[3]Sudhan Majhi

1]Department of Mathematics, IIT Patna, Patna, 801106, Bihar, India

2]Department of Informatics, Selmar Center, University of Bergen, N-5020
Bergen, Norway

[3]Department of Electrical Communication Engineering, IISC, Bangalore,
560012, Karnataka, India

# Construction of Spectrally-Null-Constrained Zero-Correlation Zone Sequences
with Flexible Support

Nishant Kumar<EMAIL_ADDRESS>Palash Sarkar<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [  *

###### Abstract

In recent years, traditional zero-correlation zone (ZCZ) sequences are being
studied due to support interference-free quasi-synchronous code division
multiple access (QS-CDMA) systems. However, in cognitive radio (CR) network,
it is desirable to design ZCZ sequences having spectral null constraint (SNC)
property to achieve low spectral density profile. This paper focuses on the
construction of SNC-ZCZ sequences having flexible support, where support
refers to a collection of indices corresponding to non-zero entries in the
sequence. The proposed SNC-ZCZ sequences reduce to traditional ZCZ sequences
when the size of the support becomes equal to the length of the sequence. To
obtain ZCZ sequences, we first propose construction of traditional/SNC-
Complete complementary codes (SNC-CCCs) using a class of extended Boolean
functions (EBFs). With the help of this class we propose another class of EBFs
that generates asymptotically optimal traditional/SNC-ZCZ sequences of prime-
power lengths with respect to Tang-Fan-Matsufuzi bound. Furthermore, a
relation between the second-order cosets of first-order generalized Reed-
Muller code and the proposed traditional ZCZ sequences is also established.

###### keywords:

Quasi-synchronous code division multiple access (QS-CDMA), spectrally null
constraint (SNC), complete complementary code (CCC), extended Boolean
functions (EBFs), ZCZ sequences, generalized Reed-Muller (GRM) code.

##  1 Introduction

The construction of sequences possessing favorable correlation properties
assumes a crucial significance in the realm of wireless communications and
radar sensing. Such sequences find widespread utility across various
applications, including but not limited to channel estimation,
synchronization, spread spectrum communications, surveillance, and ranging
[1]. The polyphase sequences having zero-correlation zone (ZCZ) properties
have been found suitable for application in quasi-synchronous-code division
multiple access (QS-CDMA) system to provide interference free environment. A
ZCZ is characterized by a region of zero auto- and cross-correlations centered
around the in-phase timing position. These traditional ZCZ sequences (having
non-zero entries) assume that there are continuous spectral bands available,
so the energy of the sequence can be distributed across all the carriers
within that spectral band. Since there’s a limited and valuable amount of
available radio spectrum for wireless communication and radar sensing,
researchers are trying to find better ways to use it efficiently [2]. This
emerging concept is referred to as cognitive radio (CR) [3] or cognitive radar
[4]. A significant challenge in cognitive radio (CR) systems is achieving
interference-free multiple access communications with a low spectral density
profile and robust anti-jamming capability. In pursuit of this goal, a
potential strategy involves the utilization of spread-spectrum CDMA
techniques, employing sequences possessing the ZCZ property. Exploiting the
correlation attribute of ZCZ sequences, a QS-CDMA system utilizing ZCZ
sequences as its spreading codes can achieve interference-free performance,
contingent on the condition that all received signals align with the ZCZ [5].
In the literature, ZCZ sequences were first proposed in [1], and have been
well developed by many researchers [6, 7, 8, 9, 10, 11]. Nonetheless,
traditional ZCZ sequences encounter a challenge when applied directly in CR
systems. This arises due to their design presumption, which typically assumes
the presence of the entire spectral band, rather than the specific non-
contiguous spectral bands dictated by the spectrum hole constraint within a CR
system. Imposing this constraint for spectral nulling jeopardizes the ZCZ
property within a traditional ZCZ sequence set, leading to potential loss or
degradation of its intended benefits. In this paper, sequences with
constraints of spectral nulls are referred to as spectrally-null-constrained
(SNC) sequences. The term ”support” pertains to a set of indices that
correspond to elements within the sequence that have non-zero values.
Therefore, SNC-Sequences can also be defined as the sequences whose size of
the support is not equal to the length of the sequence. Additionally, if the
size of the support equals to the length of the sequence, then the sequence is
referred to as a traditional sequence.

In the realm of systematic SNC sequence construction, the pioneering work of
Tsai _et al._ [12] introduced the initial SNC-ZCZ family by utilizing a comb-
like sequence distribution in the frequency domain. In 2018, Liu _et al._ [13]
achieved SNC sequences with minimal autocorrelation values through carefully
selected ternary frequency-domain duals possessing zero periodic
autocorrelation sidelobes. A recent contribution by Tian _et al._ [14]
presented an analytical construction of SNC sequences in single-channel
polyphase configurations. The work by Li _et al._ [15] proposed the
construction of both SNC-ZCZ sequences and SNC-Z-periodic complementary set
(SNC-ZPCS) sequences. Furthermore, in a recent study, Ye _et al._ [16]
introduced a method for constructing SNC-ZCZ sequences based on a circular
Florentine rectangle and an interleaving approach.

This paper is driven by the limited availability of design approaches for SNC-
ZCZ sequences in the existing literature. Due to their mathematical
properties, Extended Boolean functions (EBFs) are widely recognized in the
literature for their ability to generate sequences with flexible parameters.
This motivates us to study EBFs as a way to overcome the current limitations
in achieving flexible parameters for traditional/SNC-ZCZ sequences. In this
study, we first propose a class of EBFs to generate traditional/SNC-CCC which
further been extended to another class of EBFs to achieve traditional/SNC-ZCZ
sequences of parameter $(p^{t},(p-1)p^{n},p^{n+t+1})$. The proposed ZCZ
sequences has length in the form of prime-power and alphabet set
$\mathbb{Z}_{p}$. Furthermore, the proposed traditional/SNC-ZCZ sequences are
asymptotically optimal as the obtained parameters attain the Tang-Fan-
Matsufuzi bound asymptotically. Moreover, we also establish a relationship
between traditional ZCZ sequences and GRM codes. The remaining sections of
this paper are structured as follows.

Section 2 presents some basic definitions and notations which will be used
throughout the paper. The proposed constructions of traditional/SNC-CCC and
traditional/SNC-ZCZ sequences are provided in Section 3 and Section 4
respectively. Section 5 establishes the relation between proposed traditional
ZCZ sequences and GRM codes. In Section 6 we compare the proposed construction
with some existing construction. Lastly, we conclude the paper in Section 7.

##  2 Preliminary

###  2.1 Extended Boolean Functions (EBF) and Corresponding Sequences

For any $b\geq 2$, an extended Boolean function (EBF) $f(\mathbf{x})$ is
defined as a function $f:\mathbb{Z}_{b}^{t}\rightarrow$ $\mathbb{Z}_{b}$,
where $\mathbf{x}=\left(x_{1},x_{2},\ldots,x_{t}\right)$ is the $b$-ary
representation of the integer $x=\sum_{k=1}^{t}x_{k}b^{k-1}$. For a given EBF
$f$, let $f_{i}=f\left(i_{1},i_{2},\ldots,i_{t}\right)$, and define the
$\mathbb{Z}_{b}$-valued sequence $\xi(f)$ of length $b^{t}$ associated with
$f$ as

$\xi(f)=\left[f_{0},f_{1},\ldots,f_{b^{t}-1}\right],$ (1)

and complex valued sequence associated with $f$ as

$\psi(f)=[\omega^{f_{0}},\omega^{f_{1}},\ldots,\omega^{f_{b^{\color[rgb]{0,0,0}t}-1}}],$
(2)

where $\omega=\exp(2\pi\iota/b)$.

###  2.2 Definition and Correlation Functions

Let $\boldsymbol{a}=\left(a_{0},a_{1},\cdots,a_{N-1}\right)$ and
$\boldsymbol{b}=\left(b_{0},b_{1},\cdots,b_{N-1}\right)$ be two complex-valued
sequences of equal length $L$. For an integer $u$, define

$\gamma(\boldsymbol{a},\boldsymbol{b})(u)=\begin{cases}\sum_{i=0}^{L-1-u}a_{i}b^{*}_{(i+u)},&0\leq
u<L,\\\ \sum_{i=0}^{L+u-1}a_{(i-u)}b^{*}_{i},&-L<u<0.\end{cases}$ (3)

The functions $\gamma(\boldsymbol{a},\boldsymbol{b})$ is called aperiodic
cross-correlation function (ACCF) of $\boldsymbol{a}$ and $\boldsymbol{b}$.
Moreover, if $\boldsymbol{a}=\mathbf{b}$, then this function is called an
aperiodic auto-correlation function (AACF).

Further, the periodic cross-correlation function (PCCF) of $\boldsymbol{a}$
and $\boldsymbol{b}$ is defined as

$\phi(\boldsymbol{a},\boldsymbol{b})(u)=\begin{cases}\sum_{i=0}^{L}a_{i}b^{*}_{(i+u)\mod~{}L},&0\leq
u<L,\\\ \sum_{i=0}^{L}a_{(i-u)\mod~{}L}b^{*}_{i},&-L<u<0.\end{cases}$ (4)

When $\boldsymbol{a}=\boldsymbol{b}$, then this function is called periodic
auto-correlation function (PACF). We can also define PCCF in terms of ACCF as

${\color[rgb]{0,0,0}\phi}(\boldsymbol{a},\boldsymbol{b})(u)=\gamma(\boldsymbol{a},\boldsymbol{b})(u)+\gamma^{*}(\boldsymbol{b},\boldsymbol{a})(L-u).$
(5)

For a sequence $\boldsymbol{a}$, its polynomial representation can be given by

$p_{\boldsymbol{a}}(z)=a_{0}+a_{1}z+\ldots+a_{N-1}z^{N-1},$ (6)

where $z\in\\{e^{j2\pi t:~{}0\leq t<1}\\}$ is a complex number. Further. the
following property is easy to derive

$\arrowvert
p_{\boldsymbol{a}}(z)\arrowvert^{2}=~{}\gamma(\boldsymbol{a})(0)+\sum_{u=1}^{N-1}{(\gamma_{\boldsymbol{a}}(u)z^{-u}+\gamma^{*}_{\boldsymbol{a}}(u)z^{u})}.$
(7)

######  Definition 1.

Let $\mathbf{C}=\\{\mathbf{C}_{0},\mathbf{C}_{1},\ldots,\mathbf{C}_{K-1}\\}$
be a collection of $K$ matrices (codes) of order $M\times L$ . Define
$\mathbf{C_{\mu}}=[(\mathbf{a}_{0}^{\mu})^{T},(\mathbf{a}_{1}^{\mu})^{T},\ldots,(\mathbf{a}_{M-1}^{\mu})^{T}]^{T}$,
where $\mathbf{a}_{\nu}^{\mu}$ ($0\leq\nu\leq M-1,0\leq\mu\leq K-1$) is the
$\nu$th row sequence or $\nu$th constituent sequence and $T$ is transpose
operator. Then the ACCF of two codes
$\mathbf{C_{\mu_{1}}},\mathbf{C_{\mu_{2}}}\in\textbf{C}$ is defined as

$\gamma(\mathbf{C_{\mu_{1}}},\mathbf{C_{\mu_{2}}})(u)=\sum_{\nu=0}^{M-1}{\gamma(\mathbf{a}_{\nu}^{\mu_{1}},\mathbf{a}_{\nu}^{\mu_{2}})(u)}.$
(8)

######  Definition 2.

Let C be a code set defined as above and satisfying the following correlation
properties,

$\gamma(\mathbf{C_{\mu_{1}}},\mathbf{C_{\mu_{2}}})(u)=\begin{cases}{\color[rgb]{0,0,0}LM},&u=0,\
\mu_{1}=\mu_{2},\\\ 0,&0<|u|<L,\ \mu_{1}=\mu_{2},\\\
0,&|u|<L,\mu_{1}\neq\mu_{2}.\end{cases}$ (9)

Then C is said to be a $(K,M,L)$-MOGCS. Moreover, C is said to be CCC set if
$K=M$ and we denote it as $(K,K,L)$-CCC. Also, each code of C is called a GCS
[17].

For a GCS $\mathbf{C_{\mu}}$, following property holds [18]

$\arrowvert p_{\mathbf{a}_{0}^{\mu}}(z)\arrowvert^{2}+\arrowvert
p_{\mathbf{a}_{1}^{\mu}}(z)\arrowvert^{2}+\ldots+\arrowvert
p_{\mathbf{a}_{M-1}^{\mu}}(z)\arrowvert^{2}=~{}\text{Constant}.$ (10)

######  Definition 3.

Let
$\boldsymbol{\mathscr{Z}}=\\{\mathbf{z}_{0},\mathbf{z}_{1},\ldots,\mathbf{z}_{N-1}\\}$
be a set of $N$ sequences of length $L$ each, i.e.,
$\mathbf{z}_{i}=(\mathbf{z}_{i0},\mathbf{z}_{i1},\ldots,\mathbf{z}_{iL-1}),\ \
0\leq i\leq N-.$ Then, $\boldsymbol{\mathscr{Z}}$ is said to be $(N,Z,L)$-ZCZ
sequence set if for $0\leq i,j\leq N-1$, ${\boldsymbol{\mathscr{Z}}}$
satisfies following,

$\phi(\mathbf{z}_{i},\mathbf{z}_{j})(u)=\begin{cases}0,&i=j\ \text{and}\
1\leq|u|\leq Z,\\\ 0,&i\neq j\ \text{and}\ 0\leq|u|\leq Z,\\\ L,&i=j\
\text{and}\ u=0,\end{cases}$ (11)

where $Z$ is called ZCZ width.

######  Definition 4.

_(Tang-Fan-Matsufuji Bound[19])_ Let $\boldsymbol{\mathscr{Z}}$ be any ZCZ
sequence set with parameter $(N,Z,L)$. Then, $N(Z+1)\leq L.$ If
$\boldsymbol{\mathscr{Z}}$ achieves the Tang-Fan-Matsufuji Bound with equality
then $\boldsymbol{\mathscr{Z}}$ is said to be optimal. Further, if for the
larger value of $N$, $N(Z+1)\approx L$, then $\boldsymbol{\mathscr{Z}}$ is
said to be asymptotically optimal. However, for the binary case, it is widely
accepted that the bound is reduced to $2NZ\leq L$ [20].

###  2.3 Restricted and Squeezed Vector/Sequence

Let $p$ be a prime and $f:\mathbb{Z}_{p}^{n}\rightarrow$ $\mathbb{Z}_{p}$ be
an EBF with $n$ variables $x_{1},x_{2},\cdots,x_{n}$. Let
$J=\left(j_{1},j_{2},\cdots,j_{k}\right)$ be a list of $k$ indices with $1\leq
j_{1}<\cdots<j_{k}\leq n$ and write $\mathbf{x}_{J}=$
$\left(x_{j_{1}},x_{j_{2}},\cdots,x_{j_{k}}\right)$. Let
$\mathbf{c}=\left(c_{1},c_{2},\cdots,c_{k}\right)$ be a $p$-ary vector of
length $k$. Then we define the restricted complex valued sequence,
$\psi(\left.f\right|_{\mathbf{x}_{J}=\mathbf{c}})$, corresponding to
$\left.f\right|_{\mathbf{x}_{J}=\mathbf{c}}$ with component
$i=\sum_{j=1}^{m}i_{j}p^{j-1}$ equal to $\omega^{f_{i}}$ if
$i_{j_{\alpha}}=c_{\alpha}$ for $1\leq\alpha\leq k$, and equal to 0 otherwise.
The indices of the non-zero entries in the restricted vector is given by

$i=\sum_{\alpha=0}^{k-1}c_{\alpha}p^{j_{\alpha}}+\sum_{j\notin
J}i_{j}p^{j},\quad i_{j}\in\mathbb{Z}_{p}.$

It is clear that restricted sequences contains zero entries. So, If we remove
all the zeros from a restricted sequence then we call remaining sequence as a
squeezed sequence. It is clear that squeezed sequence has length $p^{n-k}$.

###  2.4 Spectrally-Null-Constrained (SNC) Sequences [21]

Let $\boldsymbol{a}=\left(a_{0},a_{1},\cdots,a_{N-1}\right)$ be a complex-
valued sequence, the support of $a$ is
$\Omega=\left\\{x\in\mathbb{Z}_{N}:a_{x}\neq 0\right\\}$. If
$\Omega^{c}=\left\\{x\in\mathbb{Z}_{N}:x\notin\Omega\right\\}$ is not an empty
set, $\boldsymbol{a}$ is called a spectrally null constrained (SNC) sequence;
otherwise, it is called a traditional sequence. ZCZ sequences such that each
sequence is SNC are called SNC-ZCZ sequences. Correspondingly, if there is at
least one SNC sequence in the CCC, the CCC is referred to as an SNC-CCC.

###  2.5 Generalised Reed-Muller (GRM) Code

For a prime $p$, consider the polynomial ring
$\mathbb{F}_{p}[x_{1},x_{2},\ldots,x_{l}]$ with $l$ variables where
$\mathbb{F}_{p}$ is a field with $p$ elements. GRM codes with parameters $l$
and $\vartheta$ consist of all the sequences corresponding to polynomials with
$l$ variables and degree no larger than $\vartheta$.

######  Definition 5.

The $\vartheta$-th order $\vartheta\leq p$, $p$-ary GRM code,
$GRM_{p}(l,\vartheta)$, is defined as the following set of $p$-ary vectors

$GRM_{p}(l,\vartheta)=\\{\xi(f):f\in\mathbb{F}_{p}[x_{1},x_{2},\ldots,x_{l}],~{}deg(f)<\vartheta\\}.$
(12)

Furthermore, it is noteworthy that we always have $x_{i}^{p}=x_{i}$ in
$\mathbb{F}_{p}$, so we only need to consider the polynomials in which the
degree of each $x_{i}$ is no larger than $p-1$. All such polynomials with a
degree no larger than $\vartheta$ are linear combinations of the following set
of monomials

$\mathcal{N}(r)=\Big{\\{}x_{1}^{j_{1}}x_{2}^{j_{2}},\ldots
x_{l}^{j_{l}}:j=0,1,2\ldots,p-~{}\text{and}~{}\sum_{i=1}^{l}{j_{i}}\leq\vartheta\Big{\\}}.$
(13)

Using combinatorics, we can always prove that the number of monomials in the
set is $\mid\mathcal{N}\mid=\sum_{d=0}^{\vartheta}{{l-1+d\choose d}}$. It is
well known that $\mathbb{F}_{p}[x_{1},x_{2},\ldots,x_{l}]$ forms a vector
space over the field $\mathbb{F}_{p}$. Further, it can be proved that
$GRM_{p}(l,\vartheta)$ is its subspace. Therefore, $GRM_{p}(l,\vartheta)$ is a
linear code with code length $p^{l}$ and code dimension
$\sum_{d=0}^{\vartheta}{{l-1+d\choose d}}$. Moreover, if we arrange sequences
corresponding to monomials in $\mathcal{N}$ as the rows of a matrix, then this
matrix forms a generator matrix of $GRM_{p}(l,\vartheta)$.

######  Lemma 1 ([22]).

For $\lambda=p^{n}$ and $n\geq 1$, minimum Hamming distance of
$GRM_{\lambda}(m,r)$ is $(R+1).p^{Q},$ where $R$ is the remainder and $Q$ is
the quotient resulting from dividing $m(p-1)-r$ by $p-1$.

##  3 Proposed Construction of SNC-CCC

In this section, we propose a class of EBFs which leads to the constructions
of SNC-CCC.

Let $n\geq 1,k<n,t\leq n-k$. For
$J=\\{j_{1},j_{2},\ldots,j_{k}\\}\subset\\{1,2,\ldots,n\\}$ and $1\leq
j_{1}<j_{2}<\ldots<j_{k}\leq n$, let us assume
$\mathbf{x}_{J}=(x_{j_{1}},x_{j_{2}},\ldots,x_{j_{k}})$. Consider the set
$\\{1,2,\ldots,n-k\\}$ being partitioned into $t$ sets, namely,
$V_{1},V_{2},\ldots,V_{t}$. Further, let $\pi_{r}$ be a bijective mapping from
$\\{1,2,\ldots,m_{r}\\}$ to $V_{r}$ where $m_{r}=|V_{r}|\geq 1,~{}\forall
r=1,2,\ldots,t.$ Define an EBF $f:\mathbb{Z}_{p}^{n}\rightarrow\mathbb{Z}_{p}$
as

$f\mid_{x_{J}=\mathbf{c}}=\sum_{r=1}^{t}{\sum_{\alpha=1}^{m_{r}-1}{x_{\pi_{r}(\alpha)}x_{\pi_{r}(\alpha+1)}}}+\sum_{j=1}^{n}{d_{\alpha}x_{\alpha}},$
(14)

where $d_{\alpha}\in\mathbb{Z}_{p}$. Further, let $c\in\mathbb{Z}_{p^{k}}$
such that $\mathbf{c}=(c_{1},c_{2},\ldots,c_{k})$ is the $p$-ary
representation of $c$. Now, for $k=0$, we present a result from [23].

######  Lemma 2.

For each $0\leq h\leq p^{t}-1$, consider the ordered set given by

$C_{h}=\left\\{\mathbf{c}_{h}^{s}\equiv\psi\left(f+\sum_{r=1}^{t}{x_{\pi_{r}(1)}s_{r}}+\sum_{r=1}^{t}{x_{\pi_{r}(m_{r})}h_{r}}\right):s_{r}\in\mathbb{Z}_{p}\right\\},$
(15)

where $(h_{1},h_{2},\ldots,h_{t})$ and $(s_{1},s_{2},\ldots,s_{t})$ are vector
representation of $h$ and $s$ respectively. Then, $\mathscr{C}=\\{C_{h}:0\leq
h\leq p^{t}-1\\}$ is a $(p^{t},p^{t},p^{n})$-CCC.

Now we are in the position to state one of the main results of this paper.

######  Theorem 1.

For each $0\leq h\leq p^{t}-1$, consider the ordered set given by

$C_{h}=\left\\{\mathbf{c}_{h}^{s}\equiv\psi\left(f+\sum_{r=1}^{t}{x_{\pi_{r}(1)}s_{r}}+\sum_{r=1}^{t}{x_{\pi_{r}(m_{r})}h_{r}}\Bigg{\arrowvert}_{\mathbf{x}_{J}=\mathbf{c}}\right):s_{r}\in\mathbb{Z}_{p}\right\\},$
(16)

where $(h_{1},h_{2},\ldots,h_{t})$ and $(s_{1},s_{2},\ldots,s_{t})$ are vector
representation of $h$ and $s$ respectively. Then, $\mathscr{C}=\\{C_{h}:0\leq
h\leq p^{t}-1\\}$ is a $(p^{t},p^{t},p^{n})$-SNC-CCC. Further, the support of
each sequence in each code $C_{h}$ is given by
$\Omega=\\{\sum_{i=1}^{k}{c_{i}p^{j_{i}-1}}+\sum_{\alpha=1,\alpha\notin
J}^{n}{b_{\alpha}p^{\alpha-1}}:b_{\alpha}\in\mathbb{Z}_{p}\\}$.

###### Proof.

Let $I=\\{1,2,\ldots,n\\}\setminus J\equiv\\{i_{1},i_{2},\ldots,i_{n-k}\\}$
such that $1\leq i_{1}<i_{2}<\cdots<i_{n-k}$ and
$\mathbf{x}_{I}=\\{x_{i_{1}},x_{i_{2}},\ldots,x_{i_{n-k}}\\}$. Now define a
transformation of variables as $x_{i_{\alpha}}\leftrightarrow y_{\alpha}$,
which implies $\mathbf{x}_{I}=(y_{1},y_{2},\ldots,y_{n-k})\equiv\mathbf{y}$.
Further, define $\mathscr{D}=\\{D_{h}:0\leq h\leq p^{t}-1\\}$, where

$D_{h}=\left\\{g+\sum_{r=1}^{t}{y_{r(1)}s_{r}}+\sum_{r=1}^{t}{y_{r(m_{r})}h_{r}}:s_{r}\in\mathbb{Z}_{p}\right\\},$
(17)

where
$g(\mathbf{y})=f\arrowvert_{\mathbf{x}_{J}=\mathbf{c}}(\mathbf{x}_{I}),~{}x_{\pi_{r}(1)}=y_{r(1)}$
and $x_{\pi_{r}(m_{r})}=y_{r(m_{r})}$. Then by _Lemma_ 15, $\mathscr{D}$ is a
$(p^{t},p^{t},p^{n-k})$-CCC. Let us define a set of polynomials as

$\mathscr{P}_{h}=\left\\{\omega^{d_{h}(\mathbf{y})}z^{K_{0}+\epsilon(\mathbf{y})}:d_{h}(\mathbf{y})\in
D_{h}\right\\},$ (18)

where $K_{0}=\sum_{\alpha=1}^{k}{c_{\alpha}p^{j_{\alpha}-1}}$ and
$\epsilon(\mathbf{y})=\sum_{\alpha=1}^{n-k}{y_{\alpha}p^{i_{\alpha}-1}}$.
Next, we claim following

######  Claim 1.

$\mathscr{P}_{h}$ is the polynomial representation of sequences in $C_{h}$.

To prove the claim, we need to prove following

  1. 1.

Sequences corresponding to polynomials in $\mathscr{P}_{h}$ and sequences in
$C_{h}$ have non-zero entries at the same position.

  2. 2.

Squeezed sequences corresponding to polynomials in $\mathscr{P}_{h}$ are same
as the squeezed sequences corresponding to sequences in $C_{h}$.

Using (16), (17), and (18) both the properties can be inferred directly. Hence
$\mathscr{P}_{h}$ can alternatively be written as

$\mathscr{P}_{h}=\\{p_{\mathbf{c}_{h}^{s}}(z):\mathbf{c}_{h}^{s}\in C_{h}\\}.$
(19)

Now, we prove that $\mathscr{P}_{h}$ is a GCS and a similar calculation can be
done to prove $\mathscr{P}_{h}$ and $\mathscr{P}_{h^{\prime}}$ are mutually
orthogonal for $0\leq h\neq h^{\prime}\leq p^{t}-1$. Consider

$\displaystyle\sum_{s=0}^{p^{t}-1}{p_{\mathbf{c}_{h}^{s}}(z)p_{(\mathbf{c}_{h}^{s})^{*}}(-z)}=$
$\displaystyle\sum_{d_{h}\in
D_{h}}{\sum_{y,y^{\prime}=0}^{p^{n-k}-1}{\omega^{d_{h}(\mathbf{y})-d_{h}(\mathbf{y}^{\prime})}z^{\epsilon(\mathbf{y})-\epsilon(\mathbf{y}^{\prime})}}}$
(20) $\displaystyle=$ $\displaystyle\mathscr{L}_{1}+\mathscr{L}_{2},$

where

$\mathscr{L}_{1}=\sum_{d_{h}\in
D_{h}}{\sum_{y=y^{\prime}=0}^{p^{n-k}-1}{\omega^{d_{h}(\mathbf{y})-d_{h}(\mathbf{y}^{\prime})}z^{\epsilon(\mathbf{y})-\epsilon(\mathbf{y}^{\prime})}}}$
(21)

and

$\mathscr{L}_{2}=\sum_{d_{h}\in D_{h}}{\sum_{y\neq
y^{\prime}}{\omega^{d_{h}(\mathbf{y})-d_{h}(\mathbf{y}^{\prime})}z^{\epsilon(\mathbf{y})-\epsilon(\mathbf{y}^{\prime})}}}.$
(22)

For $y=y^{\prime}$, the value of
$\omega^{d_{h}(\mathbf{y})-d_{h}(\mathbf{y}^{\prime})}z^{\epsilon(\mathbf{y})-\epsilon(\mathbf{y}^{\prime})}=1$.
Hence, $\mathscr{L}_{1}=$constant. Next, let $\beta_{r}=\sum_{d_{h}\in
D_{h}}{\sum_{y-y^{\prime}=r}{\omega^{d_{h}(\mathbf{y})-d_{h}(\mathbf{y}^{\prime})}}}$,
$1-p^{n-k}\leq r\leq p^{n-k}-1$. Then $\mathscr{L}_{2}$ can be rewritten as

$\mathscr{L}_{2}=\sum_{r\neq
0}{\beta_{r}z^{\epsilon(\mathbf{y})-\epsilon(\mathbf{y}^{\prime})}}.$ (23)

Since, $d_{h}\in D_{h}$, therefore by the property of GCS, $\beta_{r}=0$,
$\forall r$. Also, it can be noted that when $y-y^{\prime}=r$, then
$\mathbf{y}-\mathbf{y}^{\prime}$ is some fixed vector and hence,
$\epsilon(\mathbf{y}-\mathbf{y}^{\prime})=\sum_{\alpha=1}^{n-k}{(y_{\alpha}-y^{\prime}_{\alpha})p^{i_{\alpha}-1}}$
is a constant which implies that $\mathscr{L}_{2}=0$. Therefore, from (20)

$\sum_{s=0}^{p^{t}-1}{p_{\mathbf{c}_{h}^{s}}(z)p_{(\mathbf{c}_{h}^{s})^{*}}(-z)}=\text{constant}.$
(24)

Thus, $\mathscr{P}_{h}$ is a GCS. ∎

######  Example 1.

For $n=6$, $J=\\{3\\}$, let $\\{1,2,4,5,6\\}$ is partitioned into $2$ sets,
i.e., $V_{1}=\\{1,2\\}$ and $V_{2}=\\{4,5,6\\}$ with
$\pi_{1}(1)=1,~{}\pi_{1}(2)=2,~{}\pi_{2}(1)=5,~{}\pi_{2}(2)=4,$ and
$\pi_{2}(3)=6$. Let $f:\mathbb{Z}_{2}^{6}\rightarrow\mathbb{Z}_{2}$ be defined
as $f=x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{4}+x_{4}x_{5}+x_{4}x_{6}$, then
$f\arrowvert_{x_{3}=1}=x_{1}x_{2}+x_{4}x_{5}+x_{4}x_{6}+x_{2}+x_{4}$. For
$0\leq h\leq 3$, define

$C_{h}=\left\\{(f+x_{1}s_{1}+x_{5}s_{2}+x_{2}h_{1}+x_{6}h_{2})\arrowvert_{x_{3}=1}:s_{1},s_{2}\in\\{0,1\\}\right\\}.$
(25)

Then $\mathscr{C}=\\{C_{0},C_{1},C_{2},C_{3}\\}$ is a $(4,4,64)$-SNC-CCC as
shown in _Table_ 1.

Table 1: SNC-CCC corresponding to _Example_ 1 $C_{0}$ | 000011-110000-1-11-1000011-11000011-11000011-11000011-11000011-110000-1-11-1
---|---
00001-1-1-10000-111100001-1-1-100001-1-1-100001-1-1-100001-1-1-100001-1-1-10000-1111
000011-110000-1-11-10000-1-11-10000-1-11-1000011-11000011-110000-1-11-1000011-11
00001-1-1-10000-11110000-11110000-111100001-1-1-100001-1-1-10000-111100001-1-1-1
$C_{1}$ | 0000111-10000-1-1-110000111-10000111-10000111-10000111-10000111-10000-1-1-11
00001-1110000-11-1-100001-11100001-11100001-11100001-11100001-1110000-11-1-1
0000111-10000-1-1-110000-1-1-110000-1-1-110000111-10000111-10000-1-1-110000111-1
00001-1110000-11-1-10000-11-1-10000-11-1-100001-11100001-1110000-11-1-100001-111
$C_{2}$ | 000011-110000-1-11-1000011-11000011-110000-1-11-10000-1-11-10000-1-11-1000011-11
00001-1-1-10000-111100001-1-1-100001-1-1-10000-11110000-11110000-111100001-1-1-1
000011-110000-1-11-10000-1-11-10000-1-11-10000-1-11-10000-1-11-1000011-110000-1-11-1
00001-1-1-10000-11110000-11110000-11110000-11110000-111100001-1-1-10000-1111
$C_{3}$ | 0000111-10000-1-1-110000111-10000111-10000-1-1-110000-1-1-110000-1-1-110000111-1
00001-1110000-11-1-100001-11100001-1110000-11-1-10000-11-1-10000-11-1-100001-111
0000111-10000-1-1-110000-1-1-110000-1-1-110000-1-1-110000-1-1-110000111-10000-1-1-11
00001-1110000-11-1-10000-11-1-10000-11-1-10000-11-1-10000-11-1-100001-1110000-11-1-1

##  4 Proposed construction of SNC-ZCZ Sequences

In the next theorem, a construction of SNC-ZCZ sequence sets is presented.
Before presenting the theorem, let us define an EBF
$g(x_{n+1},x_{n+2},\ldots,x_{n+t+1})$ on $t+1$ variables as

$\begin{split}g(x_{n+1},x_{n+2},\ldots,x_{n+t+1})=\sum_{r=2}^{t+1}{c_{r}x_{n+r}x_{n+1}}+\sum_{2\leq\mu<\nu\leq
t}{d_{\mu\nu}x_{n+\mu}x_{n+\nu}}+\sum_{\beta=1}^{t+1}{e_{\beta}x_{n+\beta}}+e^{\prime},\end{split}$
(26)

$\text{where}\ c_{r},d_{\mu\nu},e_{\beta},e^{\prime}\in\mathbb{Z}_{p},\
{\color[rgb]{0,0,0}c_{t+1}\neq 0}\ $.

######  Remark 1.

From (5), it can be verified that $\mathscr{C}$ is also a CCC with respect to
periodic correlation. This idea has been utilized in the proof of _Theorem_ 2.

######  Theorem 2.

For $0\leq c\leq p^{k}-1$, define a set
$\boldsymbol{\mathscr{Z}}=\\{\mathbf{z}_{h}:\ h=0,1,\ldots,p^{t}-1\\}$ such
that

$\mathbf{z}_{h}=\psi\left(\left(f+g+\sum_{r=1}^{t}{x_{\pi_{r}(1)}x_{n+r}}+\sum_{r=1}^{t}{x_{\pi_{r}(m_{r})}h_{r}}\right)\Bigg{\arrowvert}_{\mathbf{x}_{J}=\mathbf{c}}\right),$
(27)

where $h=(h_{1},h_{2},\ldots,h_{t})\cdot(1,p,\ldots,p^{t-1})$ and
$c=\mathbf{c}\cdot(1,p,\ldots,p^{k-1})$. Then, the set
$\boldsymbol{\mathscr{Z}}$ is a $(p^{t},(p-1)p^{n},p^{n+t+1})$-SNC-ZCZ.
Further, the support of each sequence $\mathbf{z}_{h}$ is given by
$\Omega=\\{\sum_{i=1}^{k}{c_{i}p^{j_{i}-1}}+\sum_{\alpha=1,\alpha\notin
J}^{n+t+1}{b_{\alpha}p^{\alpha-1}}:b_{\alpha}\in\mathbb{Z}_{p}\\}$.

###### Proof.

Let $\mathbf{x}_{T}=[x_{n+1},x_{n+2},\ldots,x_{n+t+1}]$,
$\mathbf{l}=[l_{n+1},l_{n+2},\ldots,l_{n+t+1}],\mathbf{l}^{\prime}=[l_{n+1}^{\prime},l_{n+2}^{\prime},$
$\ldots,l_{n+t+1}^{\prime}]$,
$\mathbf{l}_{1}=[l_{n+1},l_{n+2},\ldots,l_{n+t}]$,
$\mathbf{l}^{\prime}_{1}=[l^{\prime}_{n+1},l^{\prime}_{n+2},\ldots,l^{\prime}_{n+t}]$.
For $0\leq h,h^{\prime}\leq p^{t}-1$, the PCCF of $\mathbf{z}_{h}$ and
$\mathbf{z}_{h^{\prime}}$ is given by

$\displaystyle P$
$\displaystyle=\phi\left(\mathbf{z}_{h},\mathbf{z}_{h^{\prime}}\right)(u)$
(28)
$\displaystyle=\phi(f+g+\sum_{r=1}^{t}x_{\pi_{r}(1)}x_{n+r}+\left.\sum_{r=1}^{t}x_{\pi_{r}\left(m_{r}\right)}h_{r}\right|_{\mathbf{x}_{J}=\mathbf{c}},$
$\displaystyle\hskip
128.0374pt\left.f+g+\sum_{r=1}^{t}x_{\pi_{r(1)}}x_{n+r}+\left.\sum_{r=1}^{t}x_{\pi_{r}\left(m_{r}\right)}h_{r}^{\prime}\right|_{\mathbf{x}_{J}=\mathbf{c}}\right)(u)$
$\displaystyle=\sum_{\mathbf{l},\mathbf{l}^{\prime}}{\omega_{p}^{\delta}\phi\left(f+\sum_{r=1}^{t}x_{\pi_{r}(1)}l_{n+r}+\left.\sum_{r=1}^{t}x_{\pi_{r}\left(m_{r}\right)}h_{r}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}},\right.}$
$\displaystyle\hskip
128.0374pt\left.f+\sum_{r=1}^{t}x_{\pi_{r(1)}}l^{\prime}_{n+r}+\left.\sum_{r=1}^{t}x_{\pi_{r}\left(m_{r}\right)}h_{r}^{\prime}\right|_{{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}^{\prime}}}\right)(u),$

where,

$\delta=\sum_{r=2}^{t+1}c_{r}(l_{n+r}l_{n+1}-l_{n+r}^{\prime}l_{n+1}^{\prime})+\sum_{2\leq\mu<\nu\leq
t}{d_{\mu\nu}(l_{n+\mu}l_{n+\nu}-l_{n+\mu}^{\prime}l_{n+\nu}^{\prime})}+\sum_{\beta=1}^{t+1}{e_{\beta}(l_{n+\beta}-l_{n+\beta}^{\prime})}.$
(29)

$P$ can further be simplified to,

$\displaystyle P=$
$\displaystyle\sum_{\mathbf{l}_{1},\mathbf{l}^{\prime}_{1}}\omega_{p}^{\delta^{\prime}}\sum_{l_{n+t+1},l_{n+t+1}^{\prime}}\omega_{p}^{c_{t+1}(l_{n+t+1}l_{n+1}-l_{n+t+1}^{\prime}l_{n+1}^{\prime})+e_{t+1}(l_{n+t+1}-l_{n+t+1}^{\prime})}$
(30)
$\displaystyle\times\phi\left(f+\sum_{r=1}^{t}x_{\pi_{r}(1)}l_{n+r}+\left.\sum_{r=1}^{t}x_{\pi_{r}\left(m_{r}\right)}h_{r}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}},\right.$
$\displaystyle\hskip
128.0374pt\left.f+\sum_{r=1}^{t}x_{\pi_{r(1)}}l^{\prime}_{n+r}+\left.\sum_{r=1}^{t}x_{\pi_{r}\left(m_{r}\right)}h_{r}^{\prime}\right|_{{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}^{\prime}}}\right)(u),$

where,

$\delta^{\prime}=\sum_{r=2}^{t}c_{r}(l_{n+r}l_{n+1}-l_{n+r}^{\prime}l_{n+1}^{\prime})+\sum_{2\leq\mu<\nu\leq
t}{d_{\mu\nu}(l_{n+\mu}l_{n+\nu}-l_{n+\mu}^{\prime}l_{n+\nu}^{\prime})}+\sum_{\beta=1}^{t}{e_{\beta}(l_{n+\beta}-l_{n+\beta}^{\prime})}.$
(31)

Now we have two cases

######  Case 1 ($l_{n+t+1}=l_{n+t+1}^{\prime}$).

Let $l_{n+1}=l^{\prime}_{n+1}$. Since, $0\leq u\leq(p-1)p^{n}$, therefore, we
have $l_{n+r}=l^{\prime}_{n+r}\ \forall r=2,3,\ldots,t$. Hence, (30) will
reduce to

$\displaystyle P$
$\displaystyle=\sum_{l_{n+t+1}}\sum_{\mathbf{l}_{1}}\phi\left(f+\sum_{r=1}^{t}x_{\pi_{r}(1)}l_{n+r}+\sum_{r=1}^{t}x_{\pi_{r}(m_{r})}h_{r}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}},$
(32) $\displaystyle\hskip
113.81102pt\left.\left.f+\sum_{r=1}^{t}x_{\pi_{r(1)}}l_{n+r}+\sum_{r=1}^{t}x_{\pi_{r}\left(m_{r}\right)}h_{r}^{\prime}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}}\right)(u).$

From _Theorem_ 1 the value of

$\displaystyle\sum_{\mathbf{l}_{1}}\phi\left(f+\sum_{r=1}^{t}x_{\pi_{r}(1)}l_{n+r}+\sum_{r=1}^{t}x_{\pi_{r}(m_{r})}h_{r}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}},$
(33) $\displaystyle\hskip
113.81102pt\left.\left.f+\sum_{r=1}^{t}x_{\pi_{r(1)}}l_{n+r}+\sum_{r=1}^{t}x_{\pi_{r}\left(m_{r}\right)}h_{r}^{\prime}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}}\right)(u)$

is zero for all non-zero time shifts. Further, if $l_{n+1}\neq
l_{n+1}^{\prime}$ then (30) will reduce to

$\displaystyle P$
$\displaystyle=\sum_{\mathbf{l}_{1},\mathbf{l}^{\prime}_{1}}\omega_{p}^{\delta^{\prime}}\sum_{l_{n+t+1}}\omega_{p}^{c_{t+1}l_{n+t+1}(l_{n+1}-l_{n+1}^{\prime})}\phi\left(f+\sum_{r=1}^{t}x_{\pi_{r}(1)}l_{n+r}+\sum_{r=1}^{t}x_{\pi_{r}(m_{r})}h_{r}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}},$
(34) $\displaystyle\hskip
142.26378pt\left.\left.f+\sum_{r=1}^{t}x_{\pi_{r(1)}}l_{n+r}+\sum_{r=1}^{t}x_{\pi_{r}(m_{r})}h_{r}^{\prime}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}^{\prime}}\right)(u)$
$\displaystyle=\sum_{\mathbf{l}_{1},\mathbf{l}^{\prime}_{1}}\omega_{p}^{\delta^{\prime}}\phi\left(f+\sum_{r=1}^{t}x_{\pi_{r}(1)}l_{n+r}+\sum_{r=1}^{t}x_{\pi_{r}(m_{r})}h_{r}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}},$
$\displaystyle\hskip
28.45274pt\left.\left.f+\sum_{r=1}^{t}x_{\pi_{r(1)}}l_{n+r}+\sum_{r=1}^{t}x_{\pi_{r}(m_{r})}h_{r}^{\prime}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}^{\prime}}\right)(u)\sum_{l_{n+t+1}}\omega_{p}^{c_{t+1}l_{n+t+1}(l_{n+1}-l_{n+1}^{\prime})}.$

Since
$\sum_{l_{n+t+1}}\omega_{p}^{c_{t+1}l_{n+t+1}(l_{n+1}-l_{n+1}^{\prime})}=0$.
Therefore, $P=0$.

######  Case 2 ($l_{n+t+1}\neq l_{n+t+1}^{\prime}$).

For $0\leq u\leq(p-1)p^{n}$, $l_{n+t+1}-l_{n+t+1}^{\prime}=-1\mod p$ and
$l_{n+1}-l_{n+1}^{\prime}=-1\mod p$. This implies that
$l_{n+t+1}l_{n+t+1}^{\prime}-l_{n+1}l_{n+1}^{\prime}=-1-l_{n+t+1}-l_{n+1}$.
Now (30) can be written as

$\displaystyle P$
$\displaystyle=\sum_{\mathbf{l}_{1},\mathbf{l}^{\prime}_{1}}\omega_{p}^{\delta^{\prime}}\sum_{l_{n+t+1}}\omega_{p}^{c_{t+1}(-1-l_{n+t+1}-l_{n+1})-e_{t+1}}\phi\left(f+\sum_{r=1}^{t}x_{\pi_{r}(1)}l_{n+r}\right.$
(35) $\displaystyle\hskip
28.45274pt\left.\left.\left.+\sum_{r=1}^{t}x_{\pi_{r}(m_{r})}h_{r}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}},f+\sum_{r=1}^{t}x_{\pi_{r(1)}}l_{n+r}+\sum_{r=1}^{t}x_{\pi_{r}(m_{r})}h_{r}^{\prime}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}^{\prime}}\right)(u)$
$\displaystyle=\sum_{\mathbf{l}_{1},\mathbf{l}^{\prime}_{1}}\omega_{p}^{\delta^{\prime}-c_{t+1}(1+l_{n+1})-e_{t+1}}\phi\left(f+\sum_{r=1}^{t}x_{\pi_{r}(1)}l_{n+r}+\sum_{r=1}^{t}x_{\pi_{r}(m_{r})}h_{r}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}},$
$\displaystyle\hskip
56.9055pt\left.\left.f+\sum_{r=1}^{t}x_{\pi_{r(1)}}l_{n+r}+\sum_{r=1}^{t}x_{\pi_{r}(m_{r})}h_{r}^{\prime}\right|_{\mathbf{x}_{J}\mathbf{x}_{T}=\mathbf{c}\mathbf{l}^{\prime}}\right)(u)\sum_{l_{n+t+1}}\omega_{p}^{-c_{t+1}l_{n+t+1}}.$

Since $\sum_{l_{n+t+1}}\omega_{p}^{-c_{t+1}l_{n+t+1}}=0$, therefore, $P=0$.

∎

######  Example 2.

Let $f:\mathbb{Z}_{2}^{3}\rightarrow\mathbb{Z}_{2}$ be an EBF defined as
$f(x_{1},x_{2},x_{3})=x_{1}x_{2}+x_{1}x_{3}+x_{1}+x_{3}$. Further, let
$J=\\{1\\}$ such that $V_{1}=\\{x_{2}\\}$ and $V_{2}=\\{x_{3}\\}$. Now, define
another EBF $g:\mathbb{Z}_{2}^{3}\rightarrow\mathbb{Z}_{2}$ as
$g(x_{4},x_{5},x_{6})=x_{4}x_{5}+x_{4}x_{6}$. For, $c=1$

$\boldsymbol{\mathscr{Z}}=\begin{bmatrix}\mathbf{z}_{0}\\\ \mathbf{z}_{1}\\\
\mathbf{z}_{2}\\\
\mathbf{z}_{3}\end{bmatrix}=\begin{bmatrix}f+g+x_{2}x_{4}+x_{3}x_{5}+x_{2}\cdot
0+x_{3}\cdot 0|_{x_{1}=1}\\\ f+g+x_{2}x_{4}+x_{3}x_{5}+x_{2}\cdot 1+x_{3}\cdot
0|_{x_{1}=1}\\\ f+g+x_{2}x_{4}+x_{3}x_{5}+x_{2}\cdot 0+x_{3}\cdot
1|_{x_{1}=1}\\\ f+g+x_{2}x_{4}+x_{3}x_{5}+x_{2}\cdot 1+x_{3}\cdot
1|_{x_{1}=1}\end{bmatrix}.$ (36)

Table 2: SNC-ZCZ sequences corresponding to _Example_ 2 $\boldsymbol{\mathscr{Z}}$ | 0-1010-1010-10-10-10-10-101010-10-10-101010-1010-10101010101010-10-1010-10-10101
---|---
0-10-10-10-10-1010-1010-10-101010-101010-10-10-10-10101010101010-10-1010-10-10101
0-101010-10-10-101010-1010-1010-10-10-10-10-101010-101010-10-1010-1010-10-10-10-10-1
0-10-101010-101010-10-10-10-10-10-1010-1010-10-10101010-10-101010101010-1010-101

Then, $\boldsymbol{\mathscr{Z}}$ is $(4,8,64)$-SNC-ZCZ sequence set as shown
in _Table_ 2

We have proposed the construction of SNC-ZCZ sequences. Next, corollary
demonstrate the relation between the proposed SNC-ZCZ sequences and
traditional ZCZ sequences.

######  Corollary 1.

In _Theorem_ 2, let $J$ be an empty set. Then the set
$\boldsymbol{\mathscr{Z}}$ is a traditional $(p^{t},(p-1)p^{n},p^{n+t+1})$-ZCZ
sequence set, i.e., $\mid\Omega\mid=p^{n+t+1}$.

######  Remark 2.

It can be verified that for the larger value of $p$, $N(Z+1)\approx L$. Hence,
proposed ZCZ sequences are asymptotically optimal.

######  Remark 3.

For the fixed $J$ and different $\mathbf{c}$, the support of corresponding
SNC-ZCZ sequences is different but have same size. On the other hand, for the
different $J$’s, the corresponding SNC-ZCZ sequences have different support
set as well as different size. This shows that the proposed SNC-ZCZ sequences
have flexible support.

##  5 Relation Between Proposed traditional ZCZ Sequences and GRM Codes

This section establishes a relation between constructed ZCZ sequences and GRM
codes. For ease of notation, we assume $n+t+1=n^{\prime}$ and $\mathbf{z}_{h}$
refers to function instead of sequence as shown in (37). Using _Corollary_ 1
we can write

$\boldsymbol{\mathscr{Z}}=\left\\{\mathbf{z}_{h}=\mathcal{Q}+\sum_{j=1}^{n}{d_{\alpha}x_{\alpha}}+\sum_{\beta=1}^{t+1}{e_{\beta}x_{n+\beta}}+\sum_{r=1}^{t}{x_{\pi_{r}(m_{r})}h_{r}}+e^{\prime}:h_{r}\in\mathbb{Z}_{p}\right\\},$
(37)

where

$\mathcal{Q}=\underbrace{\sum_{r=1}^{t}{\sum_{j=1}^{m_{r}-1}{x_{\pi_{r}(j)}x_{\pi_{r}(j+1)}}}}_{\mathcal{Q}_{1}}+\underbrace{\sum_{r=2}^{t+1}{c_{r}x_{n+r}x_{n+1}}}_{\mathcal{Q}_{2}}+\underbrace{\sum_{2\leq\mu<\nu\leq
t}{d_{\mu\nu}x_{n+\mu}x_{n+\nu}}}_{\mathcal{Q}_{3}}\\\
+\underbrace{\sum_{r=1}^{t}{x_{\pi_{r}(1)}x_{n+r}}}_{\mathcal{Q}_{4}}.$ (38)

In the following corollary, we see that ZCZ sequence sets with fixed quadratic
form $\mathcal{Q}$, partition the second order coset of the first order GRM
code, i.e., $\mathcal{Q}+GRM_{p}(n^{\prime},1)$

######  Corollary 2.

Let $\mathcal{Q}+GRM_{p}(n^{\prime},1)$ be second order coset of the first
order GRM code. Then for a fixed quadratic form $\mathcal{Q}$,
$\mathcal{Q}+GRM_{p}(n^{\prime},1)$ contains $p^{n^{\prime}+1-t}$ distinct ZCZ
sequence sets.

###### Proof.

From equation (37), it is clear that for a fixed quadratic form $\mathcal{Q}$,
the parameters which contribute to different ZCZ sequence sets are
$d_{\alpha},~{}e_{\beta},$ and $e^{\prime}$. But, whenever
$x_{\pi_{r}(m_{r})}=x_{\alpha}$, then respective $d_{\alpha}$ will no longer
contribute to generate different ZCZ sequence sets because
$\sum_{r=1}^{t}x_{\pi_{r}(m_{r})}h_{r}$ contributes to different sequences in
a particular ZCZ sequence sets. Therefore, total variables which contributes
to different ZCZ sequence sets are $n^{\prime}+1-t$. Hence total number of
different ZCZ sequence sets in $\mathcal{Q}+GRM_{p}(n^{\prime},1)$ equals to
$p^{n^{\prime}+1-t}$. ∎

It can be obtained that if we have a sequence in $\boldsymbol{\mathscr{Z}}$,
then it is a codeword belonging to second-order coset
$\mathcal{Q}+GRM_{p}(n^{\prime},1)$ where $\mathcal{Q}$ is given by (38).
Moreover, if any codeword belongs to the coset
$\mathcal{Q}+GRM_{p}(n^{\prime},1)$ then it belongs to a certain
$(p^{t},(p-1)p^{n},p^{n+t+1})$-ZCZ sequence set. Hence ZCZ sequence sets with
fixed quadratic form $\mathcal{Q}$, partition
$\mathcal{Q}+GRM_{p}(n^{\prime},1)$.

In the following corollary, we explicitly determine the number of second-order
cosets representatives having the form $\mathcal{Q}+GRM_{p}(n^{\prime},1)$.

######  Corollary 3.

The number of coset representative $\mathcal{Q}$ in the second-order coset of
the form $\mathcal{Q}+GRM_{p}(n^{\prime},1)$ where $\mathcal{Q}$ is as defined
in (38) is given by

$(p-1)p^{\frac{(t-1)t}{2}}\bigg{(}\sum_{m_{1}+m_{2}+\cdots+m_{t}=n}{\prod_{r=1}^{t}{\frac{m_{r}!}{2}}}\bigg{)}.$
(39)

###### Proof.

One can easily see that the two permutations $\pi_{r}$ and $\pi^{\prime}_{r}$,
where $\pi^{\prime}_{r}(k)=\pi_{r}(m_{r}+k-1)$ will generate the same
quadratic form. Therefore, we have $\frac{m_{r}!}{2}$ different quadratic
forms of the type

$x_{\pi_{r}(1)}x_{\pi_{r}(2)}+x_{\pi_{r}(2)}x_{\pi_{r}(3)}+\cdots+x_{\pi_{r}(m_{r}-1)}x_{\pi_{r}(m_{r})},~{}r=1,2,\ldots
t,$ (40)

where notations are the same as used in _Lemma_ 15. In addition, we have the
condition $m_{1}+m_{2}+\cdots+m_{t}=n$ where $1\leq m_{r}\leq n$. Therefore,
we can explicitly determine
$\sum_{m_{1}+m_{2}+\cdots+m_{t}=n}{\prod_{r=1}^{t}{\frac{m_{r}!}{2}}}$ number
of distinct coset representatives of the type ${\mathcal{Q}_{1}}$. It is to be
noted that once we fix ${\mathcal{Q}_{1}}$ then
${\mathcal{Q}_{2}},{\mathcal{Q}_{3}},~{}\text{and}~{}{\mathcal{Q}_{4}}$ are
also fixed except their coefficients, i.e., $c_{r}$ and $d_{\mu\nu}$ where
$c_{r},d_{\mu\nu}\in\mathbb{Z}_{p}$, $c_{t+1}\neq 0$. Since
${\mathcal{Q}_{2}}$ and ${\mathcal{Q}_{3}}$ contain $t$ and
$\frac{(t-2)(t-1)}{2}$ terms respectively and $c_{t+1}$ can take only $p-1$
values. Therefore, We have
$(p-1)p^{t-1}p^{\frac{(t-2)(t-1)}{2}}=(p-1)p^{\frac{(t-1)t}{2}}$ number of
distinct coset representative of the type
${(\mathcal{Q}_{2}}+{\mathcal{Q}_{3}}+{\mathcal{Q}_{4}})+GRM_{p}(n^{\prime},1)$
for the fixed value of ${\mathcal{Q}_{1}}$. Hence, combining all the
discussion, we can conclude that the total number of coset representative of
the form $\mathcal{Q}+GRM_{p}(n^{\prime},1)$ is

$(p-1)p^{\frac{(t-1)t}{2}}\bigg{(}\sum_{m_{1}+m_{2}+\cdots+m_{t}=n}{\prod_{r=1}^{t}{\frac{m_{r}!}{2}}}\bigg{)}.$

∎

In the next corollary, it has been concluded that the ZCZ sequence sets lying
inside the different cosets of $GRM_{p}(n^{\prime},1)$ are different.

######  Corollary 4.

Let $\boldsymbol{\mathscr{Z}}$ and $\boldsymbol{\mathscr{Z}^{\prime}}$ be two
ZCZ sequence sets obtained from _Corollary_ 1, contained in
$\mathcal{Q}+GRM_{p}(n^{\prime},1)$ and
$\mathcal{Q}^{\prime}+GRM_{p}(n^{\prime},1)$, respectively, where
$\mathcal{Q}$ and $\mathcal{Q}^{\prime}$ are different quadratic forms. Then,
$\boldsymbol{\mathscr{Z}}$ and $\boldsymbol{\mathscr{Z}^{\prime}}$ are
different ZCZ sequence sets.

###### Proof.

It Suffices to prove that the cosets $\mathcal{Q}+GRM_{p}(n^{\prime},1)$ and
$\mathcal{Q}^{\prime}+GRM_{p}(n^{\prime},1)$ are either disjoint or same. Let
us assume that there exist an element $c$ such that
$c\in\mathcal{Q}+GRM_{p}(n^{\prime},1)\cap\mathcal{Q}^{\prime}+GRM_{p}(n^{\prime},1)$.
Then $c=\mathcal{Q}+c_{1}=\mathcal{Q}^{\prime}+c_{2},$ for some
$c_{1},c_{2}\in GRM_{p}(n^{\prime},1).$ Which implies
$\mathcal{Q}-\mathcal{Q}^{\prime}=c_{2}-c_{1}$. Since, $c_{1},c_{2}\in
GRM_{p}(n^{\prime},1)$ and $GRM_{p}(n^{\prime},1)$ is linear code therefore,
$c_{2}-c_{1}\in GRM_{p}(n^{\prime},1)$, which further implies
$\mathcal{Q}-\mathcal{Q}^{\prime}\in GRM_{p}(n^{\prime},1)$. Now for some
$w\in GRM_{p}(n^{\prime},1)$ let $\mathcal{Q}-\mathcal{Q}^{\prime}=w$. Then,

$\mathcal{Q}+GRM_{p}(n^{\prime},1)=\mathcal{Q}^{\prime}+w+GRM_{p}(n^{\prime},1)=\mathcal{Q}^{\prime}+GRM_{p}(n^{\prime},1).$

Which follows the result. ∎

######  Remark 4.

From _Corollary_ 2, each cosets of $GRM_{p}(n^{\prime},1)$ comprises
$p^{n^{\prime}+1-t}$ ZCZ sequence sets. Further, from _Corollary_ 3, the
number of distinct cosets are
$(p-1)p^{\frac{(t-1)t}{2}}\bigg{(}\sum_{m_{1}+m_{2}+\cdots+m_{t}=n}{\prod_{r=1}^{t}{\frac{m_{r}!}{2}}}\bigg{)}$.
Hence, on combining we obtain
$(p-1)p^{\frac{(t-1)t}{2}}\bigg{(}\sum_{m_{1}+m_{2}+\cdots+m_{t}=n}{\prod_{r=1}^{t}{\frac{m_{r}!}{2}}}\bigg{)}p^{n^{\prime}+1-t}$
ZCZ sequence sets.

######  Example 3.

Let $p=3,n=4,t=2$, $f=x_{1}x_{2}+x_{3}x_{4}$, and $g=x_{5}x_{6}+x_{5}x_{7}$.
Further, let $\pi_{1}(1)=1,\pi_{1}(2)=2,\pi_{2}(1)=3$, and $\pi_{2}(2)=4$.
From (37),

$\boldsymbol{\mathscr{Z}}=\left\\{\mathcal{Q}+d_{1}x_{1}+d_{3}x_{3}+e_{1}x_{5}+e_{2}x_{6}+e_{3}x_{7}+(d_{2}+h_{1})x_{2}+(d_{4}+h_{2})x_{4}\right.\\\
\left.+e^{\prime}:h_{1},h_{2}\in\mathbb{Z}_{3}\right\\}.$ (41)

from the above equation, it is clear that for the different values of
$d_{1},d_{3},e_{1},e_{2},e_{3}$, and $e^{\prime}$ the ZCZ sequence set
$\boldsymbol{\mathscr{Z}}$ is different. Hence, the coset
$\mathcal{Q}+GRM_{3}(7,1)$ with $\mathcal{Q}=f+g+x_{1}x_{5}+x_{2}x_{6}$
contains $3^{6}$ distinct $(3^{4},2\cdot 3^{2},3^{7})$-ZCZ sequence sets.

######  Corollary 5.

The minimum Hamming distance of a proposed ZCZ sequence set is at least
$(p-1)p^{n^{\prime}-1}$.

###### Proof.

Using _Lemma_ 1, it can be calculated that the minimum Hamming distance of
$GRM_{p}(n^{\prime},1)$ is $(p-1)p^{n^{\prime}-1}$. From _Corollary_ 2, each
ZCZ sequence set $\boldsymbol{\mathscr{Z}}$ is a subset of a second order
coset of the first order GRM code, i.e., $\mathcal{Q}+GRM_{p}(n^{\prime},1)$.
Now, Since the coset of a code has the same minimum Hamming distance as that
of code. Therefore, minimum Hamming distance of
$\mathcal{Q}+GRM_{p}(n^{\prime},1)$ is $(p-1)p^{n^{\prime}-1}$. Hence, minimum
Hamming distance of $\boldsymbol{\mathscr{Z}}$ is at least
$(p-1)p^{n^{\prime}-1}$. ∎

##  6 Comparison with Existing Works

In this section, we compare our proposed construction of traditional/SNC-ZCZ
sequences with some existing works. A detailed comparison of traditional ZCZ
sequences can be seen in Table 3.

###  6.1 Comparison with Traditional ZCZ Sequences in [6] and [7]

In [6], Chen _et al._ proposed ZCZ sequence sets having parameters
$(2^{k},2^{m-k-1},2^{m})$, where $k\ \text{and}\ m$ are positive integers such
that $m\geq 2,$ and $k\leq m-1$. In our proposed construction, if we take
$p=2,~{}t=k,$ and $n=m-k-1$, then the parameters of our construction meet the
parameters in [6]. Moreover, in [7], authors provided ZCZ sequence sets having
parameters $(2^{k+1},2^{n},2^{n+k+2})$. Construction in [7] was based on GBFs.
Again, these parameters also appear as a special case of our parameters, i.e.,
if we take $p=2$ and $t=k+1$ in our proposed construction, then we get the
parameters of [7]. Hence, our construction covers more lengths and can be seen
as a generalization of these constructions.

Table 3: Comparison with [8],[6],[9], [7], [10], [11], and [24] Reference | Based on | Parameter ($N$,$Z,L$) | Direct/Indirect | Optimality
---|---|---|---|---
[8] | MOGCS | $(2^{n},Z_{cz}+1,2^{n+1}Z_{cz}),\ n\geq 1$ | Indirect | Optimal for binary case
[6] | GBF |  $(2^{k},2^{m-k-1},2^{m}),$ $m\geq 2,$ and $k\leq m-1$. | Direct | Optimal for binary case
[9] | PS111Perfect Sequences. |  $(2(2n+1),4k+1,4(2k+1)(2n+1)),$ $n\geq 1,\ k\geq 1$ | Indirect | Neither optimal nor almost-optimal
[7] | GBF | $(2^{k+1},2^{n},2^{n+k+2}),k\geq 0,n\geq 1$ | Direct | Optimal for binary case
[10] | PNLF222Perfect Non-linear functions. |  $(p,p-1,p^{2})$, $p$ is an odd prime | Direct | Optimal
[11] | GBnF333Generalised Bent functions. |  $(N,N-1,N^{2})$, $N$ is positive integer. | Direct | Optimal.
Theorem 1 | EBF |  ${(p^{t},(p-1)p^{n},p^{n+t+1})},\ n,t\geq 1,$ $p$ is prime and $t\leq n$ | Direct |  Optimal for binary case else asymptotically optimal

###  6.2 Comparison with Traditional ZCZ Sequences in [10] and [11]

In [10] and [11], parameters of constructed ZCZ sequence sets appear to be
$(p,p,p^{2})$ and $(N,N,N^{2})$ respectively, where $p$ is an odd prime, and
$N$ is a positive integer. In both constructions, the length parameter is
fixed, i.e., none of them can produce a ZCZ sequence of length $p^{3}$. Hence
flexibility in the length parameter is not much here. Also, once the length
parameter is fixed, it is not possible to play with ZCZ width and set size,
i.e., fixing of length parameter fixes ZCZ width and set size. On the other
hand, in our proposed construction, the length parameter is $p^{n+t+1}$, i.e.,
the length in the form of $p^{k}$, where $k\geq 3$ can be achieved. Also, for
the fixed value of $n+t,~{}n\geq 1$, and $t\leq n$, more than one combinations
of $n$ and $t$ exist. Hence, the ZCZ width and set size can be altered with
the fixed length. Thus our construction is more flexible, as it can generate
ZCZ sequences of prime-powers length, and also, it is direct.

###  6.3 Comparison with Traditional ZCZ Sequences in [25]

Recently, as a generalization of the proposed work, a new construction of
$(b^{k},(b-1)b^{m-k-1}+(b-2)b^{m-k-2},b^{m})$-ZCZ sequences using EGBFs has
been reported in [25]. The parameter of ZCZ sequence set in [25] is better
than ours. But, based on the record of the publication date of [25], the
proposed work [26] has appeared online in Arxive earlier date.

###  6.4 Comparison with SNC-ZCZ sequences in [15]

In [15], the authors proposed the construction of $(N,N,(N+e))$-SNC ZCZ
sequence set, where $N$ and $e$ are positive integers. in our construction, we
proposed $(p^{t},(p-1)p^{n},p^{n+t+1})$-SNC-ZCZ sequence set. In comparison to
[15], the proposed construction offers following benefits

  * •

The ZCZ and the number of sequences are dependent (as they are the same) in
[15], while we don’t have this type of constraint from the proposed
construction, i.e., they are independent.

  * •

The alphabet size of SNC-ZCZ sequences in [15] is $lcm(N,N+e)$, which is
larger than the alphabet size of our proposed construction, i.e., $p+1$. For
example, the SNC-ZCZ of length $27$ that can be constructed from [15] has
parameter $(3,3,27)$ and alphabet size $9$, while for the same length, our
construction can produce $(3,6,27)$-SNC-ZCZ with alphabet size $3$.

###  6.5 Comparison with SNC-ZCZ sequences in [16]

In [16], authors presented construction of SNC-ZCZ sequences with parameter
$(\tilde{F}(N),N+1,N(N+1))$, where for each positive integer $N\geq 2$,
$\tilde{F}(N)$ denotes the maximum number of rows such that an
$\tilde{F}(N)\times N$ circular Florentine rectangle (CFR) exist. In
comparison to [16], our construction gives following benefits,

  * •

The construction in [16] is heavily depends on the existence of CFR, which
makes this construction very restricted. While the proposed construction is
based on EBFs which are well known in the existing literature.

  * •

The parameters of SNC-ZCZ in [16] are interdependent (as they are all
dependent on $N$), which restrict the flexibility of parameters. On the other
hand, our construction is much more flexible.

The construction in [16] is heavily depends on the existence of CFR. which
makes this construction very limited

##  7 Conclusion

In this paper, we introduce a construction method for SNC-ZCZ sequences of
prime-power length based on EBFs having flexible support. The proposed SNC-ZCZ
sequences are asymptotically optimal. Additionally, we establish a
relationship between traditional ZCZ sequences and GRM codes.

## Declaration

Conflict of interest The authors report no conflict of interest.
Data availability Not Applicable.

## References

  *   * Fan and Darnell [1996] Fan, P., Darnell, M.: Sequence design for communications applications. New York: Wiley 2000 (1996)
  * Zhao and Sadler [2007] Zhao, Q., Sadler, B.M.: A survey of dynamic spectrum access. IEEE signal processing magazine 24(3), 79–89 (2007)
  * Haykin [2005] Haykin, S.: Cognitive radio: brain-empowered wireless communications. IEEE journal on selected areas in communications 23(2), 201–220 (2005)
  * Haykin [2006] Haykin, S.: Cognitive radar: a way of the future. IEEE signal processing magazine 23(1), 30–40 (2006)
  * Suehiro [1994] Suehiro, N.: A signal design without co-channel interference for approximately synchronized cdma systems. IEEE Journal on selected areas in communications 12(5), 837–841 (1994)
  * Chen and Wu [2018] Chen, C.-Y., Wu, S.-W.: Golay complementary sequence sets with large zero correlation zones. IEEE Trans. Commun. 66(11), 5197–5204 (2018) https://doi.org/10.1109/TCOMM.2018.2857485
  * Liu et al. [2014] Liu, Z., Guan, Y., Parampalli, U.: A new construction of zero correlation zone sequences from generalized Reed-Muller codes. In: IEEE Inf. Theory Workshop (ITW) (2014). https://doi.org/10.1109/ITW.2014.6970900
  * Tang et al. [2010] Tang, X., Fan, P., Lindner, J.: Multiple binary zero correlation zone sequence sets with good cross-correlation property based on complementary sequence sets. IEEE Trans. Inf. Theory 56, 4038–4045 (2010) https://doi.org/10.1109/TIT.2010.2050796
  * Hayashi [2009] Hayashi, T.: A class of zero-correlation zone sequence set using a perfect sequence. IEEE Signal Process. Lett. 16, 331–334 (2009) https://doi.org/10.1109/LSP.2009.2014115
  * Zhou et al. [2018] Zhou, Z., Zhang, D., Helleseth, T., Wen, J.: A construction of multiple optimal zero correlation zone sequence sets with good cross-correlation. IEEE Trans. Inf. Theory PP, 1–1 (2018) https://doi.org/10.1109/TIT.2017.2756845
  * Zhang et al. [2020] Zhang, D., Parker, M., Helleseth, T.: Polyphase zero correlation zone sequences from generalised Bent functions. Cryptography and Communications 12, 1–11 (2020) https://doi.org/10.1007/s12095-019-00413-2
  * Tsai et al. [2011] Tsai, L.-S., Chung, W.-H., Shiu, D.-s.: Lower bounds on the correlation property for ofdm sequences with spectral-null constraints. IEEE Trans. on wireless commun 10(8), 2652–2659 (2011)
  * Liu et al. [2018] Liu, Z., Guan, Y.L., Parampalli, U., Hu, S.: Spectrally-constrained sequences: Bounds and constructions. IEEE Transactions on Information Theory 64(4), 2571–2582 (2018)
  * Tian et al. [2020] Tian, L., Xu, C., Li, Y.: A family of single-channel spectrally-null-constrained sequences with low correlation. IEEE Signal Processing Letters 27, 1645–1649 (2020)
  * Li et al. [2022] Li, Y., Tian, L., Zeng, Y.: Spectrally-null-constrained zcz sequences for mimo-ofdm channel estimation over non-contiguous carriers. IEEE commun Letters 27(2), 442–446 (2022)
  * Ye et al. [2022] Ye, Z., Zhou, Z., Liu, Z., Tang, X., Fan, P.: New spectrally constrained sequence sets with optimal periodic cross-correlation. IEEE Trans. on Inf. Theory 69(1), 610–625 (2022)
  * Sarkar et al. [2020] Sarkar, P., Majhi, S., Liu, Z.: A direct and generalized construction of polyphase complementary sets with low PMEPR and high code-rate for OFDM system. IEEE Trans. Commun. 68(10), 6245–6262 (2020) https://doi.org/10.1109/TCOMM.2020.3007390
  * Shen et al. [2022] Shen, B., Yang, Y., Fan, P., Zhou, Z.: Constructions of non-contiguous complementary sequence sets and their applications. IEEE Transactions on Wireless Communications 21(7), 4871–4882 (2022) https://doi.org/10.1109/TWC.2021.3133629
  * Tang et al. [2000] Tang, X.H., Fan, P., Matsufuji, S.: Lower bounds on correlation of spreading sequence set with low or zero correlation zone. Electronics Letters 36, 551–552 (2000) https://doi.org/10.1049/el:20000462
  * Matsufuji [1999] Matsufuji, S.: Spreading sequence set for approximately synchronized CDMA system with no co-channel interference and high data capacity. WPMC’, 333–339 (1999)
  * Shen et al. [2023] Shen, B., Yang, Y., Zhou, Z., Mesnager, S.: Constructions of spectrally null constrained complete complementary codes via the graph of extended boolean functions. IEEE Trans. on Inf. Theory (2023)
  * Kasami et al. [1968] Kasami, T., Lin, S., Peterson, W.: New generalizations of the Reed-Muller codes-I: Primitive codes. IEEE Trans. on Inf. Theory 14(2), 189–199 (1968) https://doi.org/10.1109/TIT.1968.1054127
  * Sarkar et al. [2021] Sarkar, P., Liu, Z., Majhi, S.: A direct construction of complete complementary codes with arbitrary lengths. CoRR abs/2102.10517 (2021) arXiv:2102.10517
  * Tang et al. [2010] Tang, Y.S., Chen, C.Y., Chao, C.C.: A novel construction of zero correlation zone sequences based on Boolean functions. In: IEEE 11th International Symposium on Spread Spectrum Techniques and Applications, pp. 198–203 (2010). https://doi.org/10.1109/ISSSTA.2010.5654347
  * Pai et al. [2022] Pai, C.-Y., Lin, Y.-J., Chen, C.-Y.: Optimal and almost-optimal golay-zcz sequence sets with bounded paprs. IEEE Trans. on commun. (2022)
  * Kumar et al. [2021] Kumar, N., Majhi, S., Sarkar, P., Upadhyay, A.K.: A direct construction of prime-power-length zero-correlation zone sequences for qs-cdma system. arXiv preprint arXiv:2111.06675 (2021)