[table]capposition=top

# A Direct Construction of Optimal Symmetrical Z-Complementary Code Sets of
Prime Power Lengths

Praveen Kumar 1, Sudhan Majhi2, Subhabrata Paul1  1Department of Mathematics,
IIT Patna, India {praveen_2021ma03<EMAIL_ADDRESS>2Department of
Electrical Communication Engineering, IISC Bangalore, India<EMAIL_ADDRESS>
###### Abstract

This paper presents a direct construction of an optimal symmetrical
Z-complementary code set (SZCCS) of prime power lengths using a multi-variable
function (MVF). SZCCS is a natural extension of the Z-complementary code set
(ZCCS), which has only front-end zero correlation zone (ZCZ) width. SZCCS has
both front-end and tail-end ZCZ width. SZCCSs are used in developing optimal
training sequences for broadband generalized spatial modulation systems over
frequency-selective channels because they have ZCZ width on both the front and
tail ends. The construction of optimal SZCCS with large set sizes and prime
power lengths is presented for the first time in this paper. Furthermore, it
is worth noting that several existing works on ZCCS and SZCCS can be viewed as
special cases of the proposed construction.

##  I Introduction

The idea of Z-complementary pairs (ZCPs) was introduced by Fan et al. [1]. The
sum of the aperiodic auto-correlation function (AACF) of the two sequences in
a ZCP is zero within a particular zone, which is referred to as the zero-
correlation zone (ZCZ). When this ZCZ width $Z$ is equal to the sequence
length $N$, ZCP becomes a Golay complementary pair (GCP). Unlike GCPs, ZCPs
are available in arbitrary lengths with various ZCZ widths [2, 3, 4, 5, 6].

The idea of ZCPs introduced in [1] was generalized to a Z-complementary code
set (ZCCS) by Feng et al. in [7]. ZCCS only considers the front-end ZCZ of the
AACFs and aperiodic cross-correlation functions (ACCFs). Several generalized
Boolean functions (GBFs) based constructions of ZCCSs of non-power-of-two
lengths are proposed in the literature[8, 9, 10, 11, 12, 13]. A ZCCS with $K$
codes, with each code having $M$ sequences each of length $N$ and ZCZ width
$Z$ is denoted by $(K,M,N,Z)$-ZCCS. For the special case, $K=M$ and $N=Z$, it
is known as a complete complementary code (CCC) and is denoted by
$(K,K,N)$-CCC. Recently, Li et al. proposed a direct construction of multiple
CCC of prime power length and with inter set ZCZ width using a multi-variable
function (MVF) [14]. After combining these multiple CCC, optimal
$(p^{n+v},p^{n},p^{m},p^{m-v})$-ZCCS are obtained [14].

Recently, the idea of ZCCS has been extended to symmetrical-ZCCS (SZCCS),
which exhibits ZCZ properties for the front-end and the tail-end [15]. The
authors in [15] have presented a GBF based construction of
$(8,2,2^{m},2^{m-2}-1)$-SZCCS. In practice, a front-end ZCZ and a tail-end ZCZ
have a particular role in mitigating interference with small and large delays,
respectively. SZCCSs with larger set sizes are used in designing training
sequences for broadband generalized spatial modulation (GSM) systems over
frequency-selective channels [15]. The fact that SZCCS is being widely applied
in the GSM systems and also the unavailability of constructions with flexible
parameters in terms of set size, flock size, and length have inspired the
authors to propose a direct construction of SZCCSs of prime power lengths in
this paper.

The proposed MVF-based construction of SZCCS has a set size $p^{k+\delta}$,
which is much larger than the flock size of $p^{k},$ where $m\geq
3,~{}0\leq\delta<m,~{}1\leq k\leq m-\delta.$ Also, the proposed SZCCS has
large ZCZ width of $p^{m-\delta}-1$, and it achieves the optimality condition.
Many of the existing constructions of ZCCS and SZCCS appear as special cases
of the proposed construction.

The rest of the paper is structured as follows: preliminary work is covered in
Section II, and the proposed SZCCS construction based on MVF is presented in
Section III. Section IV of the paper presents a comprehensive comparison
between the proposed constructions and the existing works, providing detailed
insights. Following that, Section V concludes the paper.

##  II Preliminaries

The essential concepts, notations, and previously established findings
necessary for the proposed construction are explained in this section.

###### Definition 1

Let $\mathbf{u}$ $=(u_{0},u_{1},\ldots,u_{N-1})$ and $\mathbf{v}$
$=(v_{0},v_{1},\ldots,v_{N-1})$ be two sequences of length $N$ over
$\mathbb{Z}_{q}$. At a shift $\tau$, the ACCF is defined as

$\mathcal{C}\left({\mathbf{u},\mathbf{v}}\right)(\tau)=\begin{cases}\sum_{i=0}^{N-1-\tau}\omega_{q}^{u_{i}-v_{i+\tau}},&0\leq\tau\leq
N-1,\\\
\sum_{i=0}^{N-1+\tau}\omega_{q}^{u_{i-\tau}-v_{i}},&-N+1\leq\tau\leq-1,\\\
0,&|\tau|\geq N,\end{cases}$ (1)

where $q$ is a positive integer greater than $2$, and
$\omega_{q}=\exp(2\pi\sqrt{-1}/q)$. For the special case,
$\mathbf{u}=\mathbf{v},~{}\mathcal{C}(\mathbf{u},\mathbf{v})(\tau)$ is
referred to as the AACF of $\mathbf{u}$ and is denoted by
$\mathcal{A}(\mathbf{u})(\tau)$.

Consider a set $\mathrm{C}=\left\\{{C^{0}},{C^{1}},\ldots,{C^{K-1}}\right\\}$,
where each set ${C^{e}}$ consists of $M$ sequences, i.e.,
${C^{e}}=\left\\{\mathbf{c}_{0}^{e},\mathbf{c}_{1}^{e},\ldots,\mathbf{c}_{M-1}^{e}\right\\}$,
and length of each sequence $\mathbf{c}_{l}^{e}$ is $N$, where $0\leq e\leq
K-1$ and $0\leq l\leq M-1$.

###### Definition 2

The set $\mathrm{C}$ defined above is called a ZCCS, denoted by
$(K,M,N,Z)$-ZCCS, if the ACCF of $C^{e}$ and $C^{e^{\prime}}$ satisfies

$\displaystyle\mathcal{C}\left(C^{e},C^{e^{\prime}}\right)(\tau)$
$\displaystyle=\sum_{l=0}^{M-1}\mathcal{C}\left(\mathbf{c}_{l}^{e},\mathbf{c}_{l}^{e^{\prime}}\right)(\tau)$
(2) $\displaystyle=\begin{cases}MN,&\tau=0,e=e^{\prime},\\\
0,&0<|\tau|<Z,e=e^{\prime},\\\ 0,&|\tau|<Z,e\neq e^{\prime},\end{cases}$

where $0\leq e,e^{\prime}\leq K-1$. For a general $(K,M,N,Z)$-ZCCS, $K\leq
M\lfloor N/Z\rfloor$, whereas for the special case, $K=M\lfloor N/Z\rfloor$,
it becomes an optimal ZCCS [16]. Again, for the special case, $K=M$ and $Z=N$,
the ZCCS is called a CCC of order $K$ and length $N$, and is denoted by
$(K,K,N)$-CCC.

###### Definition 3

The set $\mathrm{C}$ defined above is known as a SZCSS, denoted by
$(K,M,N,Z)$-SZCCS, if for $\mathcal{T}_{1}=\\{1,2,\cdots,Z\\}$ and
$\mathcal{T}_{2}=\\{N-Z,N-$ $Z+1,\cdots,N-1\\}$ with $Z\leq N,$ it satisfies
the following properties

$\displaystyle
P1:\sum_{i=0}^{M-1}\mathcal{A}(\mathbf{c}_{i}^{e})(\tau)=0,\quad\text{ for all
}|\tau|\in\left(\mathcal{T}_{1}\cup\mathcal{T}_{2}\right)\cap\mathcal{T};$ (3)
$\displaystyle
P2:\sum_{i=0}^{M-1}\mathcal{C}(\mathbf{c}_{i}^{e},\mathbf{c}_{i}^{e^{\prime}})(\tau)=0,\quad\text{
for all }|\tau|\in\\{0\\}\cup\mathcal{T}_{1}\cup\mathcal{T}_{2};$

where $\mathcal{T}=\\{1,2,\ldots,N-1\\}$ and $e\neq e^{\prime}$. For a
$(K,M,N,Z)$-SZCCS, $K,M,N$ and $Z$ are known as the set size, flock size,
sequence length, and ZCZ width, respectively.

For a MVF $f:\\{0,1,\ldots,p-1\\}^{m}\rightarrow\mathbb{Z}_{q}$ in $m$ $p$-ary
variables $x_{1},x_{2},\ldots,x_{m}$, corresponding $\mathbb{Z}_{q}$-valued
sequence is denoted by $\mathbf{f}$, and calculated as
$\mathbf{f}=\left({f_{0}},{f_{1}},\ldots,{f_{p^{m}-1}}\right)$ , where
$f_{r}=f({r_{1}},{r_{2}},\ldots,r_{m})$, and
$r=({r_{1}},{r_{2}},\ldots,r_{m})$ is the $p$-ary representation of the
integer $r$, i.e., $r=\sum_{i=1}^{m}r_{i}p^{i-1}$. The complex-valued sequence
associated with $\mathbf{f}$ is produced as $\Psi(f)=$
$\left(\omega_{q}^{f_{0}},\omega_{q}^{f_{1}},\ldots,\omega_{q}^{f_{p^{m}-1}}\right)$,
where $\omega_{q}$ is the $q$-th $(q\geq 2)$ root of unity [14].

######  Lemma 1 ([15, 16])

Any unimodular $(K,M,N,Z)$-SZCCS satisfies $K\leq
M\left\lfloor\frac{N}{Z+1}\right\rfloor$, when
$K=M\left\lfloor\frac{N}{Z+1}\right\rfloor$, it is known as an optimal
$(K,M,N,Z)$-SZCCS.

##  III Proposed Construction

This section explains a MVF based construction of optimal
$(p^{k+\delta},p^{k},p^{m},p^{m-\delta}-1)$-SZCCS.

###### Theorem 1

For any positive integer $m\geq 3$ and $0\leq\delta<m$, we let the set
$\\{1,2,\ldots,m-\delta\\}$ be partitioned into $k$ sets,
$S_{1},S_{2},\ldots,S_{k}$, where $1\leq k\leq m-\delta$. Let the cardinality
of the set $S_{\beta}$ be $m_{\beta}$, and $\pi_{\beta}$ be a one-one and onto
mapping from $\\{1,2,\ldots,m_{\beta}\\}$ to $S_{\beta}$ for
$\beta=1,2,\ldots,k.$ Let us define a MVF
$g:\\{0,1,\ldots,p-1\\}^{m}\rightarrow\mathbb{Z}_{q}$ by

$g=\frac{q}{p}\sum_{\beta=1}^{k}\sum_{\gamma=1}^{m_{\beta}-1}x_{\pi_{\beta}(\gamma)}x_{\pi_{\beta}(\gamma+1)}+\sum_{l=1}^{m}\lambda_{l}x_{l}+\lambda_{0},$
(4)

where $p$ is a prime and $p\mid q$, $\lambda_{l}\in\mathbb{Z}_{q}$, $0\leq
l\leq m$. Then the set
${A}=\left\\{A^{0},A^{1},\ldots,A^{p^{k+\delta}-1}\right\\}$ is a
$(p^{k+\delta},p^{k},p^{m},p^{m-\delta}-1)$-SZCCS, where
$A^{t}=\\{\mathbf{a}^{t}_{0},\mathbf{a}^{t}_{1},\ldots,\mathbf{a}^{t}_{p^{k}-1}\\}$
and

$\displaystyle\mathbf{a}^{t}_{\sigma}=\boldsymbol{g}$
$\displaystyle+\frac{q}{p}\left(\sum_{\beta=1}^{k}\sigma_{\beta}\boldsymbol{x}_{\pi_{\beta}(1)}+\sum_{\beta=1}^{k}t_{\beta}\boldsymbol{x}_{\pi_{\beta}(m_{\beta})}\right.$
(5)
$\displaystyle\left.+\sum_{\beta=1}^{\delta}t_{k+\beta}\boldsymbol{x}_{m-\delta+\beta}\right),$

for $\sigma=0,1,\ldots,p^{k}-1$ and $t=0,1,\ldots,p^{k+\delta}-1$ with $p$-ary
vector representation $(\sigma_{1},\sigma_{2},\ldots,\sigma_{k})$ and
$(t_{1},t_{2},\ldots,t_{k},\ldots,t_{k+\delta})$, respectively.

###### Proof:

We will prove that the set
${A}=\left\\{A^{0},A^{1},\ldots,A^{p^{k+\delta}-1}\right\\}$ obtained from
Theorem 1 satisfies two conditions $P1$ and $P2$ given in Definition 3, with
$K=p^{k+\delta},M=p^{k},N=p^{m},$ and $Z=p^{m-\delta}-1$, and hence a
$(p^{k+\delta},p^{k},p^{m},p^{m-\delta}-1)$-SZCCS. Let the $p$-ary
representations of $0\leq r,s<p^{m}$ be $(r_{1},r_{2},\ldots,r_{m})$ and
$(s_{1},s_{2},\ldots,s_{m})$, respectively. Let
$\mathbf{a}^{t}_{\sigma}=(a^{t}_{\sigma,0},a^{t}_{\sigma,1},\ldots,a^{t}_{\sigma,p^{m}-1})$.

First, we prove the property $P1$ of Definition 3, i.e., for
$\mathcal{T}_{1}=\\{1,2,\ldots,p^{m-\delta}-1\\}$ and
$\mathcal{T}_{2}=\\{p^{m}-p^{m-\delta}-1,p^{m}-p^{m-\delta},\ldots,p^{m}-1\\}$
the AACF for every $A^{t}$ zero, i.e.,

$\sum_{\sigma=0}^{p^{k}-1}\mathcal{A}(\mathbf{a}_{\sigma}^{t})(\tau)=\sum_{r=0}^{p^{m}-\tau-1}\sum_{\sigma=0}^{p^{k}-1}\omega_{q}^{\left(a^{t}_{\sigma,r}-a^{t}_{\sigma,r+\tau}\right)}=0,$
(6)

for all
$|\tau|\in\left(\mathcal{T}_{1}\cup\mathcal{T}_{2}\right)\cap\mathcal{T},$
where $\mathcal{T}=\\{1,2,\ldots,p^{m}-1\\}$. Let $s=r+\tau$ for any integer
$r$. Then we consider the two cases listed below.

Case I: $r_{\pi_{\beta}(1)}\neq s_{\pi_{\beta}(1)}$ for some
$\beta\in\\{1,2,\ldots,k\\}$. Then there exist
$\mathbf{a}^{t}_{\sigma^{j}}=(a^{t}_{\sigma^{j},0},a^{t}_{\sigma^{j},1},\ldots,a^{t}_{\sigma^{j},p^{m}-1})=\mathbf{a}^{t}_{\sigma}+(jq/p)\boldsymbol{x}_{\pi_{\beta}(1)}\in
A^{t}$, where $1\leq j\leq p-1$, such that

${a^{t}_{\sigma^{j},r}-a^{t}_{\sigma,r}}=\frac{jq}{p}{r}_{\pi_{\beta}(1)},$
(7)

and

${a^{t}_{\sigma^{j},s}-a^{t}_{\sigma,s}}=\frac{jq}{p}{s}_{\pi_{\beta}(1)}.$
(8)

So from the above two equations, we get

$\left(a^{t}_{\sigma^{j},r}-a^{t}_{\sigma^{j},s}\right)-\left(a^{t}_{\sigma,r}-a^{t}_{\sigma,s}\right)=\frac{jq}{p}\left({r}_{\pi_{\beta}(1)}-{s}_{\pi_{\beta}(1)}\right).$
(9)

Raising to the power of $q$-th root of unity and then taking sum over $1\leq
j\leq p-1$, we get

$\sum_{j=1}^{p-1}\omega_{q}^{\left(a^{t}_{\sigma^{j},r}-a^{t}_{\sigma^{j},s}\right)-\left(a^{t}_{\sigma,r}-a^{t}_{\sigma,s}\right)}=\sum_{j=1}^{p-1}\omega_{p}^{j\left({r}_{\pi_{\beta}(1)}-{s}_{\pi_{\beta}(1)}\right)}=-1.$
(10)

Hence,

$\omega_{q}^{\left(a^{t}_{\sigma,r}-a^{t}_{\sigma,s}\right)}+\sum_{j=1}^{p-1}\omega_{q}^{\left(a^{t}_{\sigma^{j},r}-a^{t}_{\sigma^{j},s}\right)}=0.$
(11)

So, the sum AACF is zero. i.e.,

$\sum_{\sigma=0}^{p^{k}-1}\mathcal{A}(\mathbf{a}_{\sigma}^{t})(\tau)=0.$ (12)

Case II: $r_{\pi_{\beta}(1)}=s_{\pi_{\beta}(1)}$ for all
$\beta\in\\{1,2,\ldots,k\\}$. Then, there exist integers $\hat{\beta}$ and
$\hat{\gamma}$ such that $\hat{\beta}$ is the largest integer satisfying
$r_{\pi_{\beta}(\gamma)}=s_{\pi_{\beta}(\gamma)}$ for all
$\beta=1,2,\ldots,\hat{\beta}-1$, and $\gamma=1,2,\ldots,m_{\alpha}$, and
$\hat{\gamma}$ is the least integer satisfying
$r_{\pi_{\hat{\beta}}(\hat{\gamma})}\neq s_{\pi_{\hat{\beta}}(\hat{\gamma})}$
. If the above statement doesn’t hold, then we have $r_{i}=s_{i}$ for $i=$
$1,2,\ldots,m-\delta$ since
$\bigcup_{\alpha=1}^{k}I_{\alpha}=\\{1,2,\ldots,m-\delta\\}$. Hence the lower
bound of $\tau$ is estimated as,

$\tau=s-r=\sum_{i=m-\delta+1}^{m}\left(s_{i}-r_{i}\right)p^{i-1}\geq
p^{m-\delta},$ (13)

also the upper bound of $\tau$ is estimated as,

$\displaystyle\tau=$ $\displaystyle
s-r=\sum_{i=m-\delta+1}^{m}\left(s_{i}-r_{i}\right)p^{i-1}$ (14)
$\displaystyle\leq
p^{m-\delta}+p^{m-\delta+1}+\ldots+p^{m-1}=p^{m}-p^{m-\delta}.$

The above two equations (13) and (14) contradict the assumption that
$\tau\in\left(\mathcal{T}_{1}\cup\mathcal{T}_{2}\right)\cap\mathcal{T}$. So
this guarantees the existence of integers $\hat{\beta}$ and $\hat{\gamma}$
with the above-mentioned conditions. Let $r^{j}$ and $s^{j}$ be two integer
which differs from $r$ and $s$, respectively, in only one position
$\pi_{\hat{\beta}}(\hat{\gamma}-1)$ of their $p$-ary representation, i.e.,
$r^{j}_{\pi_{\hat{\beta}}(\hat{\gamma}-1)}=r_{\pi_{\hat{\beta}}(\hat{\gamma}-1)}-j$
and
$s^{j}_{\pi_{\hat{\beta}}(\hat{\gamma}-1)}=s_{\pi_{\hat{\beta}}(\hat{\gamma}-1)}-j$,
where $1\leq j\leq p-1$. Since $s=r+\tau$, we get $s^{j}=r^{j}+\tau$. The
difference between the terms $a^{t}_{\sigma,r}-a^{t}_{\sigma,r^{j}}$ is
calculated below as

$\displaystyle a^{t}_{\sigma,r}-a^{t}_{\sigma,r^{j}}$
$\displaystyle=g_{r}-g_{r^{j}}$ (15)
$\displaystyle=j\left(\frac{q}{p}r_{\pi_{\hat{\beta}}(\hat{\gamma}-2)}+\frac{q}{p}r_{\pi_{\hat{\beta}}(\hat{\gamma})}+\lambda_{\pi_{\hat{\beta}}(\hat{\gamma}-1)}\right).$

Similarly, it can be calculated that

$a^{t}_{\sigma,s}-a^{t}_{\sigma,s^{j}}=j\left(\frac{q}{p}s_{\pi_{\hat{\beta}}(\hat{\gamma}-2)}+\frac{q}{p}s_{\pi_{\hat{\beta}}(\hat{\gamma})}+\lambda_{\pi_{\hat{\beta}}(\hat{\gamma}-1)}\right).$
(16)

From the above two equations and using
$r_{\pi_{\hat{\beta}}(\hat{\gamma}-2)}\neq
s_{\pi_{\hat{\beta}}(\hat{\gamma}-2)}$, we get the following equality

$a^{t}_{\sigma,r^{j}}-a^{t}_{\sigma,s^{j}}-\left(a^{t}_{\sigma,r}-a^{t}_{\sigma,s}\right)=j\frac{q}{p}\left(s_{\pi_{\hat{\beta}}(\hat{\gamma})}-r_{\pi_{\hat{\beta}}(\hat{\gamma})}\right).$
(17)

Raising to the power of $q$-th root of unity and then taking sum over $1\leq
j\leq p-1$, we get the following expression

$\sum_{j=1}^{p-1}\omega_{q}^{\left(a^{t}_{\sigma,r^{j}}-a^{t}_{\sigma,s^{j}}\right)-\left(a^{t}_{\sigma,r}-a^{t}_{\sigma,s}\right)}=\sum_{j=1}^{p-1}\omega_{p}^{j\left(s_{\pi_{\hat{\beta}}(\hat{\gamma})}-r_{\pi_{\hat{\beta}}(\hat{\gamma})}\right)}=-1.$
(18)

Hence,

$\omega_{q}^{\left(a^{t}_{\sigma,r}-a^{t}_{\sigma,s}\right)}+\sum_{j=1}^{p-1}\omega_{q}^{\left(a^{t}_{\sigma,r^{j}}-a^{t}_{\sigma,s^{j}}\right)}=0.$
(19)

So, the AACF is zero. Combining Cases I and II, we get AACF of $A^{t}$ is zero
for $\tau\in\left(\mathcal{T}_{1}\cup\mathcal{T}_{2}\right)\cap\mathcal{T}$.

Next, in the following part, we will demonstrate that any two distinct sets
$A^{t_{1}}$ and $A^{t_{2}}$ have zero ACCF for all
$|\tau|\in\\{0\\}\cup\mathcal{T}_{1}\cup\mathcal{T}_{2}$, i.e.,

$\sum_{\sigma=0}^{p^{k}-1}\mathcal{C}(\mathbf{a}_{\sigma}^{t_{1}},\mathbf{a}_{\sigma}^{t_{2}})(\tau)=\sum_{r=0}^{p^{m}-\tau-1}\sum_{\sigma=0}^{p^{k}-1}\omega_{q}^{\left(a^{t_{1}}_{\sigma,r}-a^{t_{2}}_{\sigma,r+\tau}\right)}=0.$
(20)

Similar to the first part, we let $s=r+\tau$ for any integer $r$ and consider
two cases.

Case I: Suppose $r_{\pi_{\beta}(1)}\neq s_{\pi_{\beta}(1)}$ for some
$\beta\in\\{1,2,\ldots,k\\}$. In the same manner as Case I in the first part
there exist
$\mathbf{a}^{t}_{\sigma^{j}}=(a^{t}_{\sigma^{j},0},a^{t}_{\sigma^{j},1},\ldots,a^{t}_{\sigma^{j},p^{m}-1})=\mathbf{a}^{t}_{\sigma}+(jq/p)\boldsymbol{x}_{\pi_{\beta}(1)}\in
A^{t}$, for $t=t_{1},t_{2}$, where $1\leq j\leq p-1$, such that

$\sum_{j=1}^{p-1}\omega_{q}^{\left(a^{t_{1}}_{\sigma^{j},r}-a^{t_{2}}_{\sigma^{j},s}\right)-\left(a^{t_{1}}_{\sigma,r}-a^{t_{2}}_{\sigma,s}\right)}=\sum_{j=1}^{p-1}\omega_{p}^{j\left({r}_{\pi_{\beta}(1)}-{s}_{\pi_{\beta}(1)}\right)}=-1.$
(21)

Hence, similar to the first part, the ACCF becomes zero, i.e.,

$\sum_{\sigma=0}^{p^{k}-1}\omega_{q}^{\left(a^{t_{1}}_{\sigma,r}-a^{t_{2}}_{\sigma,s}\right)}=0.$
(22)

Case II: Suppose we have $r_{\pi_{\beta}(1)}=s_{\pi_{\beta}(1)}$ for all
$\beta\in\\{1,2,\ldots,k\\}$. As argued in Case II in the first part, a
similar result can be obtained, i.e.,

$\omega_{q}^{\left(a^{t_{1}}_{\sigma,r}-a^{t_{2}}_{\sigma,s}\right)}+\sum_{j=1}^{p-1}\omega_{q}^{\left(a^{t_{1}}_{\sigma,r^{j}}-a^{t_{2}}_{\sigma,s^{j}}\right)}=0.$
(23)

So, from Case I and Case II, we get
$\sum_{\sigma=0}^{p^{k}-1}\mathcal{C}(\mathbf{a}_{\sigma}^{t_{1}},\mathbf{a}_{\sigma}^{t_{2}})(\tau)=0$,
for $|\tau|\in\mathcal{T}_{1}\cup\mathcal{T}_{2}$. It only suffices to show
that

$\sum_{\sigma=0}^{p^{k}-1}\mathcal{C}(\mathbf{a}_{\sigma}^{t_{1}},\mathbf{a}_{\sigma}^{t_{2}})(0)=\sum_{\sigma=0}^{p^{k}-1}\sum_{r=0}^{p^{m}-1}\omega_{q}^{\left(a^{t_{1}}_{\sigma,r}-a^{t_{2}}_{\sigma,r}\right)}=0.$
(24)

Let $\oplus$ denotes modulo-$p$ addition; let
$\left(t_{11},t_{12},\ldots,t_{1k+\delta}\right)$ and
$\left(t_{21},t_{22},\ldots,t_{2k+\delta}\right)$ denote the $p$-ary
representations of $t_{1}$ and $t_{2}$, respectively. Then, for
$\sigma=0,1,\ldots,p^{k}-1$, we have
$\mathbf{a}_{\sigma}^{t_{1}}-\mathbf{a}_{\sigma}^{t_{2}}\equiv(q/p)\mathbf{d}\
(\mathrm{mod}\ q)\quad$ where $\quad\mathbf{d}=$ $\left(t_{11}\oplus
t_{21}\right)\boldsymbol{x}_{\pi_{1}\left(m_{1}\right)}\oplus\left(t_{12}\oplus
t_{22}\right)\boldsymbol{x}_{\pi_{2}\left(m_{2}\right)}\oplus\cdots\oplus\left(t_{1k}\oplus
t_{2k}\right)\boldsymbol{x}_{\pi_{k}\left(m_{k}\right)}\oplus$
$\left(t_{1k+1}\oplus
t_{1k+1}\right)\boldsymbol{x}_{m-\delta+1}\oplus\cdots\oplus\left(t_{1k+\delta}\oplus
t_{2k+\delta}\right)\boldsymbol{x}_{m}$. For every $1\leq\beta\leq k$,
$\boldsymbol{x}_{\pi_{\beta}\left(m_{\beta}\right)}$ takes each value from
$\\{0,1,\ldots,p-1\\}$ exactly $p^{m-1}$ times. Also for
$1\leq\beta\leq\delta$, $\boldsymbol{x}_{m-\delta+\beta}$ takes each value
from $\\{0,1,\ldots,p-1\\}$ exactly $p^{m-1}$ times. So, $\mathbf{d}$ also
takes each value from $\\{0,1,\ldots,p-1\\}$ exactly $p^{m-1}$ times, i.e., it
is balanced. So,

$\sum_{r=0}^{p^{m}-1}\omega_{q}^{\left(a^{t_{1}}_{\sigma,r}-a^{t_{2}}_{\sigma,r}\right)}=\sum_{r=0}^{p^{m}-1}\omega_{p}^{d_{r}}=0.$
(25)

Therefore, from (24) and (25) the ACCF at shift $\tau=0$ is zero, i.e.,

$\sum_{\sigma=0}^{p^{k}-1}\mathcal{C}(\mathbf{a}_{\sigma}^{t_{1}},\mathbf{a}_{\sigma}^{t_{2}})(0)=0.$
(26)

∎

The proposed SZCCS in Theorem 1, is optimal since
$K=p^{k+\delta}=p^{k}(p^{m}/p^{m-\delta})=M(N/(Z+1))$.

###### Remark 1

For $\delta=0$, the proposed construction generates $(p^{k},p^{k},p^{m})$-CCC.
So, the construction of $(p^{k},p^{k},p^{m})$-CCC, in [17, 18] become
particular cases of the proposed construction.

###### Remark 2

For $p=2$, the proposed construction generates
$(2^{k+\delta},2^{k},2^{m},2^{m-\delta})$-ZCCS, so the available construction
of ZCCS provided in [19] is a special case of the proposed construction.

###### Remark 3

In [20], direct construction of $(p^{\delta+1},p,p^{m},p^{m-\delta})$-ZCCS is
provided, the proposed construction with $k=1$ generates the ZCCS with the
same parameter.

###### Remark 4

Since every SZCCS is a ZCCS, the construction of
$(p^{n+v},p^{n},p^{m},p^{m-v})$-ZCCS in [14] occurs as a special case of the
proposed construction in Theorem 1.

###### Remark 5

The construction of $(8,2,2^{m},2^{m-2}-1)$-SZCCS is given is [15], which
appears as a special case of the proposed construction when
$p=2,\delta=2,k=1.$

We provide the following example to illustrate how optimal SZCCS is obtained
from Theorem 1.

###### Example 1

For $m=3,\delta=1$, let $\pi_{1}=\pi$ is the identity permutation of
$\\{1,2\\}$, i.e. $\pi(1)=1$ and $\pi(2)=2$. Further let us take $p=3$ and
$q=3$ and define the MVF $g:\\{0,1,2\\}^{3}\rightarrow\mathbb{Z}_{3}$ as
$g(x_{1},x_{2},x_{3})=x_{1}x_{2}$. Also let us define the sets
$A^{t}=\\{\mathbf{f}+\sigma_{1}x_{1}+t_{1}x_{2}+t_{2}x_{3}:\sigma_{1}\in\\{0,1,2\\}\\}$,
for $t=0,1,\ldots,8$ with $p$-ary representation $(t_{1},t_{2})$. So, from
Theorem 1, $\mathrm{A}=\\{A^{0},A^{1},\ldots,A^{8}\\}$ is a
$(9,3,27,8)$-SZCCS. The codes $A^{0},A^{1},\ldots,A^{8}$ are listed in Table I
explicitly, where by integer $i$, we mean $\omega^{i}$ where
$\omega=exp(2\pi\sqrt{-1}/3)$, and AACF graph of $A^{0}$ is plotted in Fig. 1,
and ACCF graph of $A^{2}$ and $A^{8}$ in Fig. 2.

TABLE I: Different codes of the SZCSS obtained from Example 1

$A^{0}$ | $A^{1}$
---|---
$\begin{matrix}000012021000012021000012021\\\ 000120210000120210000120210\\\ 000201102000201102000201102\end{matrix}$ | $\begin{matrix}012021000012021000012021000\\\ 012102222012102222012102222\\\ 012210111012210111012210111\end{matrix}$
$A^{2}$ | $A^{3}$
$\begin{matrix}021000012021000012021000012\\\ 021111201021111201021111201\\\ 021222120021222120021222120\end{matrix}$ | $\begin{matrix}000012021111120102222201210\\\ 000120210111201021222012102\\\ 000201102111012210222120021\end{matrix}$
$A^{4}$ | $A^{5}$
$\begin{matrix}012021000120102111201210222\\\ 012102222120210000201021111\\\ 012210111120021222201102000\end{matrix}$ | $\begin{matrix}021000012102111120210222201\\\ 021111201102222012210000120\\\ 021222120102000201210111012\end{matrix}$
$A^{6}$ | $A^{7}$
$\begin{matrix}000012021222201210111120102\\\ 000120210222012102111201021\\\ 000201102222120021111012210\end{matrix}$ | $\begin{matrix}012021000201210222120102111\\\ 012102222201021111120210000\\\ 012210111201102000120021222\end{matrix}$
$A^{8}$
$\begin{matrix}021000012210222201102111120\\\ 021111201210000120102222012\\\
021222120210111012102000201\end{matrix}$

Figure 1: AACF of $A^{0}$. Figure 2: ACCF of $A^{2}$ and $A^{8}$.

##  IV Comparison With Existing Works

In the literature, research on SZCCS is relatively recent, with only
constructions available for $(8,2,2^{m},2^{m-2}-1)$-SZCCS [15] and
$(2,2,2^{m-1}+2^{v},2^{v}-1)$-SZCCS [21]. Optimal SZCCS are utilized in the
design of optimal training sequences for GSM systems, which can achieve
superior channel estimation performance compared to other sequence classes
[15, 21]. The proposed construction in Theorem 1 produces optimal SZCCS with
variable set size $(p^{k+\delta})$, flock size $(p^{k})$, sequence length
$(p^{m})$, and ZCZ width $(p^{m-\delta}-1)$. Thus, due to the flexibility of
parameters, the proposed SZCCS offers adaptability in generating optimal
training matrices for GSM systems with multiple active transmit antennas [21].

Based on definitions 2 and 3, it is clear that a ZCCS can be regarded as a
particular case of an SZCCS, where $\mathcal{T}_{2}=\phi$ and
$\mathcal{T}_{1}=\\{1,2,\ldots,Z-1\\}$. Moreover, according to Lemma 1,
optimality of SZCCS implies optimality of ZCCS. Consequently, as detailed in
Remark 1-4, the existing constructions of CCC in [17, 18] and ZCCS in [19, 20,
14] can be viewed as special cases of the proposed construction.

##  V Conclusion

An optimal SZCCS of prime power length has been constructed directly using MVF
in this paper. Since SZCCS have both front-end and tail-end ZCZ width, they
are used in designing optimal training sequences for broadband GSM systems
over frequency-selective channels. The proposed MVF-based construction
generates optimal $(p^{k+\delta},p^{k},p^{m},p^{m-\delta}-1)$-SZCCS, which
generalizes many of the existing works of ZCCS and SZCCS.

## References

  * [1] P. Fan, W. Yuan, and Y. Tu, “Z-complementary binary sequences,” _IEEE Signal Process. Lett._ , vol. 14, no. 8, pp. 509–512, Aug. 2007.
  * [2] C.-Y. Pai, S.-W. Wu, and C.-Y. Chen, “Z-complementary pairs with flexible lengths from generalized Boolean functions,” _IEEE Commun. Lett._ , vol. 24, no. 6, pp. 1183–1187, 2020.
  * [3] C.-Y. Chen, “A novel construction of Z-complementary pairs based on generalized Boolean functions,” _IEEE Signal Process. Lett._ , vol. 24, no. 7, pp. 987–990, 2017.
  * [4] A. R. Adhikary, P. Sarkar, and S. Majhi, “A direct construction of $q$-ary even length Z-complementary pairs using generalized Boolean functions,” _IEEE Signal Process. Lett._ , vol. 27, pp. 146–150, 2020.
  * [5] A. R. Adhikary, S. Majhi, Z. Liu, and Y. L. Guan, “New sets of optimal odd-length binary Z-complementary pairs,” _IEEE Trans. Inf. Theory_ , vol. 66, no. 1, pp. 669–678, 2020.
  * [6] P. Kumar, P. Sarkar, S. Majhi, and S. Paul, “A direct construction of even length ZCPs with large ZCZ ratio,” _Cryptogr. Commun._ , vol. 15, pp. 85–94, 2023.
  * [7] L. Feng, P. Fan, X. Tang, and K. K. Loo, “Generalized pairwise Z-complementary codes,” _IEEE Signal Process. Lett._ , vol. 15, pp. 377–380, 2008.
  * [8] S.-W. Wu, A. Şahin, Z.-M. Huang, and C.-Y. Chen, “Z-complementary code sets with flexible lengths from generalised Boolean functions,” _IEEE Access_ , vol. 9, pp. 4642–4652, 2021.
  * [9] 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.
  * [10] B. Shen, Y. Yang, P. Fan, and Z. Zhou, “New Z-complementary/complementary sequence sets with non-power-of-two length and low PAPR,” _Cryptogr. Commun._ , 2022.
  * [11] G. Ghosh, S. Majhi, P. Sarkar, and A. K. Upadhaya, “Direct construction of optimal Z-complementary code sets with even lengths by using generalized Boolean functions,” _IEEE Signal Process. Lett._ , vol. 29, pp. 872–876, 2022.
  * [12] C. Xie, Y. Sun, and Y. Ming, “Constructions of optimal binary Z-complementary sequence sets with large zero correlation zone,” _IEEE Signal Process. Lett._ , vol. 28, pp. 1694–1698, 2021.
  * [13] L. Tian, Y. Li, Z. Zhou, and C. Xu, “Two classes of Z-complementary code sets with good cross-correlation subsets via paraunitary matrices,” _IEEE Trans. Commun._ , vol. 69, no. 5, pp. 2935–2947, 2021.
  * [14] X. Men and Y. Li, “New construction of multiple complete complementary codes with inter-set zero cross-correlation zone,” _IEEE Signal Process. Lett._ , vol. 29, pp. 1958–1962, 2022.
  * [15] Y. Zhou, Z. Zhou, Z. Liu, Y. Yang, P. Yang, and P. Fan, “Symmetrical Z-complementary code sets for optimal training in generalized spatial modulation,” in _10th Int. Workshop Signal Des. Appl. Commun. (IWSDA)_ , 2022, pp. 1–5.
  * [16] L. Feng, P. Fan, and X. Zhou, “Lower bounds on correlation of Z-complementary code sets.” _Wireless Pers Commun_ , p. 1475–1488, 2013\.
  * [17] P. Sarkar, Z. Liu, and S. Majhi, “Multivariable function for new complete complementary codes with arbitrary lengths,” 2021. [Online]. Available: https://arxiv.org/abs/2102.10517
  * [18] N. Kumar, S. Majhi, and A. K. Upadhyay, “A direct construction of complete complementary code with zero correlation zone property for prime-power length,” 2022. [Online]. Available: https://arxiv.org/abs/2207.13642
  * [19] S.-W. Wu and C.-Y. 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.
  * [20] B. Shen, H. Meng, Y. Yang, and Z. Zhou, “New constructions of Z-complementary code sets and mutually orthogonal complementary sequence sets,” _Des. Codes Cryptogr._ , 2022.
  * [21] Y. Zhou, Z. Zhou, Z. Liu, Y. Yang, P. Yang, and P. Fan, “Symmetrical z-complementary code sets for optimal training in generalized spatial modulation,” _Signal Processing_ , vol. 208, p. 108990, 2023. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0165168423000646