 blocks. Then the
inequality $d\geq\frac{1}{2}$ is satisfied. Then the proof is complete. ∎

##  4 Super-simple directed group divisible designs with block size $4$ and
index $2$

In this section, we construct some super-simple DGDDs with $d\geq\frac{1}{2}$,
that we need them in our main result.

######  Lemma 2.

There exists a super-simple $(4,2)$-DGDD of type $3^{t}$ for $t\in\\{6,7,8,9,$
$13\\}$ with $d\geq\frac{1}{2}$.

###### Proof.

Let the point set be $X=Z_{3t}$ and the group set be
$\mathcal{G}=\\{\\{0,t,2t\\}+i\,|\,\ 0\leq i\leq t-1\\}$. The base blocks are
listed below. All the bellow base blocks are developed by mod 3t.

$t=6$:  (2,0,5,9) (0,1,2,16) (7,10,0,2) (11,4,1,0) (1,5,0,10)

This super-simple DGDD has 90 blocks, in the first column it contains 18
cyclical trades of volume 3 and in the second column 18 disjoint directed
trades of volume 2. Then each defining set for this super-simple DGDD contains
at least $\frac{3+1}{2}\times 18+18=54$ blocks. So $d\geq\frac{1}{2}$.

$t=7$:  (0,5,1,13) (0,11,17,19) (0,1,4,6) (11,10,1,5) (12,11,0,19) (7,12,4,1)

This super-simple DGDD has 126 blocks, each of three columns has 21 disjoint
directed trades of volume 2. Therefore each defining set for this super-simple
DGDD contains at least $21\times 3=63$ blocks. So $d\geq\frac{1}{2}$.

$t=8$:  (0,1,13,6) (0,4,10,15) (12,7,0,2) (2,1,0,4) (21,0,4,18) (2,0,9,15)
(1,2,11,22)

This super-simple DGDD contains 168 blocks, the first column contains 24
cyclical trades of volume 3 and each of two other columns has 24 disjoint
directed trades of volume 2. Then each defining set for this super-simple DGDD
contains at least $\frac{3+1}{2}\times 24+2\times 24=96$ blocks. So the
desired inequality follows.

$t=9$:  (0,6,1,13) (1,0,2,5) (0,6,2,17) (3,13,1,23) (3,11,0,24) (0,14,8,4)
(17,1,6,4) (1,8,0,20)

This super-simple DGDD has 216 blocks, each of two first column contains 27
cyclical trades of volume 3 and the last column contains 27 disjoint directed
trades of volume 2. So each defining set for this super-simple DGDD contains
at least $2(\frac{3+1}{2}\times 27)+27=135$ blocks. So the desired inequality
follows.

$t=13$:

( 3,0,12,21 ) | ( 2,1,0,4 ) | ( 8,0,2,19 ) | ( 0,16,22,5 ) | ( 5,0,11,1 )
---|---|---|---|---
( 19,0,3,35 ) | ( 4,0,32,18 ) | ( 2,0,29,14 ) | ( 0,17,1,8 ) | ( 0,5,15,36 )
(5,12,0,20) | ( 14,0,33,24 ) |  |  |

This super-simple DGDD has 468 blocks, each of two first columns of this
super-simple DGDD contains 39 cyclical trades of volume 3 and each of the next
three columns has 39 disjoint directed trades of volume 2. So each defining
set for this super-simple DGDD has at least $2(\frac{3+1}{2}\times 39)+3\times
39=273$ blocks. Therefore $d\geq\frac{1}{2}$. ∎

######  Lemma 3.

There exists a super-simple $(4,2)$-DGDD of type $t^{4}$ for
$t\in\\{4,5,6,13,\\\ 19,22\\}$ with $d\geq\frac{1}{2}$.

###### Proof.

Let $X=Z_{4t}$, $\mathcal{G}=(\\{0,4,...,4(t-1)\\}+i\,|\,\ 0\leq i\leq 3\\}$
and the base blocks are listed below, all the bellow base blocks are developed
by mod 4t.

$t=4$:  (0,1,3,10) (0,5,2,11) (2,0,3,13) (0,15,14,5)

This super-simple DGDD has 64 blocks, each of two columns has 16 disjoint
directed trades of volume 2. So each defining set for this super-simple DGDD
must contain at least $16\times 2=32$ blocks. Then the desired inequality
follows.

$t=5$:  (1,0,10,3) (0,7,2,5) (0,1,18,11) (3,0,14,9) (1,6,0,7)

This super-simple DGDD has 100 blocks, the first column of this super-simple
DGDD contains 20 cyclical trades of volume 3 and the second column has 20
disjoint directed trades of volume 2, so each defining set contains at least
$\frac{3+1}{2}\times 20+20=60$ blocks. So the desired inequality follows.

$t=6$:  (0,1,2,7) (2,0,5,19) (1,0,14,11) (0,2,15,21) (14,0,23,17) (0,15,22,9)

This super-simple DGDD has 144 blocks, each of two columns of this super-
simple DGDD contains 24 cyclical trades of volume 3, so each defining set for
this super-simple DGDD contains at least $2(\frac{3+1}{2}\times 24)=96$
blocks. So the desired inequality follows.

$t=13$:  (0,10,15,1) (1,7,0,2) (3,0,25,6) (0,5,39,26) (1,0,18,27) (0,29,19,2)
(0,13,7,30) (7,10,0,21) (0,29,43,18) (5,15,22,0)

This super-simple DGDD has 520 blocks, each of two first columns of this
super-simple DGDD contains 52 cyclical trades of volume 3, the third column
has 52 disjoint directed trades of volume 2, and two last columns are cyclical
trades of volume 52. So each defining set for this super-simple DGDD contains
at least $2(\frac{3+1}{2}\times 52)+52+2\times 26=312$ blocks. Then the
inequality $d\geq\frac{1}{2}$ follows.

$t=19$:  (30,0,3,9) (22,0,35,1) (22,11,0,29) (0,47,33,70) (6,25,0,51)
(0,2,13,43) (5,15,0,22) (0,11,1,26) (9,26,55,0) (0,57,15,34) (2,0,33,7)
(0,11,1,26) (7,17,54,0) ( 5,0,14,39 ) (10,1,0,3) (1,19,0,6) (3,33,0,62)
(2,23,0,37) (13,0,31,58) (18,3,0,41)

This super-simple DGDD has 1520 blocks, each of four first columns contains 76
cyclical trades of volume 3 and one cyclical trade of volume 76, and the last
column has 152 disjoint directed trades of volume 2. So each defining set for
this super-simple DGDD contains at least $4(\frac{3+1}{2}\times 76)+4\times
38+152=912$ blocks. Then the inequality $d\geq\frac{1}{2}$ follows.

$t=22$:

(0,6,19,1) | (2,27,5,0) | (2,39,0,25) | (35,0,2,29) | (0,30,7,69)
---|---|---|---|---
(6,0,15,33) | (39,0,5,26) | (3,18,37,0) | (12,27,0,10) | (0,13,39,46)
(15,0,62,37) | (25,0,14,3) | (9,19,0,54) | (10,0,3,41) | (7,0,65,50)
(0,57,74,79) | (42,0,11,53) | (38,1,0,67) | (1,0,43,2) |
(17,0,35,58) | (21,10,0,55) | (0,29,38,59) |  |

This super-simple DGDD has 1936 blocks, each of three first columns contains
88 cyclical trades of volume 3 and 88 disjoint directed trades of volume 2,
the fourth column contains 88 cyclical trades of volume 3 and a cyclical trade
of volume 88 and the last column contains 88 cyclical trades of volume 3. So
each defining set for this super-simple DGDD contains at least
$5(\frac{3+1}{2}\times 88)+3\times 88+44=1188$ blocks. Then the desired
inequality follows. ∎

######  Lemma 4.

There exists a super-simple $(4,2)$-DGDD of type $9^{t}$ for $t\in\\{4,5\\}$
with $d\geq\frac{1}{2}$.

###### Proof.

Let the point set be $X=Z_{9t}$ and the group set be
$\mathcal{G}=\\{\\{0,t,2t,...,8t\\}+i\,|\,\ 0\leq i\leq t-1\\}$. The base
blocks are listed below. All the bellow base blocks are developed by mod 9t.

$t=4$:  (1,0,2,7) (0,1,10,19) (3,0,21,10) (2,0,5,31) (3,14,0,17) (1,0,23,14)
(9,0,15,2) (10,0,31,25)

This super-simple DGDD has 288 blocks, each of four columns of this DGDD has
36 disjoint directed trades of volume 2. So each defining set for this super-
simple DGDD contains at least $36\times 4=144$ blocks. Therefore
$d\geq\frac{1}{2}$.

$t=5$:  (0,1,2,4) (2,0,11,19) (6,13,27,0) (11,0,42,33) (0,26,3,19) (1,0,8,14)
(4,0,37,13) (1,7,0,24) (0,16,27,39) (17,3,0,29) (16,0,43,34) (0,4,41,28)

This super-simple DGDD has 540 blocks, each of two first columns of this
super-simple DGDD contains 45 cyclical trades of volume 3 and each of the next
three columns has 45 disjoint directed trades of volume 2. So each defining
set for this super-simple DGDD has at least $2(\frac{3+1}{2}\times 45)+3\times
45=315$ blocks. Therefore $d\geq\frac{1}{2}$. ∎

######  Lemma 5.

There exists a super-simple $(4,2)$-DGDD of type $6^{5}$ with
$d\geq\frac{1}{2}$.

###### Proof.

Let $X=Z_{30}$, $\mathcal{G}=\\{\\{0,5,10,15,20,25\\}+i\,|\,\ 0\leq i\leq
4\\}$, the set base blocks are listed below. They are developed by mod $30$.

(7,9,0,1) | (19,1,12,0) | (1,2,4,0) | (3,9,0,17)
---|---|---|---
(3,0,7,16) | (6,13,0,19) | (4,8,16,0) |
(2,0,18,21) |  |  |

This super-simple DGDD has 240 blocks, the first column contains 30 cyclical
trades of volume 3 and each of the second and third column has 30 disjoint
directed trades of volume 2 and the last column is a cyclical trade of volume
30. So each defining set for this super-simple DGDD contains at least
$\frac{3+1}{2}\times 30+2\times 30+15=135$ blocks. Then the desired inequality
follows. ∎

##  5 Proof of Main Theorem

This section is devoted to find super-simple $(v,4,2)$DDs for some admissible
values of $v$ by recursive constructions presented in Section 2 and using
super-simple DGDDs obtained in Section 4. First we present some lemmas which
are generalized form of the lemmas in [3] to directed designs. Finally, we
conclude the main result in the end of this section.

######  Lemma 6.

There exists a super-simple $(v,4,2)$DD for $v\in\\{37,46,52,76,88\\}$ with
$d\geq\frac{1}{2}$.

###### Proof.

By Lemma 4, we know that there exists a super-simple $(4,2)-$DGDD of group
type $9^{t}$ for $t\in\\{4,5\\}$. Since $|G|+1=9+1=10$ for each group $G$ of
the super-simple DGDD, by Lemma 1 and Construction 2, we conclude that there
exists a super-simple $(v,4,2)$DD that $v\in\\{37,46\\}$ with
$d\geq\frac{1}{2}$.

By Lemma 3, there exists a super-simple $(4,2)-$DGDD of group type $t^{4}$ for
$t\in\\{13,19,22\\}$ with $d\geq\frac{1}{2}$. Since $|G|=13,19$ or $22$, for
each group $G$ of the super-simple DGDD, by Lemma 1 and Construction 2 it
follows there exists a super-simple $(v,4,2)$DD for $v\in\\{52,76,88\\}$ with
$d\geq\frac{1}{2}$. ∎

######  Lemma 7.

There exists a super-simple $(v,4,2)$DD with $d\geq\frac{1}{2}$ for
$v\in\\{49,61,73\\\ ,97,121\\}$.

###### Proof.

By using a super-simple $(4,2)-$DGDD of group type $t^{4}$, $4\leq t\leq 6$ by
Lemma 3 with $d\geq\frac{1}{2}$ and applying Construction 1 by using a
$TD(4,3)$ as an input design, we obtain a super-simple $(4,2)-$DGDD of group
type $(3t)^{4}$ with $d\geq\frac{1}{2}$. On the other hand, for $4\leq t\leq
6$ there exists a super-simple $(3t+1,4,2)$DD with $d\geq\frac{1}{2}$. So by
Construction 2, there exists a super-simple $(12t+1,4,2)$DD with
$d\geq\frac{1}{2}$ for $4\leq t\leq 6$.

By considering a super-simple $(4,2)-$DGDD of group type $6^{4}$ with
$d\geq\frac{1}{2}$, obtained in Lemma 3 and applying Construction 1 by using a
$TD(4,m)$ in which $m\in\\{4,5\\}$ as an input design, we obtain a super-
simple $(4,2)-$DGDD of group type ${(6m)}^{4}$ with $d\geq\frac{1}{2}$. By
considering the existence of super-simple $(6m+1,4,2)$DD from Lemma 1 and
using Construction 2, we conclude that there exists a super-simple
$(24m+1,4,2)$DD with $d\geq\frac{1}{2}$. ∎

######  Lemma 8.

There exists a super-simple $(v,4,2)$DD for $v\in\\{64,100,112\\}$ with
$d\geq\frac{1}{2}$.

###### Proof.

By using a super-simple $(4,2)-$DGDD of group type $4^{4}$ with
$d\geq\frac{1}{2}$ from Lemma 3 and applying Construction 1 by using a
$TD(4,m)$ in which $m\in\\{4,7\\}$ as an input design, we obtain a super-
simple $(4,2)-$DGDD of group type ${(4m)}^{4}$ with $d\geq\frac{1}{2}$. By
Lemma 1, there exists a super-simple $(4m,4,2)$DD with $d\geq\frac{1}{2}$ for
$m\in\\{4,7\\}$ . Then by Construction 2, we obtain a super-simple
$(16m,4,2)$DD.

By considering a super-simple $(4,2)-$DGDD of group type $5^{4}$ with
$d\geq\frac{1}{2}$ from Lemma 3 and applying Construction 1 by using a
$TD(4,5)$ as an input design, we obtain a super-simple $(4,2)-$DGDD of group
type ${(25)}^{4}$ with $d\geq\frac{1}{2}$. Since by Lemma 1 there exists a
super-simple $(25,4,2)$DD with $d\geq\frac{1}{2}$, then by Construction 2, we
obtain a super-simple $(100,4,2)$DD with $d\geq\frac{1}{2}$. ∎

######  Lemma 9.

There exists a super-simple $(v,4,2)$DD for $v\in\\{70,127\\}$ with
$d\geq\frac{1}{2}$.

###### Proof.

We have a $4-$GDD of group type $2^{7}$ with set $X=Z_{14}$, group set
$\mathcal{G}=\\{\\{0,7\\}+i\,|\,\ 0\leq i\leq 6\\}$ and block set
$\mathcal{B}=\\{(0,1,4,6)$ mod $14\\}$. Applying Constraction 1 and using a
super-simple $(4,2)-$DGDD of group type $5^{4}$ from Lemma 3 with
$d\geq\frac{1}{2}$ as an input design, we obtain a super-simple $(4,2)-$DGDD
of group type $10^{7}$ with $d\geq\frac{1}{2}$. By Lemma 1 and Construction 2
we obtain a super-simple $(70,4,2)$DD with $d\geq\frac{1}{2}$.

By using a $4-$GDD of group type $2^{7}$, applying Construction 1 and using a
super-simple $(4,2)-$DGDD of group type $9^{4}$ (see Lemma 4) with
$d\geq\frac{1}{2}$ as an input design, we get a super-simple $(4,2)-$DGDD of
group type $18^{7}$ with $d\geq\frac{1}{2}$. Since $18+1=19$, by Lemma 1 and
Construction 2, we obtain a super-simple $(127,4,2)$DD with
$d\geq\frac{1}{2}$. ∎

######  Lemma 10.

There exists a super-simple $(v,4,2)$DD for $v\in\\{55,82,85,106,109,$
$118\\}$ with $d\geq\frac{1}{2}$.

###### Proof.

By Lemma 2, we know that there exists a super-simple $(4,2)-$DGDD of group
type $3^{t}$ for $t\in\\{6,7,9,13\\}$ with $d\geq\frac{1}{2}$. Applying
Construction 1 by using $TD(4,m)$ in which $m\in\\{3,4\\}$ as an input design
we obtain a super-simple $(4,2)-$DGDD of type ${(3m)}^{t}$ with
$d\geq\frac{1}{2}$. By considering the existence of super-simple
$(3m+1,4,2)$DD and by Construction 2, we conclude that there exists a super-
simple $(3mt+1,4,2)$DD where $m\in\\{4,3\\}$ and $t\in\\{6,7,9,13\\}$ with
$d\geq\frac{1}{2}$.

By using a super-simple $(4,2)-$DGDD of group type $3^{7}$ with
$d\geq\frac{1}{2}$, (see Lemma 2), applying Construction 1 and using a
$TD(4,5)$ as an input design we obtain a super- simple $(4,2)-$DGDD of group
type $15^{7}$ with $d\geq\frac{1}{2}$. By Lemma 1, there exists a super-simple
$(15+1,4,2)$DD, then by Contruction 2, we obtain a super-simple
$(105+1,4,2)$DD with $d\geq\frac{1}{2}$. ∎

######  Lemma 11.

There exists a super-simple $(v,4,2)$DD for $v\in\\{115,133\\}$ with
$d\geq\frac{1}{2}$.

###### Proof.

For $v=115$, we delete one point from the last group of a $TD(5,4)$ to obtain
a $(\\{4,5\\})-$ GDD of group type $4^{4}3^{1}$. By applying Construction 1
and using a super-simple $(4,2)-$DGDDs of group type $6^{4}$ and $6^{5}$ with
$d\geq\frac{1}{2}$ from Lemma 3 and Lemma 5, as input designs, we get a super-
simple $(4,2)-$DGDD of group type $24^{4}18^{1}$, with $d\geq\frac{1}{2}$.
Since for each group G, $|G|+1\in\\{19,25\\}$, by Lemma 1 and Construction 2,
there exsits a super-simple$(115,4,2)$DD with $d\geq\frac{1}{2}$.

For $v=133$, we delete three points from the last group of a $TD(5,5)$ to
obtain a $(\\{4,5\\},1)-$GDD of group type $5^{4}2^{1}$. Applying Construction
1 and using super-simple $(4,2)-$DGDDs of group types $6^{4}$ and $6^{5}$ as
input designs, we get a super-simple $(4,2)-$DGDD of group type
$30^{4}12^{1}$. Since for each group G, $|G|+1\in\\{13,31\\}$, by Lemma 1 and
Construction 2, there exsits a super-simple $(133,4,2)$DD. ∎

Let $N(m)$ be the number of mutually orthogonal latin squares of order $m$.
The following lemmas are generalized forms of Lemmas 4.27 and 4.28 in [3].

######  Lemma 12.

Let $N(m)\geq 6$, ${a,b}\in[0,m]$. If there exists a super-simple $(v,4,2)$DD
for $v\in\\{3m+1,3a+1,3b+1\\}$ with $d\geq\frac{1}{2}$, then there exists a
super-simple $(18m+3a+3b+1,4,2)$DD with $d\geq\frac{1}{2}$.

###### Proof.

Delete $m-a$ and $m-b$ points from the last two groups of a $TD(8,m)$ to
obtain a $(\\{6,7,8\\})-$GDD of group type $m^{6}a^{1}b^{1}$. Apply
Construction 1 and use super-simple $(4,2)-$DGDDs of group type $3^{k}$ for
$k\in\\{6,7,8,9\\}$ as input designs. We obtain a super-simple $(4,2)-$DGDD of
group type $(3m)^{6}(3a)^{1}(3b)^{1}$. Since by Lemma 1 there exists a super-
simple $(v,4,2)$DD for $v\in\\{3m+1,3a+1,3b+1\\}$, we conclude that there
exists a super-simple$(18m+3a+3b+1,4,2)$DD. This completes the proof. ∎

######  Lemma 13.

Let $N(m)\geq 7$, ${a,b,c}\in[0,m]$. If there exists a super-simple
$(v,4,2)$DD for $v\in\\{3m+1,3a+1,3b+1,3c+1\\}$ with $d\geq\frac{1}{2}$, then
there exists a super-simple $(18m+3a+3b+3c+1,4,2)$DD with $d\geq\frac{1}{2}$.

###### Proof.

The proof is similarly to the proof of Lemma 12. ∎

######  Lemma 14.

There exists a super-simple $(v,4,2)$DD for $v\in M=\\{136,139,$
$277,280,283,292,295,298,301,409,412,415,424,427,430,433,436,439,442,445,$
$448,454,457,460,463,478,481,496,499,553\\}$ with $d\geq\frac{1}{2}$.

###### Proof.

For $v\in\\{136,139\\}$, apply Lemma 12 with $m=7$ and $a+b\in\\{3,4\\}$.
For $v\in M\setminus\\{136,139\\}$ apply Lemma 13 with the parameters shown in
following table. We conclude that for these values of $v$ there exists a
super-simple $(v,4,2)$DD.

Let $[a,b]_{3}^{1}$ denotes the set $\\{v|\ v\equiv 1$ (mod $3),\ \ a\leq
v\leq b\\}$.

$v$ | $m$ | $a+b+c\subseteq[3,m]\cup\\{0\\}$
---|---|---
$[277,298]_{3}^{1}$ | 11 |  $a+b+c\in$[26,33]$\setminus${29,30}
301 | 16 |  $a+b+c=$4
$[409,433]_{3}^{1}$ $\setminus${418,421} | 16 |  $a+b+c\in$[40,48]$\setminus${43,44}
$[436,463]_{3}^{1}$ $\setminus${451} | 19 |  $a+b+c\in$[31,40]$\setminus${36}
{478,481} | 19 |  $a+b+c\in$ {45,46}
{496,499} | 19 |  $a+b+c\in$ {51,52}
553 | 19 |  $a+b+c=$ 22

This completes the proof. ∎

######  Lemma 15.

[10] There exists a $PBD(n,(6,7,8,9\\})$ for all $n$ except $n\in[10,30],$
$[32,41],[45,47],[93,95],[98,101],[137,139],[142,150],[152,155],\\{160,161,166,$
$167,185\\}$.

######  Corollary 1.

There exists a $\\{6,7,8,9\\}-$GDD with group type $5^{a}6^{b}7^{c}8^{d}$,
where a,b,c,d are nonnagative integers and $5a+6b+7c+8d=m$ for all $m$, except
$m\in[9,29],[31,40],[44,46],[92,94],[97,100],[136,138],[141,149],[151,154],\\{159,160,$
$165,166,184\\}$.

Now, we are in a position to conclude the main result.
Main Theorem. For all $v\equiv 1$ (mod 3 ) and $v\geq 10$, there exists a
super-simple $(v,4,2)$DD with $d\geq\frac{1}{2}$.

###### Proof.

By Corollary 1, there exists $\\{6,7,8,9\\}-$GDD of order $m$ and group type
$5^{a}6^{b}7^{c}8^{d}$ where $a,b,c$ and $d$ are nonnegative integers and
$m=5a+6b+7c+8d$ for values of $m$ satisfied in Corollary 1. We apply
Construction 1 to this GDD by using a weight 3 to get a super-simple
$(4,2)-$DGDD of type $15^{a}18^{b}21^{c}24^{d}$. Finally, by Lemma 1 and
Construction 2, we get a super-simple $(3m+1,4,2)$DD. For the remaining values
of $m$, the existence of super-simple $(3m+1,4,2)$DD has been proved in Lemmas
1 and 6-14. The proof is complete. ∎

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