# New Construction of Z-Complementary Code Sets and Mutually Orthogonal
Complementary Sequence Sets

Bingsheng Shen, Yang Yang, Zhengchun Zhou

###### Abstract

Due to the zero nontrivial aperiodic correlation of complete complementary
code (CCCs), it is used in asynchronous multi carrier code division multiple
access (MC-CDMA) communication to provide zero interference performance.
However, there is no CCC for some lengths. In this case, if the system pays
more attention to the correlation, it can use mutually orthogonal
complementary sequence sets (MOCSSs), and if the system pays more attention to
the set size, it can use Z-complementary code sets (ZCCSs). In this paper, we
propose a direct construction of $q$-ary $(q^{v+1},q,q^{m},q^{m-v})$-ZCCS. In
addition, based on the known CCCs, we propose a new construction of MOCSSs and
CCCs respectively.

###### Index Terms:

Complete complementary codes (CCCs), multi carrier code division multiple
access (MC-CDMA), Z-complementary code sets (ZCCSs), mutually orthogonal
complementary sequence sets (MOCSSs).

##  I Introduction

Golay proposed a pair of sequences, which the sum of their aperiodic auto-
correlation function (AACF) is zero everywhere except at the zero-shift
position, known as Golay complementary pair [1]. GCPs have found numerous
applications in modern day communication systems, radar, image processing
[2]–[5]. For the specified alphabet set, the length of GCP is very limited.
For example, the length of known binary GCPs all take the form
$2^{a}10^{b}26^{c}$, where $a,~{}b,~{}c$ are non-negative integers. Therefore,
Tseng and Liu extended the idea of GCP to complementary sequence sets (CSSs),
each set contains multiple sequences, and the sum of their AACF is also zero
for all nonzero time shifts [6]. In addition, they also proposed mutually
orthogonal complementary sequence sets (MOCSSs). A $(M,N,L)$-MOCSS is a family
of $M$ $(N,L)$-sequence sets, where $M$ denotes the set size (i.e., the number
of users), $N$ denotes the flock size (i.e., the number of sub-carriers) and
$L$ denotes the sequence length [6]. In 1988, Suehiro and Hatori given an
upper bound of the set size of the $(M,N,L)$-MOCSS, i.e., $M\leq N$ [7]. When
the equal sign holds, MOCSS is called a complete complementary code (CCC).
Owing to their ideal correlation properties, MOCSSs or CCCs have found
applications in synthetic aperture imaging systems [8], OFDM-CDMA systems [9],
especially the application multi-carrier code division multiple access (MC-
CDMA) systems [10]–[12], because zero multi-path interference (MPI) and zero
multi-user interference (MUI) performance can be achieved. For more research
on CCCs and MOCSSs, please refer to [12]–[17].

A major disadvantage of CCC is that the number of supported users is limited
by the number of row sequences in each complementary matrix. This problem can
be solved by Z-complementary code sets (ZCCS), which there is a zero
correlation zone (ZCZ) in the aperiodic cross- and auto-correlation, and set
size is several times that of CCC [18]. A $(M,N,L,Z)$-ZCCS, the upper bound of
its set size is given by [19], i.e., $M\leq N\lfloor L/Z\rfloor$. When the
equal sign holds, ZCCS is called optimal. If all the received multi-user
signals are roughly synchronized within the ZCZ width, the ZCZ property of
ZCCS can alleviate multiple-access interference. In addition to their
applications in MC-CDMA, ZCCS have also been employed as optimal training
sequences in multiple-input multiple-output (MIMO) communications [20, 21].

In terms of the construction of ZCCS, it is currently divided into the
following categories: The first is the direct construction based on
generalized Boolean functions (GBFs), and the parameters of this kind of
construction are all related to two. Such as, Wu _et. al._ proposed a
construction of $(2^{k+v},2^{k},2^{m},2^{m-v})$-ZCCS based on second order
GBFs, where $v\leq m$, $k\leq m-v$ [22]. Sarkar _et. al._ proposed a
construction of $(2^{k+p+1},2^{k+1},2^{m},2^{m-p})$-ZCCS based on second order
GBFs in 2019, where $k+p\leq m$ [23]. Sarkar explained their construction from
the perspective of graph theory of GBF, so the number of ZCCS constructed is
more than [22]. Later on, Sarkar _et. al._ present a construction of
$(2^{n+p},2^{n},2^{m},2^{m-p})$-ZCCS based on higher order $(\geq 2)$ GBFs in
2021, where $p\leq m$ [24]. In addition, Sarkar _et. al._ also proposed a
construction of ZCCS with non-power-of-two lengths based on GBFs. Wu _et. al._
also put forward a construction of ZCCS with non-power-of-two lengths, which
is not optimal, but its length is very flexible [16]. The second is based on
the construction of the Z-paraunitary matrices (ZPU) and unitary matrices [27,
28], this construction depends on the existence of the ZPU and unitary matrix.
The third is based on GCP and Z-complementary pair (ZCP), which can be
referred to [29],[30].

In this paper, we first propose a construction of $q$-ary
$(q^{v+1},q,q^{m},q^{m-v})$-ZCCS based on extended Boolean function (EBF). EBF
is a generalization of Boolean function, defined as a mapping from
$\mathbb{Z}_{q}^{m}$ to $\mathbb{Z}_{q}$. As far as we know, the parameters of
the known direct constructions based on GBFs are always related to $2$.
However, due to the arbitrariness of $q$, the construction of this paper
breaks through this limitation. In addition, we generalize the result of
literature [17] so that it can construct $(M_{1}M_{2}/2,2L_{1}L_{2})$-CCC
based on $(M_{1},L_{1})$-CCC and $(M_{2},L_{2})$-CCC. Finally, we construct a
$(M,2M,L_{1}+L_{2})$-MOCSS from $(M,L_{1})$-CCC and $(M,L_{2})$-CCC. Although
this construction is simple, it can summarize all current constructions in
terms of sequence length.

The rest of this paper is organized as follows. In Section II, we review some
definitions. We will provide a construction of ZCCS in Section III, and
propose a novel construction of CCC and MOCSS in Section VI. In Section V, we
will compare our results with the existing works, and conclude this paper.

##  II Preliminary

Before we begin, let us define some notations, which will be used throughout
this paper.

  * •

$\xi=e^{\sqrt{-1}\frac{2\pi}{q}}$ denotes the $q$-th root of unity;

  * •

$\mathbb{Z}_{q}=\\{0,1,\cdots,q-1\\}$ is an additive group of order $q$;

  * •

Bold small letter $\mathbf{a}$ denotes a sequence of length $L$, i.e.,
$\mathbf{a}=(a_{0},a_{1},\dots,a_{L-1})$;

  * •

$\mathcal{I}(\mathbf{a},\mathbf{b})=(a_{0},b_{0},a_{1},b_{1},\cdots,a_{L-1},b_{L-1})$
denotes the interleaved sequence of $\mathbf{a}$ and $\mathbf{b}$;

  * •

$\mathbf{a}\oplus d=(a_{0}\oplus d,a_{1}\oplus d,\cdots,a_{L-1}\oplus d)$,
where $\oplus$ denotes the addition over $\mathbb{Z}_{q}$, and
$d\in\mathbb{Z}_{q}$.

  * •

$\mathbf{a}|\mathbf{b}=(a_{0},a_{1},\cdots,a_{L-1},b_{0},\cdots,b_{L-1})$
denotes the concatenation of $\mathbf{a}$ and $\mathbf{b}$;

###  II-A Definitions of Correlations and Sequences

A sequence $\mathbf{a}=(a_{0},a_{1},\cdots,a_{L-1})$ is called $q$-ary
sequence if $a_{i}\in\mathbb{Z}_{q}$ for all $i\in\\{0,1,\cdots,L-1\\}$. A
sequence set (SS) $\mathbf{A}$ contains $N$ sequences of length $L$, denoted
by $(N,L)$-SS, is usually represented by an $N\times L$ matrix, namely
$\mathbf{A}=[\mathbf{a}_{0}^{T},\mathbf{a}_{1}^{T},\cdots,\mathbf{a}_{N-1}^{T}]^{T}$,
where $\mathbf{a}_{n}~{}(0\leq n\leq N-1)$ is the $(n+1)$-th row sequence of
$\mathbf{A}$.

######  Definition 1.

Given two length-$L$ $q$-ary sequences $\mathbf{a}$ and $\mathbf{b}$, their
aperiodic cross-correlation function (ACCF) is defined as

$\displaystyle
R_{\mathbf{a},\mathbf{b}}(\tau)=\begin{cases}\sum\limits_{i=0}^{L-1-\tau}\xi^{a_{i}-b_{i+\tau}},&0\leq\tau\leq
L-1,\\\
\sum\limits_{i=0}^{L-1+\tau}\xi^{a_{i-\tau}-b_{i}},&1-L\leq\tau\leq-1,\\\
0,&\mbox{otherwise.}\end{cases}$ (1)

When $\mathbf{a}=\mathbf{b}$, $R_{\mathbf{a},\mathbf{b}}(\tau)$ is called the
aperiodic auto-correlation function (AACF). For simplicity, the AACF of
$\mathbf{a}$ will be written as $R_{\mathbf{a}}(\tau)$.

According to the definition of aperiodic correlation function, we have

$\displaystyle
R_{\mathbf{b},\mathbf{a}}(-\tau)=R_{\mathbf{a},\mathbf{b}}^{*}(\tau),$ (2)

where $(\cdot)^{*}$ denotes the conjugate of complex.

######  Definition 2.

Let $\mathbf{A}$ and $\mathbf{B}$ be two $(N,L)$-SSs. The ACCF between
$\mathbf{A}$ and $\mathbf{B}$ is defined as

$\displaystyle
R_{\mathbf{A},\mathbf{B}}(\tau)=\sum\limits_{n=1}^{N}R_{\mathbf{a}_{n},\mathbf{b}_{n}}(\tau),~{}0\leq|\tau|\leq
L-1.$ (3)

When $\mathbf{A}=\mathbf{B}$, $R_{\mathbf{A},\mathbf{B}}(\tau)$ is called the
AACF, denoted by $R_{\mathbf{A}}(\tau)$ for short.

######  Definition 3.

Let $\mathbf{A}$ be a $(N,L)$-SS. $\mathbf{A}$ is a called complementary
sequence set (CSS) if $R_{\mathbf{A}}(\tau)=0$ for all $\tau\neq 0$, denoted
by $(N,L)$-CSS.

######  Definition 4.

Let $\mathbf{A}$ be a $(N,L)$-SS. $\mathbf{A}$ is called an even-shift
complementary sequence sets (ESCSS) if $R_{\mathbf{A}}(\tau)=0$ for all even
number $1<|\tau|\leq L-1$, denoted by $(N,L)$-ESCSS.

######  Definition 5.

Let $\mathcal{A}=\\{\mathbf{A}_{0},\mathbf{A}_{1},\cdots,\mathbf{A}_{M-1}\\}$
be a family of $M$ matrices, each of size $N\times L$, i.e.,

$\displaystyle\mathbf{A}_{m}=\begin{bmatrix}\mathbf{a}_{0}^{m}\\\
\mathbf{a}_{1}^{m}\\\ \vdots\\\ \mathbf{a}_{N-1}^{m}\end{bmatrix}_{N\times
L},$ (4)

The family $\mathcal{A}$ is called a Z-complementary code set (ZCCS), denoted
by $(M,N,L,Z)$-ZCCS, if

$\displaystyle
R_{\mathbf{A}_{i},\mathbf{A}_{j}}(\tau)=\begin{cases}NL,&i=j,\tau=0;\\\
0,&i=j,1\leq|\tau|\leq Z-1;\\\ 0,&i\neq j,0\leq|\tau|\leq Z-1.\end{cases}$ (5)

When $Z=L$, the ZCCS is called mutually orthogonal complementary sequence set
(MOCSS), denoted by $(M,N,L)$-MOCSS.

######  Lemma 1 ([7]).

For a $(M,N,L)$-MOCSS, the upper bound of family size satisfies

$\displaystyle M\leq N.$ (6)

When the equal sign holds, MOCSS is called a complete complementary code
(CCC), denoted by $(M,L)$-CCC.

######  Lemma 2 ([19]).

For any $(M,N,L,Z)$-ZCCS, there exists an inequality given as

$\displaystyle M\leq N\left\lfloor\frac{L}{Z}\right\rfloor.$ (7)

When the equal sign holds, the ZCCS is said to be optimal.

###  II-B Extended Boolean function

A extended Boolean function (EBF) $f(\mathbf{x})$ is defined as a mapping
$f:\mathbb{Z}_{q}^{m}\rightarrow\mathbb{Z}_{q}$, where
$\mathbf{x}=(x_{1},x_{2},\cdots,x_{m})$ be the $q$-ary representation of the
integer $x=\sum_{k=1}^{m}x_{k}q^{k-1}$. Given $f(\mathbf{x})$, let
$f_{i}=f(i_{1},i_{2},\cdots,i_{m})$, and define the associated sequence
$\mathbf{f}$ of length $q^{m}$ as

$\displaystyle\mathbf{f}:=(f_{0},f_{1},\cdots,f_{q^{m}-1}).$ (8)

######  Example 1.

Let $m=2$ and $q=3$. The associated sequences for extended Boolean functions
of $x_{1},x_{2},x_{1}x_{2}+1$ are

$\displaystyle\begin{split}\mathbf{x}_{1}&:=(0,1,2,0,1,2,0,1,2),\\\
\mathbf{x}_{2}&:=(0,0,0,1,1,1,2,2,2),\\\
\mathbf{x}_{1}\mathbf{x}_{2}+\mathbf{1}&:=(1,1,1,1,2,0,1,0,2),\end{split}$ (9)

respectively.

##  III Construction of ZCCSs Based on EBFs

In this section, we propose a new construction of ZCCS. Before giving the main
theorem, we first introduce a construction of ternary complementary sequence
set.

######  Lemma 3 ([31]).

Let

$\displaystyle f(\mathbf{x})$
$\displaystyle=\alpha\sum\limits_{k=1}^{m-1}x_{\pi(k)}x_{\pi(k+1)}+\sum_{k=1}^{m}c_{k}x_{k}^{2}+\sum_{k=1}^{m}d_{k}x_{k}+d_{0},$
(10) $\displaystyle g(\mathbf{x})$ $\displaystyle=f(\mathbf{x})+x_{\pi(1)},$
(11) $\displaystyle h(\mathbf{x})$ $\displaystyle=f(\mathbf{x})+2x_{\pi(1)}.$
(12)

where $\alpha\in\mathbb{Z}_{3}^{*}$, $c_{k},d_{k}\in\mathbb{Z}_{3}$, $\pi$ is
a permutation of the set $\\{1,2,\cdots,m\\}$, $m$ is a positive integer. Then
the set $\\{\mathbf{f},\mathbf{g},\mathbf{h}\\}$ form a ternary CSS of length
$3^{m}$.

Generalizing the construction of Lemma 3, we obtain the following Theorem 1.

######  Theorem 1.

Let

$\displaystyle f(\mathbf{x})$
$\displaystyle=\alpha\sum\limits_{k=1}^{m-1}x_{\pi(k)}x_{\pi(k+1)}+\sum_{l=1}^{q-1}\sum_{k=1}^{m}c_{l,k}x_{k}^{l}+c_{0},$
(13)

where $\alpha\in\mathbb{Z}_{q}^{*}$ and $\alpha$ is coprime with $q$,
$c_{0},c_{l,k}\in\mathbb{Z}_{q}$, $\pi$ is a permutation of the set
$\\{1,2,\cdots,m\\}$, $m$ is a positive integer. Then the set
$\mathbf{F}=\\{\mathbf{f}_{n}\\}_{n=0}^{q-1}$ form a $q$-ary CSS of length
$q^{m}$, where

$\displaystyle f_{n}(\mathbf{x})=f(\mathbf{x})+nx_{\pi(1)}.$ (14)

###### Proof.

The case $m=1$ is easily checked by hand, so assume $m\geq 2$ and fix
$\tau\neq 0$. By the definition of AACF, we have

$\displaystyle\begin{split}R_{\mathbf{F}}(\tau)=\sum_{n=0}^{q-1}\sum_{i=0}^{q^{m}-1-\tau}\xi^{f_{n,i}-f_{n,i+\tau}}\end{split}$
(15)

where $f_{n,i}$ is the $i$-th element of sequence $\mathbf{f}_{n}$. For a
given integer $i$, set $j=i+\tau$ and let $(i_{1},i_{2},\cdots,i_{m})$ and
$(j_{1},j_{2},\cdots,j_{m})$ be the $q$-ary representation of $i$ and $j$,
respectively.

_Case 1:_ $i_{\pi(1)}\neq j_{\pi(1)}$, then

$\displaystyle\begin{split}R_{\mathbf{F}}(\tau)&=\sum_{i=0}^{q^{m}-1-\tau}\left[\xi^{f_{i}-f_{j}}\left(\sum_{n=0}^{q-1}\xi^{n(i_{\pi(1)}-j_{\pi(1)})}\right)\right]\\\
&=\sum_{i=0}^{q^{m}-1-\tau}\left(\xi^{f_{i}-f_{j}}\times\frac{1-(\xi^{i_{\pi(1)}-j_{\pi(1)}})^{q}}{1-\xi^{i_{\pi(1)}-j_{\pi(1)}}}\right)\\\
&=0.\end{split}$ (16)

where $f_{i}$ is the $i$-th element of sequence $\mathbf{f}$.

_Case 2:_ $i_{\pi(1)}=j_{\pi(1)}$. Since $i\neq j$, we can define $v$ to be
the smallest integer for $i_{\pi(v)}\neq j_{\pi(v)}$. Let $i^{(n)}$ and
$j^{(n)}$ be integers which $q$-ary representation are different from $i$ and
$j$ in only one position $\pi(v-1)$, i.e., $i_{\pi(v-1)}^{(n)}=n\oplus
i_{\pi(v-1)}$ and $j_{\pi(v-1)}^{(n)}=n\oplus j_{\pi(v-1)}$ for
$n=1,2,\cdots,q-1$. Then

$\displaystyle\begin{split}&~{}~{}~{}~{}\xi^{f_{i}-f_{j}}+\xi^{f_{i^{(1)}}-f_{j^{(1)}}}+\cdots+\xi^{f_{i^{(q-1)}}-f_{j^{(q-1)}}}\\\
&=\xi^{f_{i}-f_{j}}\Big{(}1+\xi^{f_{i^{(1)}}-f_{j^{(1)}}-f_{i}+f_{j}}\\\
&\hskip
80.00012pt+\cdots+\xi^{f_{i^{(q-1)}}-f_{j^{(q-1)}}-f_{i}+f_{j}}\Big{)}\\\
&=\xi^{f_{i}-f_{j}}\Big{(}1+\xi^{\alpha(i_{\pi(v)}-j_{\pi(v)})}\\\ &\hskip
80.00012pt+\cdots+\xi^{(q-1)\alpha(i_{\pi(v)}-j_{\pi(v)})}\Big{)}\\\
&=\xi^{f_{i}-f_{j}}\times\frac{1-\xi^{q\alpha(i_{\pi(v)}-j_{\pi(v)})}}{1-\xi^{\alpha(i_{\pi(v)}-j_{\pi(v)})}}\\\
&=0.\end{split}$ (17)

That is to say
$\xi^{f_{n,i}-f_{n,j}}+\xi^{f_{n,i^{(1)}}-f_{n,j^{(1)}}}+\cdots+\xi^{f_{n,i^{(q-1)}}-f_{n,j^{(q-1)}}}=0$
for all $n=0,1,2,\cdots,q-1$.

Combining both cases, we see that $R_{\mathbf{F}}(\tau)=0$. Therefore,
sequence set $\mathbf{F}$ is a $q$-ary CSS. ∎

In the following, we will give an example to illustrate Theorem 1.

######  Example 2.

Let $q=5$, $m=2$, $\alpha=1$, $\pi(1)=2$, $\pi(2)=1$, $c_{0}=0$,
$(c_{1,1},c_{1,2})=(1,3)$, $(c_{2,1},c_{2,2})=(2,4)$,
$(c_{3,1},c_{3,2})=(1,1)$, $(c_{4,1},c_{4,2})=(0,3)$. Then we can get a CSS
with size $5$ and of length $25$ based on Theorem 1, i.e.,

$\displaystyle\mathbf{f}_{1}$ $\displaystyle=(0,4,3,3,0,1,1,1,2,0,3,4,0,2,1,$
$\displaystyle\hskip 100.00015pt4,1,3,1,1,2,0,3,2,3),$
$\displaystyle\mathbf{f}_{2}$ $\displaystyle=(0,4,3,3,0,2,2,2,3,1,0,1,2,4,3,$
$\displaystyle\hskip 100.00015pt2,4,1,4,4,1,4,2,1,2),$
$\displaystyle\mathbf{f}_{3}$ $\displaystyle=(0,4,3,3,0,3,3,3,4,2,2,3,4,1,0,$
$\displaystyle\hskip 100.00015pt0,2,4,2,2,0,3,1,0,1),$
$\displaystyle\mathbf{f}_{4}$ $\displaystyle=(0,4,3,3,0,4,4,4,0,3,4,0,1,3,2,$
$\displaystyle\hskip 100.00015pt3,0,2,0,0,4,2,0,4,0),$
$\displaystyle\mathbf{f}_{5}$ $\displaystyle=(0,4,3,3,0,0,0,0,1,4,1,2,3,0,4,$
$\displaystyle\hskip 100.00015pt1,3,0,3,3,3,1,4,3,4).$

It is easy to verify that sequence set
$\mathbf{F}=\\{\mathbf{f}_{n}\\}_{n=1}^{5}$ is a $(5,25)$-CSS.

Based on Theorem 1, we obtain the following construction of ZCCS.

######  Theorem 2.

Let $q\geq 2$, $m\geq 2$ and $v\leq m$ be a nonnegative integer, $\pi$ be a
partition of $\\{1,2,\cdots,m-v\\}$. Let

$\displaystyle f(\mathbf{x})$
$\displaystyle=\alpha\sum\limits_{k=1}^{m-v-1}x_{\pi(k)}x_{\pi(k+1)}+\sum_{l=1}^{q-1}\sum_{k=1}^{m}c_{l,k}x_{k}^{l}+c_{0},$
(18)

where $\alpha\in\mathbb{Z}_{q}^{*}$ and $\alpha$ is coprime with $q$,
$c_{0},c_{l,k}\in\mathbb{Z}_{q}$. Then,
$\mathcal{S}=\\{\mathbf{S}^{0},\mathbf{S}^{1},\cdots,\mathbf{S}^{q^{v+1}-1}\\}$
is a $(q^{v+1},q,q^{m},q^{m-v})$-ZCCS, where
$\mathbf{S}^{p}=\\{\mathbf{s}_{0}^{p},\mathbf{s}_{1}^{p},\cdots,\mathbf{s}_{q-1}^{p}\\}$
and

$\displaystyle\begin{split}s_{n}^{p}(\mathbf{x})&=f(\mathbf{x})+nx_{\pi(1)}\\\
&\hskip
10.00002pt+\beta\left({p_{1}x_{\pi(m-v)}}+\sum_{k=1}^{v}p_{k+1}x_{m-v+k}\right)\end{split}$
(19)

for $n=0,1,\cdots,q-1$ and $p=0,1,\cdots,q^{v+1}-1$ with $q$-ary
representation $(p_{1},p_{2},\cdots,p_{v+1})$, where
$\beta\in\mathbb{Z}_{q}^{*}$ and $\beta$ is coprime with $q$.

###### Proof.

For any $\mathbf{S}^{p}$, each sequence $\mathbf{s}_{n}^{p}$
$(n=0,1,\cdots,q-1)$ is divided into $q^{v}$ sub-sequences, i.e.,

$\displaystyle\mathbf{s}_{n}^{p}=\mathbf{s}_{n,0}^{p}|\mathbf{s}_{n,1}^{p}|\cdots|\mathbf{s}_{n,q^{v}-1}^{p}$
(20)

More precisely, starting from the second sub-sequence, each sub-sequence can
be obtained by adding a number to the first sub-sequence, i.e.,

$\displaystyle\mathbf{s}_{n}^{p}=\mathbf{s}_{n,0}^{p}|(\mathbf{s}_{n,0}^{p}\oplus
d_{1})|\cdots|(\mathbf{s}_{n,0}^{p}\oplus d_{q^{v}-1})$ (21)

where $d_{i}\in\mathbb{Z}_{q}$. According to Theorem 1, we know that sequence
set
$\\{\mathbf{s}_{0,0}^{p},\mathbf{s}_{1,0}^{p},\cdots,\mathbf{s}_{q-1,0}^{p}\\}$
forms a CSS. Combine Eq. (21), we can obtain that $\mathbf{S}^{p}$ has zero
auto-correlation zone for $0<\tau<q^{m-v}$.

For any two different sequence sets $\mathbf{S}^{r}$ and $\mathbf{S}^{t}$,
like Eq. (21), we can get

$\displaystyle\mathbf{s}_{n}^{r}=\mathbf{s}_{n,0}^{r}|(\mathbf{s}_{n,0}^{r}\oplus
x_{1})|\cdots|(\mathbf{s}_{n,0}^{r}\oplus x_{q^{v}-1}),$ (22)
$\displaystyle\mathbf{s}_{n}^{t}=\mathbf{s}_{n,0}^{t}|(\mathbf{s}_{n,0}^{t}\oplus
y_{1})|\cdots|(\mathbf{s}_{n,0}^{t}\oplus y_{q^{v}-1}),$ (23)

where $x_{i},~{}y_{i}\in\mathbb{Z}_{q}$. Let $(r_{1},r_{2},\cdots,r_{m})$ and
$(t_{1},t_{2},\cdots,t_{m})$ be $q$-ary representation of $r$ and $t$
respectively.

  * •

Case 1: $r_{1}=t_{1}$, then $\mathbf{s}_{n,0}^{r}=\mathbf{s}_{n,0}^{t}$. For
any $0<\tau<q^{m-v}$, we have

$\displaystyle R_{\mathbf{S}^{r},\mathbf{S}^{t}}(\tau)$
$\displaystyle=\sum_{n=0}^{q-1}R_{\mathbf{s}_{n}^{r},\mathbf{s}_{n}^{t}}(\tau)$
$\displaystyle=q^{v}\sum_{n=0}^{q-1}R_{\mathbf{s}_{n,0}^{r}}(\tau)+\sum_{k=1}^{q^{v}-1}\xi^{x_{k}-y_{k}}\sum_{n=0}^{q-1}R_{\mathbf{s}_{n,0}^{r}}(\tau)$
$\displaystyle=\left(q^{v}+\sum_{k=1}^{q^{v}-1}\xi^{x_{k}-y_{k}}\right)\sum_{n=0}^{q-1}R_{\mathbf{s}_{n,0}^{r}}(\tau)$
$\displaystyle=0.$

  * •

Case 2: $r_{1}\neq t_{1}$, for any $0<\tau<q^{m-v}$, we have

$\displaystyle R_{\mathbf{S}^{r},\mathbf{S}^{t}}(\tau)$
$\displaystyle=\sum_{n=0}^{q-1}R_{\mathbf{s}_{n}^{r},\mathbf{s}_{n}^{t}}(\tau)$
$\displaystyle=\left(q^{v}+\sum_{k=1}^{q^{v}-1}\xi^{x_{k}-y_{k}}\right)\sum_{n=0}^{q-1}R_{\mathbf{s}_{n,0}^{r},\mathbf{s}_{n,0}^{t}}(\tau).$

Similar to the proof of Theorem 1, we can get
$\sum_{n=0}^{q-1}R_{\mathbf{s}_{n,0}^{r},\mathbf{s}_{n,0}^{t}}(\tau)=0$, so
$R_{\mathbf{S}^{r},\mathbf{S}^{t}}(\tau)=0$.

Now, it only suffices to show that $R_{\mathbf{S}^{r},\mathbf{S}^{t}}(0)=0$.
Our calculations yield the following result:

$\displaystyle\hskip 10.00002ptR_{\mathbf{S}^{r},\mathbf{S}^{t}}(0)$
$\displaystyle=\sum_{n=0}^{q-1}\sum_{i=0}^{q^{m}-1}\xi^{s_{n,i}^{r}-s_{n,i}^{t}}$
$\displaystyle=\sum_{n=0}^{q-1}\sum_{i=0}^{q^{m}-1}\xi^{\beta\left[(r_{1}-t_{1})i_{\pi(m-v)}+\sum_{k=1}^{v}(r_{k+1}-t_{k+1})i_{m-v+k}\right]}$
$\displaystyle=0.$

To sum up,
$\mathcal{S}=\\{\mathbf{S}^{0},\mathbf{S}^{1},\cdots,\mathbf{S}^{q^{v+1}-1}\\}$
is a $(q^{v+1},q,q^{m},q^{m-v})$-ZCCS. This completes the proof. ∎

######  Remark 1.

The ZCCSs from Theorem 2 are optimal since
$M=q^{v+1}=q\cdot(q^{m}/q^{m-v})=N\cdot\lfloor L/Z\rfloor$, which achieves the
theoretical bound given in Eq. $(\ref{eq-bound-zccs})$.

######  Remark 2.

The parameters of ZCCS obtained by Theorem 2 can be either odd or even, which
can not be realized by GBF before. In addition, our constructed ZCCSs are
reduced to $(q,q^{m})$-CCCs when $v=0$, the ZCCSs can have set size larger
than the flock size by taking $v>0$.

Next, we give an example to illustrate Theorem 2.

######  Example 3.

Taking $q=3$, $m=3$, $v=1$, $\pi(1)=1,~{}\pi(2)=2$, $\alpha=2$, $\beta=1$,
$c_{0}=0$, $(c_{1,1},c_{1,2},c_{1,3})=(1,2,1)$,
$(c_{2,1},c_{2,2},c_{2,3})=(0,1,2)$. Then, in Table I, we obtain a family of
sequence set
$\mathcal{S}=\\{\mathbf{S}^{0},\mathbf{S}^{1},\cdots,\mathbf{S}^{8}\\}$ by
Theorem 2, it is easy to verify that $\mathcal{S}$ is a ternary
$(9,3,27,9)$-ZCCS.

TABLE I: $(9,3,27,9)$-ZCCS in Example 3 $\mathbf{S}^{0}$
---
$\begin{bmatrix}\mathbf{\mathbf{s}_{0}^{0}}\\\ \mathbf{\mathbf{s}_{1}^{0}}\\\
\mathbf{\mathbf{s}_{2}^{0}}\end{bmatrix}=\begin{bmatrix}0,1,2,0,0,0,2,1,0,0,1,2,0,0,0,2,1,0,1,2,0,1,1,1,0,2,1\\\
0,2,1,0,1,2,2,2,2,0,2,1,0,1,2,2,2,2,1,0,2,1,2,0,0,0,0\\\
0,0,0,0,2,1,2,0,1,0,0,0,0,2,1,2,0,1,1,1,1,1,0,2,0,1,2\end{bmatrix}$
$\mathbf{S}^{1}$
$\begin{bmatrix}\mathbf{\mathbf{s}_{0}^{1}}\\\ \mathbf{\mathbf{s}_{1}^{1}}\\\
\mathbf{\mathbf{s}_{2}^{1}}\end{bmatrix}=\begin{bmatrix}0,1,2,1,1,1,1,0,2,0,1,2,1,1,1,1,0,2,1,2,0,2,2,2,2,1,0\\\
0,2,1,1,2,0,1,1,1,0,2,1,1,2,0,1,1,1,1,0,2,2,0,1,2,2,2\\\
0,0,0,1,0,2,1,2,0,0,0,0,1,0,2,1,2,0,1,1,1,2,1,0,2,0,1\end{bmatrix}$
$\mathbf{S}^{2}$
$\begin{bmatrix}\mathbf{\mathbf{s}_{0}^{2}}\\\ \mathbf{\mathbf{s}_{1}^{2}}\\\
\mathbf{\mathbf{s}_{2}^{2}}\end{bmatrix}=\begin{bmatrix}0,1,2,2,2,2,0,2,1,0,1,2,2,2,2,0,2,1,1,2,0,0,0,0,1,0,2\\\
0,2,1,2,0,1,0,0,0,0,2,1,2,0,1,0,0,0,1,0,2,0,1,2,1,1,1\\\
0,0,0,2,1,0,0,1,2,0,0,0,2,1,0,0,1,2,1,1,1,0,2,1,1,2,0\end{bmatrix}$
$\mathbf{S}^{3}$
$\begin{bmatrix}\mathbf{\mathbf{s}_{0}^{3}}\\\ \mathbf{\mathbf{s}_{1}^{3}}\\\
\mathbf{\mathbf{s}_{2}^{3}}\end{bmatrix}=\begin{bmatrix}0,1,2,0,0,0,2,1,0,1,2,0,1,1,1,0,2,1,0,1,2,0,0,0,2,1,0\\\
0,2,1,0,1,2,2,2,2,1,0,2,1,2,0,0,0,0,0,2,1,0,1,2,2,2,2\\\
0,0,0,0,2,1,2,0,1,1,1,1,1,0,2,0,1,2,0,0,0,0,2,1,2,0,1\end{bmatrix}$
$\mathbf{S}^{4}$
$\begin{bmatrix}\mathbf{\mathbf{s}_{0}^{4}}\\\ \mathbf{\mathbf{s}_{1}^{4}}\\\
\mathbf{\mathbf{s}_{2}^{4}}\end{bmatrix}=\begin{bmatrix}0,1,2,1,1,1,1,0,2,1,2,0,2,2,2,2,1,0,0,1,2,1,1,1,1,0,2\\\
0,2,1,1,2,0,1,1,1,1,0,2,2,0,1,2,2,2,0,2,1,1,2,0,1,1,1\\\
0,0,0,1,0,2,1,2,0,1,1,1,2,1,0,2,0,1,0,0,0,1,0,2,1,2,0\end{bmatrix}$
$\mathbf{S}^{5}$
$\begin{bmatrix}\mathbf{\mathbf{s}_{0}^{5}}\\\ \mathbf{\mathbf{s}_{1}^{5}}\\\
\mathbf{\mathbf{s}_{2}^{5}}\end{bmatrix}=\begin{bmatrix}0,1,2,2,2,2,0,2,1,1,2,0,0,0,0,1,0,2,0,1,2,2,2,2,0,2,1\\\
0,2,1,2,0,1,0,0,0,1,0,2,0,1,2,1,1,1,0,2,1,2,0,1,0,0,0\\\
0,0,0,2,1,0,0,1,2,1,1,1,0,2,1,1,2,0,0,0,0,2,1,0,0,1,2\end{bmatrix}$
$\mathbf{S}^{6}$
$\begin{bmatrix}\mathbf{\mathbf{s}_{0}^{6}}\\\ \mathbf{\mathbf{s}_{1}^{6}}\\\
\mathbf{\mathbf{s}_{2}^{6}}\end{bmatrix}=\begin{bmatrix}0,1,2,0,0,0,2,1,0,2,0,1,2,2,2,1,0,2,2,0,1,2,2,2,1,0,2\\\
0,2,1,0,1,2,2,2,2,2,1,0,2,0,1,1,1,1,2,1,0,2,0,1,1,1,1\\\
0,0,0,0,2,1,2,0,1,2,2,2,2,1,0,1,2,0,2,2,2,2,1,0,1,2,0\end{bmatrix}$
$\mathbf{S}^{7}$
$\begin{bmatrix}\mathbf{\mathbf{s}_{0}^{7}}\\\ \mathbf{\mathbf{s}_{1}^{7}}\\\
\mathbf{\mathbf{s}_{2}^{7}}\end{bmatrix}=\begin{bmatrix}0,1,2,1,1,1,1,0,2,2,0,1,0,0,0,0,2,1,2,0,1,0,0,0,0,2,1\\\
0,2,1,1,2,0,1,1,1,2,1,0,0,1,2,0,0,0,2,1,0,0,1,2,0,0,0\\\
0,0,0,1,0,2,1,2,0,2,2,2,0,2,1,0,1,2,2,2,2,0,2,1,0,1,2\end{bmatrix}$
$\mathbf{S}^{8}$
$\begin{bmatrix}\mathbf{\mathbf{s}_{0}^{8}}\\\ \mathbf{\mathbf{s}_{1}^{8}}\\\
\mathbf{\mathbf{s}_{2}^{8}}\end{bmatrix}=\begin{bmatrix}0,1,2,2,2,2,0,2,1,2,0,1,1,1,1,2,1,0,2,0,1,1,1,1,2,1,0\\\
0,2,1,2,0,1,0,0,0,2,1,0,1,2,0,2,2,2,2,1,0,1,2,0,2,2,2\\\
0,0,0,2,1,0,0,1,2,2,2,2,1,0,2,2,0,1,2,2,2,1,0,2,2,0,1\end{bmatrix}$

##  IV Construction of CCCs and MOCSSs Based on Concatenation

In this section, a new construction of CCCs and MOCSSs will be proposed, which
can make the lengths of CCCs and MOCSSs more extensive. Before giving a new
construction of CCCs, we first give the following lemma.

######  Lemma 4 ([17]).

Let $\mathbf{P}$ and $\mathbf{Q}$ be two mutually orthogonal
$(N,L_{1})$-ESCSSs, i.e., $R_{\mathbf{P},\mathbf{Q}}(\tau)=0$ for all $\tau$,
and let
$\mathcal{C}=\\{\mathbf{C}^{0},\mathbf{C}^{1},\dots,\mathbf{C}^{M-1}\\}$ be a
$(M,L_{2})$-CCC, where $M$ is even, and $\mathbf{C}^{i}$ is given by

$\mathbf{C}^{i}=\begin{bmatrix}\mathbf{c}^{i}_{0}\\\ \mathbf{c}^{i}_{1}\\\
\vdots\\\ \mathbf{c}^{i}_{M-1}\end{bmatrix}.$ (24)

For each $0\leq k\leq M-1$, define an $MN\times L_{1}L_{2}$ matrix
$\mathbf{S}_{k}$

$\mathbf{S}^{k}=\begin{bmatrix}\mathbf{S}^{k,0}\\\ \mathbf{S}^{k,1}\\\
\vdots\\\ \mathbf{S}^{k,N-1}\end{bmatrix},$ (25)

where $\mathbf{S}^{k,t}$ is an $M\times L_{1}L_{2}$ matrix, and the $j$-th row
of $\mathbf{S}^{k,t}$ is given by

$\displaystyle\mathbf{s}_{j}^{k,t}=\begin{cases}\phi(\mathbf{p}_{t},\mathbf{c}_{j}^{2k},\mathbf{c}_{j}^{2k+1}),&\text{if~{}}0\leq
k\leq M/2-1,\\\
\phi(\mathbf{q}_{t},\mathbf{c}_{j}^{2k-M},\mathbf{c}_{j}^{2k-M+1}),&\text{if~{}}M/2\leq
k\leq M-1,\end{cases}$ (26)

where

$\phi(\mathbf{d},\mathbf{a},\mathbf{b})=(\mathbf{a}\oplus
d_{0})|(\mathbf{b}\oplus d_{1})|(\mathbf{a}\oplus d_{2})|(\mathbf{b}\oplus
d_{3})|\cdots$ (27)

Then $\mathcal{S}=\\{\mathbf{S}_{1},\mathbf{S}_{2},\cdots,\mathbf{S}_{M}\\}$
is a $(M,MN,L_{1}L_{2})$-MOCSS.

######  Remark 3.

The condition of Lemma 4 can be weakened, that is, $\mathbf{P}$, $\mathbf{Q}$
are two ESCSSs and $R_{\mathbf{P},\mathbf{Q}}(\tau)=0$ for all even $\tau$.

######  Theorem 3.

Let $\mathcal{S}=\\{\mathbf{S}^{0},\mathbf{S}^{1},\cdots,\mathbf{S}^{M-1}\\}$
be a $(M,L)$-CCC, where $M$ is a nonnegative even number. Each of
$\mathbf{S}^{m}$ is a $(M,L)$-CSS, i.e.,

$\displaystyle\mathbf{S}^{m}=\begin{bmatrix}\mathbf{s}_{0}^{m}\\\
\mathbf{s}_{1}^{m}\\\ \vdots\\\ \mathbf{s}_{M-1}^{m}\end{bmatrix}_{M\times
L},$ (28)

Let

$\displaystyle\mathbf{A}^{m}=\begin{bmatrix}\mathcal{I}(\mathbf{s}_{0}^{m},\mathbf{s}_{1}^{m})\\\
\mathcal{I}(\mathbf{s}_{2}^{m},\mathbf{s}_{3}^{m})\\\ \vdots\\\
\mathcal{I}(\mathbf{s}_{M-2}^{m},\mathbf{s}_{M-1}^{m})\end{bmatrix}_{M/2\times
2L},$ (29)

where $0\leq m\leq M-1$. Then, any two $\mathbf{A}^{m_{1}},\mathbf{A}^{m_{2}}$
($m_{1}\neq m_{2}$), $R_{\mathbf{A}^{m_{1}},\mathbf{A}^{m_{2}}}(\tau)=0$ for
all even $\tau$.

###### Proof.

Let $\tau=2t$ be an even number, for any $m_{1}\neq m_{2}$, we have

$\displaystyle\begin{split}R_{\mathbf{A}^{m_{1}},\mathbf{A}^{m_{2}}}(\tau)&=\sum\limits_{k=0}^{M/2-1}\left[R_{\mathbf{s}_{2k}^{m_{1}},\mathbf{s}_{2k}^{m_{2}}}(t)+R_{\mathbf{s}_{2k+1}^{m_{1}},\mathbf{s}_{2k+1}^{m_{2}}}(t)\right]\\\
&=R_{\mathbf{S}^{m_{1}},\mathbf{S}^{m_{2}}}(t)\\\ &=0,\end{split}$ (30)

where last equal sign is due to that $\mathcal{S}$ is a CCC. ∎

######  Remark 4.

According to the result of [32], each $\mathbf{A}^{m}$ in Theorem 3 is a
$(M/2,2L)$-ESCSS.

Based on Theorem 3, we extend Lemma 4 to construct CCCs.

######  Theorem 4.

Let
$\mathcal{A}=\\{\mathbf{A}^{0},\mathbf{A}^{1},\cdots,\mathbf{A}^{M_{1}-1}\\}$
be a family of set derived from Theorem 3 and $\mathcal{B}$ be a
$(M_{2},L_{2})$-CCC, where $M_{1},~{}M_{2}$ are two even numbers,
$\mathbf{A}^{m_{1}}$ ($0\leq m_{1}\leq M_{1}-1$) is an
$(M_{1}/2,2L_{1})$-ESCSS. Suppose $\mathbf{P}=\mathbf{A}^{2k}$,
$\mathbf{Q}=\mathbf{A}^{2k+1}$ and $\mathcal{C}=\mathcal{B}$, we can get
$M_{1}/2$ $(M_{2},M_{1}M_{2}/2,2L_{1}L_{2})$-MOCSSs
$\\{\mathcal{S}^{0},\mathcal{S}^{1},\cdots,\mathcal{S}^{M_{1}/2-1}\\}$ through
Lemma 4, where $0\leq k\leq M_{1}/2-1$. Then

$\displaystyle\mathcal{S}=\bigcup\limits_{i=0}^{M_{1}/2-1}\mathcal{S}^{i}$
(31)

form a $(M_{1}M_{2}/2,2L_{1}L_{2})$-CCC.

The proof of Theorem 4 is the same as the proof of Theorem 6 of [17]. Due to
the limitation of space, we will not repeat it here.

In Theorem 4, we construct $(M_{1}M_{2}/2,2L_{1}L_{2})$-CCC based on two known
$(M_{1},L_{1})$-CCC and $(M_{2},L_{2})$-CCC. We aim to construct CCC of length
$L_{1}+L_{2}$, but it is very difficult, therefore, we look for construction
that are not as attractive as the original aim. In the following theorem, we
propose a construction of MOCSS with length $L_{1}+L_{2}$, and the ratio $M/N$
is $1/2$.

######  Theorem 5.

Let $\mathcal{A}$ be a $(M,L_{1})$-CCC and $\mathcal{B}$ be a $(M,L_{2})$-CCC.
Let

$\displaystyle\mathbf{C}^{m}=\begin{bmatrix}\mathbf{A}^{m}|\mathbf{B}^{m}\\\
\mathbf{A}^{m}|-\mathbf{B}^{m}\end{bmatrix}_{2M\times(L_{1}+L_{2})}$ (32)

be an matrix of size $2M\times(L_{1}+L_{2})$, where $-\mathbf{B}^{m}$ means
that the additive inverse of $\mathbf{B}^{m}$ over $\mathbb{Z}_{q}$, and

$\displaystyle\mathbf{A}^{m}|\mathbf{B}^{m}=\begin{bmatrix}\mathbf{a}_{0}^{m}|\mathbf{b}_{0}^{m}\\\
\mathbf{a}_{1}^{m}|\mathbf{b}_{1}^{m}\\\ \vdots\\\
\mathbf{a}_{M-1}^{m}|\mathbf{b}_{M-1}^{m}\end{bmatrix}_{M\times(L_{1}+L_{2})}.$
(33)

Then $\mathcal{C}=\\{\mathbf{C}^{0},\mathbf{C}^{1},\cdots,\mathbf{C}^{M-1}\\}$
be a $(M,2M,L_{1}+L_{2})$-MOCSS.

###### Proof.

Referring to [33], it is a simple observation that $\mathbf{C}^{m}$ is a
$(2M,L_{1}+L_{2})$-CSS for all $0\leq m\leq M-1$. After that, we just need to
prove that $R_{\mathbf{C}^{m_{1}},\mathbf{C}^{m_{2}}}(\tau)=0$ for any $\tau$
and $0\leq m_{1}\neq m_{2}\leq M-1$. Let us assume that $L_{1}\leq L_{2}$.

When $0\leq\tau<L_{1}$, we have

$\displaystyle\begin{split}&\hskip
10.00002ptR_{\mathbf{C}^{m_{1}},\mathbf{C}^{m_{2}}}(\tau)\\\
&=\sum\limits_{n=1}^{M-1}\Big{[}R_{\mathbf{a}_{n}^{m_{1}},\mathbf{a}_{n}^{m_{2}}}(\tau)+R_{\mathbf{b}_{n}^{m_{1}},\mathbf{b}_{n}^{m_{2}}}(\tau)\\\
&\hskip
10.00002pt+R_{\mathbf{b}_{n}^{m_{2}},\mathbf{a}_{n}^{m_{1}}}^{*}(L_{1}-\tau)+R_{\mathbf{a}_{n}^{m_{1}},\mathbf{a}_{n}^{m_{2}}}(\tau)\\\
&\hskip
10.00002pt+R_{-\mathbf{b}_{n}^{m_{1}},-\mathbf{b}_{n}^{m_{2}}}(\tau)+R_{-\mathbf{b}_{n}^{m_{2}},\mathbf{a}_{n}^{m_{1}}}^{*}(L_{1}-\tau)\Big{]}\\\
&=2\left[R_{\mathbf{A}^{m_{1}},\mathbf{A}^{m_{2}}}(\tau)+R_{\mathbf{B}^{m_{1}},\mathbf{B}^{m_{2}}}(\tau)\right]\\\
&=0.\end{split}$ (34)

When $L_{1}\leq\tau<L_{2}$, we have

$\displaystyle\begin{split}&\hskip
10.00002ptR_{\mathbf{C}^{m_{1}},\mathbf{C}^{m_{2}}}(\tau)\\\
&=\sum\limits_{n=1}^{M-1}\Big{[}R_{\mathbf{b}_{n}^{m_{1}},\mathbf{b}_{n}^{m_{2}}}(\tau)+R_{\mathbf{b}_{n}^{m_{2}},\mathbf{a}_{n}^{m_{1}}}^{*}(L_{1}-\tau)\\\
&\hskip
10.00002pt+R_{-\mathbf{b}_{n}^{m_{1}},-\mathbf{b}_{n}^{m_{2}}}(\tau)+R_{-\mathbf{b}_{n}^{m_{2}},\mathbf{a}_{n}^{m_{1}}}^{*}(L_{1}-\tau)\Big{]}\\\
&=2R_{\mathbf{B}^{m_{1}},\mathbf{B}^{m_{2}}}(\tau)\\\ &=0.\end{split}$ (35)

When $L_{2}\leq\tau<L_{1}+L_{2}$, we have

$\displaystyle\begin{split}&\hskip
10.00002ptR_{\mathbf{C}^{m_{1}},\mathbf{C}^{m_{2}}}(\tau)\\\
&=\sum\limits_{n=1}^{M-1}\Big{[}R_{\mathbf{b}_{n}^{m_{2}},\mathbf{a}_{n}^{m_{1}}}^{*}(L_{1}-\tau)+R_{-\mathbf{b}_{n}^{m_{2}},\mathbf{a}_{n}^{m_{1}}}^{*}(L_{1}-\tau)\Big{]}\\\
&=0.\end{split}$ (36)

In conclusion, we prove that
$R_{\mathbf{C}^{m_{1}},\mathbf{C}^{m_{2}}}(\tau)=0$ for any $\tau$ and $0\leq
m_{1}\neq m_{2}\leq M-1$. Therefore, $\mathcal{C}$ is a
$(M,2M,L_{1}+L_{2})$-MOCSS. ∎

######  Remark 5.

When the ratio $M/N$ is $1/2$, compared with the constructions in [17] and
[16], Theorem 5 can construct MOCSS with more lengths. For example, Theorem 5
can construct binary $(2,4,11)$-MOCSS, but [17] and [16] cannot.

##  V Comparison and Conclusion

###  V-A Comparison

At present, constructions of ZCCSs are mainly divided into direct construction
and indirect construction. The direct constructions are mainly based on
generalized Boolean functions [22, 23, 25, 26, 24]. There are many indirect
construction methods, such as constructions based on ZPU matrices [27],
constructions based on GCPs, ZCPs and unitary matrices [29, 30] etc. In Table
II, we compare the parameters of our proposed ZCCSs with that of the previous
works. Compared with the previous constructions, our proposed construction has
new parameters and is a direct construction.

TABLE II: Summary of Existing ZCCS Source | Based on | Parameters | Conditions | Optimality | Remark
---|---|---|---|---|---
[22] | GBF | $(2^{k+v},2^{k},2^{m},2^{m-v})$ | $v\leq m,k\leq m-v$ | Optimal | Direct
[23] | GBF | $(2^{k+p+1},2^{k+1},2^{m},2^{m-p})$ | $k+p\leq m$ | Optimal | Direct
[24] | GBF | $(2^{n+p},2^{n},2^{m},2^{m-p})$ | $p\leq m$ | Optimal | Direct
[25] | GBF |  $(2^{n},2^{n},2^{m-1}+2,$ $2^{m-2}+2^{\pi(m-3)}+1)$ | $m\geq 3$ | Optimal | Direct
[26] | GBF | $(2^{k+v},2^{k},2^{m},2^{m-v})$ | $v\leq m,k\leq m-v$ | Optimal | Direct
[26] | GBF | $(2^{\hat{k}},2^{k},2^{m-1}+2^{h},Z)$ |  $v,m_{1}$ are two non-negative integer, $m_{1}\leq m$, $k\leq m-v$, $\hat{k}\leq k,m_{1}\leq h\leq m-v$ | Not Optimal | Direct
[27] |  Butson-type Hadamard Matrices | $(K,M,K,M)$ | $K,M\geq 2$ | Optimal | Indirect
[27] |  Optimal ZPU Matrices | $(K,M,M^{N+1}P,M^{N+1})$ | $K,M\geq 2$ | Optimal | Indirect
[29] | GCP | $(rZ,L,rs,s)$ | $r,s\geq 2,s|Z$ | Not Optimal | Indirect
[30] | ZCP | $(2^{m},2^{m},L,Z)$ | $Z\geq\lceil\frac{L}{2}\rceil$ | Optimal | Indirect
This Paper | EBF | $(q^{v+1},q,q^{m},q^{m-v})$ | $v\leq m$ | Optimal | Direct

For the comparison of MOCSS, we mainly refer to [16] and [17], both of which
propose the construction of MOCSS with $M/N$ ratio of $1/2$. The construction
in [16] is based on the second order GBF, which is a direct construction, but
the length of the generated sequence is relatively limited. Literature [17]
constructs MOCSS with flexible lengths based on known CCCs, even-shift
complementary sequence sets (ESCSSs) and concatenation operator, the rules of
connection are controlled by sequences in ESCSSs. In Table III, we give the
length $(L\leq 40)$ of binary MOCSS with set size $M=2$ of flock size $N=4$
constructed by [16], [17] and this paper, respectively. The new length
obtained by our proposed construction is marked in red.

TABLE III: Binary $(2,4)$-MOCSS for Various Lengths Source | Lengths
---|---
[16] |  $4,6,8,10,12,16,18,20,24,32,34,36,40$
[17] |  $3,4,5,6,7,8,9,10,12,14,16,17,18,$ $20,21,22,24,26,28,32,33,34,36,40$
This Paper |  $3,4,5,6,7,8,9,10,{\color[rgb]{1,0,0}11},12,14,16,17,18,20,$ $21,22,24,26,{\color[rgb]{1,0,0}27},28,32,33,34,36,40$

###  V-B Conclusion

In this paper, based on the work in [31], we first proposed a direct
construction of $q$-ary optimal ZCCS. Compared with the previous construction,
our proposed construction can fill some of the gaps in ZCCS parameters.
Secondly, we generalized the result of [17] to construct CCC. Finally, we
proposed a construction of MOCSS with $M/N$ ratio of $1/2$, which is very
simple, but many new lengths can be constructed.

## References

  * [1] M. J. E. Golay, “Complementary series,” _IRE Trans. Inf. Theory_ , vol. 7, no. 2, pp. 82–87, 1961.
  * [2] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences and Reed-Muller codes,” _IEEE Trans. Inf. Theory_ , vol. 45, no. 7, pp. 2397–2417, 1999.
  * [3] K. G. Paterson, “Generalized Reed-Muller codes and power control in OFDM modulation,” _IEEE Trans. Inf. Theory_ , vol. 46, no. 1, pp. 104–120, 2000.
  * [4] P. Spasojevic and C. N. Georghiades, “Complementary sequences for ISI channel estimation,” _IEEE Trans. Inf. Theory_ , vol. 47, no. 3, pp. 1145–1152, 2001.
  * [5] A. Pezeshki, A. R. Calderbank, W. Moran and S. D. Howard, “Doppler resilient Golay complementary waveforms,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 9, pp. 4254–4266, 2008.
  * [6] C. C. Tseng and C. Liu, “Complementary sets of sequences,” _IEEE Trans. Inf. Theory_ , vol. 18, no. 5, pp. 644–652, 1972.
  * [7] N. Suehiro and M. Hatori, “$N$-shift cross-orthogonal sequences,” _IEEE Trans. Inf. Theory_ , vol. 34, no. 1, pp. 143–146, 1988.
  * [8] I. Trots, “Mutually orthogonal Golay complementary sequences in synthetic aperture imaging systems,” _Arch. Acoust._ , vol. 40, no. 2, pp. 283–289, 2015.
  * [9] Z. Zhang, F. Tian, F. Zeng, L. Ge and G. Xuan, “Mutually orthogonal complementary pairs for OFDM-CDMA systems,” _12th Int. conf. Signal Process._ , HangZhou, pp. 1761–1765, 2014.
  * [10] S. M. Tseng and M. R. Bell, “Asynchronous multicarrier DS-CDMA using mutually orthogonal complementary sets of sequences,” _IEEE Trans. Commun._ , vol. 48, no. 1, pp. 53–59, 2000.
  * [11] J. Aparicio and T. Shimura, “Asynchronous detection and identification of multiple users by multi-carrier modulated complementary set of sequences,” _IEEE Access_ , vol. 6, pp. 22054–22069, 2018.
  * [12] Z. Liu, Y. Guan and U. Parampalli, “New complete complementary codes for peak-to-mean power control in multi-carrier CDMA,” _IEEE Trans. Commun._ , vol. 26, no. 3, pp. 1105–1113, 2014.
  * [13] A. Rathinakumar and A. K. Chaturvedi, “Complete mutually orthogonal Golay complementary sets from Reed Muller codes,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 3, pp. 1339–1346, 2008.
  * [14] S. Das, S. Budišin, S. Majhi, Z. Liu, and Y. L. Guan, “A multiplier-free generator for polyphase complete complementary codes,” _IEEE Trans. Signal Process._ , vol. 66, no. 5, pp. 1184–1196, 2018.
  * [15] S. Das, S. Majhi, and Z. Liu, “A novel class of complete complementary codes and their applications for APU matrices,” _IEEE Signal Process. Lett._ , vol. 25, no. 9, pp. 1300–1304, 2018.
  * [16] W. Wu, C. Chen and Z. Liu, “How to construct mutually orthogonal complementary sets with non-power-of-two lengths?,” _IEEE Trans. Inf. Theory_ , Early Access Article, 2021.
  * [17] B. Shen, Y. Yang, Y. Feng and Z. Zhou, “A generalised construction of mutually orthogonal complementary sequence sets with non-power-of-two lengths,” _IEEE Trans. Commun._ , Early Access Article, 2021.
  * [18] Z. Liu, Y. Guan, B. C. Ng and H. Chen, “Correlation and set size bounds of complementary sequences with low correlation zone,” _IEEE Trans. Commun._ , vol. 59, no. 12, pp. 3285–3289, 2011.
  * [19] L. Feng, P. Fan and X. Zhou, “Lower bounds on correlation of Zcomplementary code sets,” _Wireless Pers. Commun._ , vol. 72, no. 2, pp. 1475–1488, 2013.
  * [20] H. Wang, X. Gao, B. Jiang, X. You and W. Hong, “Efficient MIMO channel estimation using complementary sequences,” _IET Commun._ , vol. 1, no. 5, pp. 962–969, 2007.
  * [21] W. Yuan, Y. Tu and P. Fan, “Optimal training sequences for cyclicprefix- based single-carrier multi-antenna systems with space-time blockcoding,” _IEEE Trans. Wireless Commun._ , vol. 7, no. 11, pp. 4074–4050, 2008.
  * [22] S. Wu and C. Chen, “Optimal Z-complementary sequence sets with good peak-to-average power-ratio property,” _IEEE Signal Process. Lett._ , vol. 25, no. 10, pp. 1500–1504, 2018.
  * [23] P. Sarkar, S. Majhi and Z. Liu, “Optimal Z-complementary code set from generalized Reed-Muller codes,” _IEEE Trans. Commun._ , vol. 67, no. 3, pp. 1783–1796, 2019.
  * [24] P. Sarkar and S. Majhi, “A direct construction of optimal ZCCS with maximum column sequence PMEPR two for MC-CDMA system,” _IEEE Commun. Lett._ , Early Access Article, 2021.
  * [25] P. Sarkar, A. Roy and S. Majhi, “Construction of Z-complementary code sets with non-power-of-two lengths based on generalized Boolean functions,” _IEEE Commun. Lett._ , vol. 24, no. 8, pp. 1607–1611, 2020.
  * [26] S. Wu, A. Sahin, Z. Huang and C. Chen, “Z-complementary code sets with flexible lengths from generalized Boolean functions,” _IEEE Access_ , vol. 9, pp. 4642–4652, 2021.
  * [27] S. Das, U. Parampalli, S. Majhi, Z. Liu and S. Budišin, “New optimal Z-complementary code sets based on generalized paraunitary matrices,” _IEEE Trans. Commun._ , vol. 68, pp. 5546–5558, 2020.
  * [28] X. Chen G. Li and H. Li, “Two constructions of zero correlation zone aperiodic complementary sequence sets,” _IET Commun._ , vol. 14, no. 4, pp. 556–560, 2020.
  * [29] Y. Li and C. Xu, “ZCZ aperiodic complementary sequence sets with low column sequence PMEPR,” _IEEE Commun. Lett._ , vol. 19, no. 8, pp. 1303–1306, 2015.
  * [30] A. R. Adhikary and S. Majhi, “New construction of optimal aperiodic Z-complementary sequence sets of odd-lengths,” _Electronics Lett._ , vol. 55, no. 19, pp. 1043–1045, 2019.
  * [31] Z. Wang, D. Ma, G. Gong and E. Xue, “New construction of complementary sequence (or array) sets and complete complementary codes,” _arXiv_ , vol. 2001.04898v1, 2020.
  * [32] B. Shen, Y. Yang and Z. Zhou, “A construction of binary golay complementary sets based on even-shift complementary pairs,” _IEEE Access_ , vol. 8, pp. 29882–29890, 2020.
  * [33] G. Wang, A. R. Adhikary Z. Zhou and Y. Yang, “Generalized constructions of complementary sets of sequences of lengths non-power-of-two,” _IEEE Signal Proces. Lett._ , vol. 27, pp. 136–140, 2020.