# A New Construction of Optimal Symmetrical ZCCS

Rajen Kumar  Department of Mathematics
Indian Institute of Technology
Patna, India
<EMAIL_ADDRESS>Prashant Kumar Srivastava  Department of
Mathematics
Indian Institute of Technology
Patna, India
<EMAIL_ADDRESS>Sudhan Majhi The work of Rajen Kumar is supported in part by
the CSIR, India, under award letter $09/1023(0034)/2019$-EMR-I.The work of
Sudhan Majhi is supported by the SERB, Govt. of India, under grant no.
CRG$/2022/000529$ and EEQ$/2022/001018$. Department of ECE
Indian Institute of Science
Bangalore, India
<EMAIL_ADDRESS>
###### Abstract

We propose new constructions for a two-dimensional ($2$D) perfect array,
complete complementary code (CCC), and multiple CCCs as an optimal symmetrical
$Z$-complementary code set (ZCCS). We propose a method to generate a two-
dimensional perfect array and CCC. By utilising mutually orthogonal sequences,
we developed a method to extend the length of a CCC without affecting the set
or code size. Additionally, this concept is extended to include the
development of multiple CCCs, and the correlation characteristics of these
multiple CCCs are identical with the characteristics of optimal symmetrical
ZCCS.

##  I Introduction

The widespread investigation of high-dimensional signals with impulse-like
high-dimensional periodic auto-correlation has been driven by the application
of synchronisation[1, 2], phased-array antennas [3], communication[4],
radar[5], sonar, and other fields. Numerous methods have been presented to
generate such signals. There are two significant groups of such signals. The
first one is known as a high-dimensional perfect array, while the second one
is referred to as a high-dimensional complementary array set. The high-
dimensional periodic auto-correlation function (PACF) is derived by summing
the auto-correlation functions of all sub-arrays inside it. Considering the
ease of implementing signals, two-dimensional ($2$D) signals are favoured.

Multi-carrier code division multiple access (MC-CDMA) systems that employ
Complete Complementary Codes (CCC) have various advantages compared to
traditional CDMA systems. These advantages include offset stacked spreading,
being able to provide results in lower power spectral density, multi-access
interference-free transmission, more resistance to jamming and offering more
freedom in positioning code elements within a channel [6]. In MC-CDMA, each
code is allocated to a unique user, resulting in a need for a substantial
quantity of codes. The rise in the number of codes in CCC results in an
expansion of the code size. CCC with various parameters can be found in [7, 8,
9]. The set size of the $Z$-complementary code set (ZCCS) is considerably
larger than that of CCC, which promotes the ability of a ZCCS based MC-CDMA
system to support a larger number of users, especially in a quasi-synchronous
environment [10].

The idea of a type-I ZCCS is proposed to overcome this drawback by reducing
the dependence on ideal correlation from all delays to a zero correlation zone
(ZCZ) in the front end, which results in the set size of type-I ZCCS being
several times higher than that of CCCs [11]. Recently, Kumar et al. proposed a
construction for type-II ZCCS with ZCZ at the tail end, providing more codes
than the type-I ZCCS with the same flock size [12]. Another type, symmetrical
ZCCS, was introduced by Zhau et al. with ZCZ at both the front and tail end
[13]. Multiple CCCs (MCCC) with favourable inter-set cross-correlation
attributes have received recent attention from researchers for their ability
to resist inter-cell interference caused by multiple cells in multi-cell MC-
CDMA systems [14]. The primary objective of this work is to explore
construction techniques that incorporate flexible factors. We have made
devoted attempts to fulfil this objective.

The remaining portion of the work is structured as follows. Section II
presents preliminary information on the notations and definitions. In Section
III, we presented proposed constructions combined with a complete explanation
and an illustrative example. Section IV provides a comparison to previous
research. Finally, in Section V, a conclusion has been made.

##  II Preliminaries

In this section, we provide a few important definitions and symbols that will
be utilized throughout the paper.

###### Definition 1

Let

$C_{k}=\begin{bmatrix}c_{1,1}^{k}&c_{1,2}^{k}&\cdots&c_{1,N}^{k}\\\
c_{2,1}^{k}&c_{2,2}^{k}&\cdots&c_{2,N}^{k}\\\ \vdots&\vdots&\vdots&\vdots\\\
c_{M,1}^{k}&c_{M,2}^{k}&\cdots&c_{M,N}^{k}\end{bmatrix}$ (1)

be a code of size $M\times N$. Then $2$D PACF
$\mathcal{P}(C_{k})(\tau_{1},\tau_{2})$ is defined as follows

$\mathcal{P}(C_{k})(\tau_{1},\tau_{2})=\sum\limits_{i_{1}=1}^{M}\sum\limits_{i_{2}=1}^{N}c_{i_{1},i_{2}}^{k}(c_{i_{1}+\tau_{1},i_{2}+\tau_{2}}^{k})^{*},$
(2)

where the symbol “$*$” denotes the complex conjugate and shifts
$|\tau_{1}|<M,|\tau_{2}|<N$ subscript addition $i_{1}+\tau_{1}$ and
$i_{2}+\tau_{2}$ are calculated modulo $M$ and $N$, respectively.

###### Definition 2

Let $C$ be a code defined in (1) and referred to as a perfect array if its
$2$D PACF has

$\mathcal{P}(C)(\tau_{1},\tau_{2})=\left\\{\begin{array}[]{cc}NM&(\tau_{1},\tau_{2})=(0,0)\\\
0&(\tau_{1},\tau_{2})\neq(0,0)\end{array}\right..$ (3)

###### Definition 3

Let $C_{s}$ and $C_{t}$ be any two codes of order $M\times N$ as defined in
(1), then ACCS between $C_{s}$ and $C_{t}$ is defined by

$\mathcal{C}\left({C}_{s},{C}_{t}\right)(\tau)=\left\\{\begin{array}[]{ll}\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N-\tau}c_{i,j}^{s}\left(c^{t}_{i,j+\tau}\right)^{*},&\tau\geq
0\\\
\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N-\tau}c_{i,j-\tau}^{s}\left(c^{t}_{i,j}\right)^{*},&\tau<0.\end{array}\right.$
(4)

When ${C}_{s}={C}_{t}$, $\mathcal{C}(C_{s},C_{t})(\tau)$ is called AACS of
${C}_{s}$ and is denoted as $\mathcal{C}(C_{s})(\tau)$.

###### Definition 4

Let $\mathbf{C}=\\{C_{1},C_{2},\ldots,C_{K}\\}$ be a set of $K$ codes, each
having order $M\times N$ and it follows

  * C1:

$\mathcal{C}(C_{k})(\tau)=0$, for
$|\tau|\in\mathcal{T}_{1}\bigcup\mathcal{T}_{2}$ and $1\leq k\leq K$,

  * C2:

$\mathcal{C}(C_{k_{1}},C_{k_{2}})(\tau)=0$, for
$|\tau|\in\mathcal{T}_{1}\bigcup\mathcal{T}_{2}\bigcup\\{0\\}$, $k_{1}\neq
k_{2}$ and $1\leq k_{1},k_{2}\leq K$,

where $\mathcal{T}_{1}=\\{1,2,\ldots,Z\\}$ and
$\mathcal{T}_{2}=\\{N-Z,N-Z+1,\ldots,N-1\\}$ then set $\mathbf{C}$ is called a
symmetrical $(K,M,N,Z)$-ZCCS. And when $K=\lfloor\frac{MN}{Z+1}\rfloor$,
$\mathbf{C}$ is called an optimal symmetrical $(K,M,N,Z)$-ZCCS [13].

###### Definition 5

Let $\mathbf{C}=\\{C_{1},C_{2},\ldots,C_{M}\\}$ be a set of $M$ codes of order
$M\times N$ and it follows

$\mathcal{C}(C_{k_{1}},C_{k_{2}})(\tau)=\left\\{\begin{array}[]{cc}MN&k_{1}=k_{2},\tau=0\\\
0&\text{otherwise.}\end{array}\right.$ (5)

Then $\mathbf{C}$ is called a $(M,N)$-CCC.

###### Definition 6

Let $\mathbf{C}_{1},\mathbf{C}_{2},\ldots,\mathbf{C}_{K}$ be collection of
$(M,N)$-CCCs is referred as $(K,M,N,Z)$-MCCC if it follows

$\mathcal{C}(C_{k_{1}},C_{k_{2}})(\tau)=0,$ (6)

where $C_{k_{1}}\in\mathbf{C}_{i}$, $C_{k_{1}}\in\mathbf{C}_{j}$ for $i\neq j$
and $|\tau|<Z$.

Below, we present a lemma that is necessary in order to complete the proof.

######  Lemma 1 ([15])

For $M(\geq 2),s\in\mathbb{N}$, such that $M\nmid s$. Let
$\zeta=\exp{\frac{2\pi\sqrt{-1}}{M}}$, then

$\sum_{i=1}^{M}\zeta^{s\cdot i}=0.$ (7)

##  III Proposed Construction

###  III-A Construction using multiplication matrix

The term "multiplication matrix" refers to a matrix in which the elements are
determined by the product of subscripts.

###### Construction 1

Let $s$ be any positive integer. Now, for a $2\leq M\in\mathbb{N}$, we define
a matrix of size $M\times M$ as $\mathcal{M}(M,s,x)=\mathcal{M}$ where each
entries of matrix is defined as

$[\mathcal{M}]_{i,j}=\zeta^{x+s(i-1)(j-1)},$ (8)

for $i,j\in\\{1,2,\ldots,M\\}$, $x\in\mathbb{Z}$ and
$\zeta=\frac{2\pi\sqrt{-1}}{M}$.

###### Theorem 1

$\mathcal{M}(M,s,x)$ is a perfect array for $M,s,x\in\mathbb{Z}$, $M\geq 2$
and $M\nmid s$ ($M$ does not devides $s$).

###### Proof:

Let $M,s,x\in\mathbb{Z}$, $M\geq 2$ and $\gcd(M,s)=1$. Then

$\mathcal{P}(\mathcal{M})(\tau_{1},\tau_{2})=\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{M}\mathcal{M}_{i,j}(\mathcal{M}_{i+\tau_{1},j+\tau_{2}})^{*},$
(9)

where $\mathcal{M}_{i,j}=\zeta^{x+s(i-1)(j-1))}$ and subscript addition
$i+\tau_{1}$ and $j+\tau_{2}$ are calculated over modulo $M$. When
$(\tau_{1},\tau_{2})=(0,0)$ implies
$\mathcal{M}_{i,j}(\mathcal{M}_{i+\tau_{1},j+\tau_{2}})^{*}=\mathcal{M}_{i,j}(\mathcal{M}_{i,j})*=1$
and $\mathcal{P}(\mathcal{M})(0,0)=M^{2}$. Let $(\tau_{1},\tau_{2})\neq(0,0)$

$\displaystyle\mathcal{M}_{i,j}(\mathcal{M}_{i+\tau_{1},j+\tau_{2}})^{*}=$
$\displaystyle\zeta^{x+s(i-1)(j-1))-x-s(i+\tau_{1}-1)(j+\tau_{2}-1))}$ (10)
$\displaystyle=\zeta^{s(-i\tau_{2}-j\tau_{1}+\tau_{1}+\tau_{2}-\tau_{1}\tau_{2})}.$

Let $\Omega^{1}(\tau_{1},\tau_{2})=\tau_{1}+\tau_{2}-\tau_{1}\tau_{2}$. Now,
we have

$\begin{aligned}
\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{M}\mathcal{M}_{i,j}(\mathcal{M}_{i+\tau_{1},j+\tau_{2}})^{*}=&\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{M}\zeta^{-s(i\tau_{2}+j\tau_{1}-\Omega^{1}(\tau_{1},\tau_{2}))}\\\
=&\zeta^{s\Omega^{1}(\tau_{1},\tau_{2})}\sum\limits_{i=1}^{M}\zeta^{-si\tau_{2}}\sum\limits_{j=1}^{M}\zeta^{-sj\tau_{1}}\end{aligned}.$
(11)

From (11) and (9), we have

$\displaystyle\mathcal{P}(\mathcal{M})(\tau_{1},\tau_{2})$
$\displaystyle=\zeta^{s\Omega^{1}(\tau_{1},\tau_{2})}\left(\sum\limits_{i=1}^{M}\zeta^{-si\tau_{2}}\right)\left(\sum\limits_{j=1}^{M}\zeta^{-sj\tau_{1}}\right)$
(12) $\displaystyle=0,\text{from {Lemma} \ref{Lem:sum_zero}.}$

This complete the proof. ∎

For $u\in\\{0,1,\ldots,M-1\\}$, $\mathcal{M}^{u}$ denotes the matrix obtained
by repeating the cyclic shift with respect to the rows for $u$ times.

###### Theorem 2

Let $M\geq 2$ be an arbitrary integer, $M\nmid s$ and $x\in\mathbb{Z}$. Then
$\mathbf{C}=\\{\mathcal{M}^{u}(M,s,x):u\in{0,1,\ldots M-1}\\}$ is a
$(M,M)$-CCC.

###### Proof:

For any $u\in\\{0,1,\ldots,M-1\\}$ each entries of $\mathcal{M}^{u}$ is
defined as

$[\mathcal{M}^{u}]_{i,j}=[\mathcal{M}]_{i+u,j},$ (13)

where subscript addition $i+u$ is calculated modulo $M$ and $0$ is equivalent
to $M$.

Let $u_{1},u_{2},\tau\in\\{0,1,\ldots,M-1\\}$, then

$\displaystyle{}_{i,j}[\mathcal{M}^{u_{2}}]^{*}_{i,j+\tau}=$
$\displaystyle[\mathcal{M}]_{i+u_{1},j}[\mathcal{M}]^{*}_{i+u_{2},j+\tau}$
(14) $\displaystyle=$
$\displaystyle\zeta^{s(i+u_{1}-1)(j-1)-s(i+u_{2}-1)(j+\tau-1)}$
$\displaystyle=$ $\displaystyle\zeta^{s(j(u_{1}-u_{2})-i\tau+\tau-\tau
u_{2}+u_{2})}$ $\displaystyle=$
$\displaystyle\zeta^{sj(u_{1}-u_{2})}\zeta^{-si\tau}\zeta^{\Omega^{2}(\tau,u_{1},u_{2})},$

where $\Omega^{2}(\tau,u_{1},u_{2})=s(\tau-\tau u_{2}+u_{2})$.

$\displaystyle\mathcal{C}(\mathcal{M}^{u_{1}},\mathcal{M}^{u_{2}})(\tau)=$
$\displaystyle\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N-\tau}[\mathcal{M}^{u_{1}}]_{i,j}[\mathcal{M}^{u_{2}}]^{*}_{i,j+\tau}$
(15) $\displaystyle=$
$\displaystyle\zeta^{\Omega^{2}}\sum\limits_{i=1}^{M}\zeta^{-si\tau}\sum\limits_{j=1}^{N-\tau}\zeta^{sj(u_{1}-u_{2})}$

Let $\tau=0$ and $u_{1}\neq u_{2}$, from (15), we have

$\displaystyle\mathcal{C}(\mathcal{M}^{u_{1}},\mathcal{M}^{u_{2}})(0)=$
$\displaystyle\zeta^{\Omega^{2}}M\sum\limits_{j=1}^{M}\zeta^{sj(u_{1}-u_{2})}$
(16) $\displaystyle=$ $\displaystyle 0,\text{from {Lemma}
\ref{Lem:sum_zero}.}$

Now, consider $\tau>0$, from (15), we have

$\displaystyle\mathcal{C}(\mathcal{M}^{u_{1}},\mathcal{M}^{u_{2}})(\tau)=$
$\displaystyle\zeta^{\Omega^{2}}\sum\limits_{j=1}^{N-\tau}\zeta^{sj(u_{1}-u_{2})}\sum\limits_{i=1}^{M}\zeta^{-si\tau}$
(17) $\displaystyle=$ $\displaystyle 0,\text{from {Lemma}
\ref{Lem:sum_zero}.}$

Similarly, for $\tau<0$,
$\mathcal{C}(\mathcal{M}^{u_{1}},\mathcal{M}^{u_{2}})(\tau)=0$. This completes
the proof. ∎

###  III-B Construction using mutually orthogonal sequences

###### Construction 2

Let $C_{1},C_{2},\ldots,C_{P}$ be a set of $M\times N$ matrices and
$\mathbf{b}=(b_{1},b_{2},\ldots,b_{P})$ be sequence of length $P$, then we
define

$\mathcal{R}(C_{1},C_{2},\ldots,C_{P};\mathbf{b})=[A_{1}||A_{2}||\cdots||A_{P}],$
(18)

where, $A_{i}=b_{i}\otimes C_{i}$ for $1\leq i\leq P$, $\otimes$ represents
Kronecker product and $||$ represents concatenation of two matrices.

Let
$C_{i_{1}},C_{i_{2}},\ldots,C_{i_{P}},C_{k_{1}},C_{k_{2}},\ldots,C_{k_{P}}$ be
a set of $M\times N$ matrices,
$\mathbf{b}^{j_{1}}=(b^{j_{1}}_{1},b^{j_{1}}_{2},\ldots,b^{j_{1}}_{P})$ and
$\mathbf{b}^{j_{2}}=(b^{j_{2}}_{1},b^{j_{2}}_{2},\ldots,b^{j_{2}}_{P})$ be
sequence of length $P$. Now, we construct two code as
$B_{1}=\mathcal{R}(C_{i_{1}},C_{i_{2}},\ldots,C_{i_{P}};\mathbf{b}^{j_{1}})$
and
$B_{2}=\mathcal{R}(C_{k_{1}},C_{k_{2}},\ldots,C_{k_{P}};\mathbf{b}^{j_{2}})$,
then ACCS of $B_{1}$ and $B_{2}$ is obtained by

$\displaystyle\mathcal{C}(B_{1},B_{2})(uN+v)=\sum_{\alpha=1}^{P-u}b^{j_{1}}_{\alpha}b^{{j_{2}}^{*}}_{\alpha+u}\mathcal{C}(C_{i_{\alpha}},C_{k_{\alpha+u}})(v)$
(19)
$\displaystyle+\sum_{\alpha=1}^{P-u-1}b^{j_{1}}_{\alpha}b^{{j_{2}}^{*}}_{\alpha+u+1}\left(\mathcal{C}(C_{i_{\alpha}},C_{k_{\alpha+u+1}})(v-N)\right),$

where $0\leq v<N$ and $-P\leq u<P$.

###### Theorem 3

Let $\mathbf{C}=\\{C_{1},C_{2},\ldots,C_{M}\\}$ be a $(M,N)$-CCC. Let $P$ and
$S$ be two positive integers such that $PS=M$, for $1<P\leq M$. Let
$\mathbf{b}^{1},\mathbf{b}^{2},\ldots,\mathbf{b}^{P}$ be mutually orthogonal
sequences of length $P$. Define

$B_{iP+j}=\mathcal{R}(C_{\pi(iP+1)},C_{\pi(iP+2)},\ldots,C_{\pi(iP+P)};\mathbf{b}^{j}),$
(20)

where $0\leq i<S$, $1\leq j\leq P$ and $\pi$ is permutation of
$\\{1,2,\ldots,M\\}$. Then the obtained code set
$\mathbf{B}=\\{B_{1},B_{2},\ldots,B_{M}\\}$ is a $(M,NP)$-CCC.

###### Proof:

First, we prove that each code $B_{k}$ is a Golay complementary set (GCS), for
$1\leq k\leq M$. Let $k=iP+j$, then AACF of $B_{k}$ is obtained by

$\displaystyle\mathcal{C}(B_{k})(uN+v)=\sum_{\alpha=1}^{P-u}b^{j}_{\alpha}b^{j^{*}}_{\alpha+u}\mathcal{C}(C_{\pi(iP+\alpha)},C_{\pi(iP+\alpha+u)})(v)$
(21)
$\displaystyle+\sum_{\alpha=1}^{P-u-1}b^{j}_{\alpha}b^{j^{*}}_{\alpha+u+1}\mathcal{C}(C_{\pi(iP+\alpha)},C_{\pi(iP+\alpha+u+1)})(v-N).$

Now, we discuss AACF for non-zero time shifts, i.e., excluding $u=v=0$.

Let $u=0$, implies $v\neq 0$ and $N-v\neq 0$. Putting $u=0$ in (21), we have

$\mathcal{C}(B_{k})(0+v)=\sum_{\alpha=1}^{P}b^{j}_{\alpha}b^{j^{*}}_{\alpha}\mathcal{C}(C_{\pi(iP+\alpha)})(v).$
(22)

Each $C_{\pi(iP+\alpha)}$, for $\alpha\in\\{1,2,\ldots,P\\}$ is a GCS, hence
$\mathcal{C}(C_{\pi(iP+\alpha)})(v)=0$, for $v\neq 0$, which makes (22) equals
to zero.

Now, consider $u\neq 0$. As $C_{\pi(iP+\alpha)}$ is complementary mate with
$C_{\pi(iP+\alpha+u)}$ (& $C_{\pi(iP+\alpha+u+1)}$) for $1\leq\alpha\leq
P-u\;(\&P-u-1)$, it implies
$\mathcal{C}(C_{\pi(iP+\alpha)},C_{\pi(iP+\alpha+u)})(v)$ and
$\mathcal{C}(C_{i,\pi(\alpha)},C_{i,\pi(\alpha+u+1-P)})(v-N)$ are zero in
(21). Therefore, each code $B_{k}$ is a GCS.

Let $1\leq k_{1}\neq k_{2}\leq M$, then $k_{1}=i_{1}P+j_{1}$ and
$k_{2}=i_{2}P+j_{2}$ such that $(i_{1},j_{1})\neq(i_{2},j_{2})$. Then ACCS
between $B_{k_{1}}$ and $B_{k_{2}}$ is as given in (23).

$\displaystyle\mathcal{C}(B_{k_{1}},B_{k_{2}})(uN+v)=$
$\displaystyle\sum_{\alpha=1}^{P-u}b^{j_{1}}_{\alpha}b^{{j_{2}}^{*}}_{\alpha+u}\mathcal{C}(C_{\pi(i_{1}P+\alpha)},C_{\pi(i_{2}P+\alpha+u)})(v)$
(23)
$\displaystyle+\sum_{\alpha=1}^{P-u-1}b^{j_{1}}_{\alpha}b^{{j_{2}}^{*}}_{\alpha+u+1}\left(\mathcal{C}(C_{\pi(i_{1}P+\alpha)},C_{\pi(i_{2}P+\alpha+u+1)})(v-N)\right),$

To complete the proof, we divided this case into two cases.

Case 1. When $i_{1}=i_{2}$, i.e., $j_{1}\neq j_{2}$.

Subcase (i): For $u=v=0$,

$\displaystyle\rho(B_{k_{1}},B_{k_{2}})(0N+0)=$
$\displaystyle\sum_{\alpha=1}^{P}b^{j_{1}}_{\alpha}b^{{j_{2}}^{*}}_{\alpha}\rho(C_{\pi(i_{1}P+\alpha)})(0)$
(24) $\displaystyle=$ $\displaystyle
MN\sum_{\alpha=1}^{P}b^{j_{1}}_{\alpha}b^{{j_{2}}^{*}}_{\alpha}=0.$

Subcase (ii): For $v=0$ but $u\neq 0$, $C_{\pi(i_{1}P+\alpha)}$ and
$C_{\pi(i_{2}+\alpha+u)}$ are complementary mate for $1\leq\alpha\leq P$,
which makes (23) equals to zero.

Subcase (iii): $v\neq 0$. In this case $C_{\pi(i_{1}P+\alpha)}$ is either
complementary mate of $C_{\pi(i_{2}P+\alpha+u)}$ or the same, for
$1\leq\alpha\leq P$. In both cases
$\rho(C_{\pi(i_{1}P+\alpha)},C_{\pi(i_{2}P+\alpha+u)})(v)=0$ for $v\neq 0$,
which makes (23) equals to zero.

Case 2. When $i_{1}\neq i_{2}$. In this case, $C_{\pi(i_{1}P+\alpha)}$ and
$C_{\pi(i_{2}P+\alpha+u)}$ are complementary mate, for $1\leq\alpha\leq P-u$,
similarly, $C_{\pi(i_{1}P+\alpha)}$ and $C_{\pi(i_{2}P+\alpha+u+1)}$ are
complementary mate, for $1\leq\alpha<P-u-1$, which implies
$\rho(C_{\pi(i_{1}P+\alpha)},C_{\pi(i_{2}P+\alpha+u)})(v)=0$ and
$\rho(C_{\pi(i_{1}P+\alpha)},C_{\pi(i_{2}P+\alpha+u+1)})(N-v)=0$, which makes
(23) equals to zero.

Combining Case $1$ and Case $2$, it is proved that
$\rho(B_{k_{1}},B_{k_{2}})(\tau)=0$, for any value of $\tau$ and $k_{1}\neq
k_{2}$. Therefore, $\mathbf{B}$ is a $(M,PN)$-CCC. ∎

$\displaystyle\mathcal{C}(B_{k_{1}M+i_{1}P+j_{1}},B_{k_{2}M+i_{2}P+j_{2}})(uN+v)=$
$\displaystyle\sum_{\alpha=1}^{P-u}b^{j_{1}}_{\alpha}b^{{j_{2}}^{*}}_{\alpha+u}\mathcal{C}(C_{\pi_{k_{1}}(i_{1}P+\alpha)},C_{\pi_{k_{2}}(i_{2}P+\alpha+u)})(v)$
(25)
$\displaystyle+\sum_{\alpha=1}^{P-u-1}b^{j_{1}}_{\alpha}b^{{j_{2}}^{*}}_{\alpha+u+1}\left(\mathcal{C}(C_{\pi_{k_{1}}(i_{1}P+\alpha)},C_{\pi_{k_{2}}(i_{2}P+\alpha+u+1)})(v-N)\right).$

###### Theorem 4

Let $\mathbf{C}=\\{C_{1},C_{2},\ldots,C_{M}\\}$ be a $(M,N)$-CCC. Let $P$ and
$S$ be two positive integers such that $PS=M$, for $1<P\leq M$. Let
$\mathbf{b}^{1},\mathbf{b}^{2},\ldots,\mathbf{b}^{P}$ be mutually orthogonal
sequences of length $P$. Define

$B_{kM+iP+j}=\mathcal{R}(C_{\pi_{k}(iP+1)},C_{\pi_{k}(iP+2)},\ldots,C_{\pi_{k}(iP+P)};\mathbf{b}^{j}),$
(26)

where $0\leq i<S$, $1\leq j\leq P$, $0\leq k<P-1$ and $\pi_{k}$ are
permutation of $\\{1,2,\ldots,M\\}$. Each $\pi_{k}$ should be follow
$\pi_{k_{1}}(i_{1}P+j)\neq\pi_{k_{2}}(i_{2}P+j)$ for $k_{1}\neq k_{2}$, $0\leq
i_{1},i_{2}<S$ and $1\leq j\leq P$. Then, each
$\mathbf{B}_{1}=\\{B_{1},B_{2},\ldots,B_{M}\\},$
$\mathbf{B}_{2}=\\{B_{M+1},B_{M+2},\ldots,B_{2M}\\},$ $\ldots$
$\mathbf{B}_{P}=\\{B_{(P-1)M+1},B_{(P-1)M+2},\ldots,B_{PM}\\}$ is a
$(M,NP)$-CCC. Combining these sets
$\mathbf{B}=\\{\mathbf{B}_{1}\bigcup\mathbf{B}_{2}\bigcup\cdots\bigcup\mathbf{B}_{P}\\}$,
is a MCCC also an optimal symmetrical $(PM,M,PN,N-1)$-ZCCS.

###### Proof:

From Theorem 3, for a fixed $\pi_{k}$, corresponding $\mathbf{B}_{k}$ is a
$(M,PN)$-CCC for $1\leq k\leq P$. Now, we show that codes from two different
sets have an inter-set correlation zone. We observe the ACCF between
$B_{k_{1}M+i_{1}P+j_{1}}$ and $B_{k_{2}M+i_{2}P+j_{2}}$, for $k_{1}\neq
k_{2}$. From (19) ACCF between $B_{k_{1}M+i_{1}P+j_{1}}$ and
$B_{k_{2}M+i_{2}P+j_{2}}$ is as given in (25).

The right hand side of (25) is non-zero only for $v=0$,
$\pi_{k_{1}}(i_{1}P+\alpha)=\pi_{k_{2}}(i_{2}P+\alpha+u)$ and $u\neq 0$,
otherwise it is zero. And for $v=0$ and $u\neq 0$, the ACCF value maybe non-
zero at only $|\tau|\in\\{N,2N,\ldots,(P-1)N\\}$. This completes the proof. ∎

###### Example 1

Let $+$ represents $1$ and $-$ represents $-1$, now
$C_{1}=\begin{bmatrix}+++\\\ ++-\\\ ++-\\\
-+-\end{bmatrix},C_{2}=\begin{bmatrix}+-+\\\ ++-\\\ \--+\\\
+++\end{bmatrix},C_{3}=\begin{bmatrix}+--\\\ +++\\\ -+-\\\ +--\end{bmatrix}$
and $C_{4}=\begin{bmatrix}+--\\\ +-+\\\ +++\\\ -++\end{bmatrix}$ be a
$(4,3)$-CCC [9]. Here, $M=4$, $N=3$, assuming $P=2$, $\pi_{1}=(1,2,3,4)$,
$\pi_{2}=(2,1,4,3)$ $\mathbf{b}^{1}=(--)$ and $\mathbf{b}^{2}=(+-)$. Now, from
Theorem 4, we have $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ as two $(4,6)$-CCC
given in Table I, which is also a $(2,4,6,3)$-MCCC and an optimal
$(8,4,6,2)$-SZCCS. With the given parameter no CCC, MCCC and SZCCS are
available.

TABLE I: $(8,4,6,2)$-SZCCS | $B_{1}$ | $B_{2}$ | $B_{3}$ | $B_{4}$
---|---|---|---|---
$\mathbf{B}_{1}$ | $\begin{smallmatrix}----+-\\\ \--+--+\\\ \--+++-\\\ +-+---\par\end{smallmatrix}$ | $\begin{smallmatrix}+++-+-\\\ ++---+\\\ ++-++-\\\ -+----\end{smallmatrix}$ | $\begin{smallmatrix}-++-++\\\ \----+-\\\ +-+---\\\ -+++--\end{smallmatrix}$ | $\begin{smallmatrix}+---++\\\ +++-+-\\\ -+----\\\ +--+--\par\end{smallmatrix}$
| $B_{5}$ | $B_{6}$ | $B_{7}$ | $B_{8}$
$\mathbf{B}_{2}$ | $\begin{smallmatrix}-+----\\\ \--+--+\\\ ++---+\\\ \---+-+\end{smallmatrix}$ | $\begin{smallmatrix}+-+---\\\ ++---+\\\ \--+--+\\\ ++++-+\end{smallmatrix}$ | $\begin{smallmatrix}-++-++\\\ -+----\\\ \---+-+\\\ +---++\end{smallmatrix}$ | $\begin{smallmatrix}+---++\\\ +-+---\\\ ++++-+\\\ -++-++\end{smallmatrix}$

##  IV Comparison

In this section, we compare the proposed work with existing work.

Source | Code set | Constraints
---|---|---
[14] |  $(2^{v},2^{k+1},2^{m},2^{m-v})$-MCCC | $0\leq v\leq k\leq m-1$
[16] |  $(p^{v},p^{n},p^{m},p^{m-v})$-MCCC |  $0\leq v\leq m$, $n\geq k+v$,
|  |  $1\leq k\leq m-v$, $p$ is prime
Proposed |  $(P,M,PN,N)$-MCCC |  $(M,N)$-CCC available, $P|M$
[13] |  $(2,2,2^{m}+2^{\delta},2^{\delta}-1)$-SZCCS |  $m\geq 3$, $1\leq\delta\leq m-1$
|  $(8,2,2^{m+1},2^{m-1}-1)$-SZCCS |
[17] |  $(p^{k+\delta},p^{k},p^{m},p^{m-\delta}-1)$-SZCCS |  $m\geq 3$, $1\leq\delta\leq m-1$
|  |  $p$ is prime
Proposed |  $(PM,M,PN,N-1)$-SZCCS |  $(M,N)$-CCC available, $P|M$
TABLE II: Comparison with existing MCCC and SZCCS

In the available work, the code size and set size appear to follow the power
of prime, as the proposed construction accommodates any existing CCC, it
results with additional parameters.

##  V Conclusion

In this paper, we introduce unique constructions for 2D perfect array and
$(M,M)$-CCC utilising multiplication. In addition, new constructions have been
developed for $(M,NP)$-CCC, $(P,M,PN,N)$ multiple CCC, and optimal
$(PM,M,PN,N-1)$-SZCCS. These constructions utilise a $(M,N)$-CCC and mutually
orthogonal sequences. The code set obtained offers more flexibility in
comparison to the existing literature.

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