# A Direct Construction of Complete Complementary Code with Zero Correlation
Zone property for Prime-Power Length

Nishant Kumar, Sudhan Majhi, , and A.K. Upadhyay

###### Abstract

In this paper, we propose a direct construction of a novel type of code set,
which has combined properties of complete complementary code (CCC) and zero-
correlation zone (ZCZ) sequences and called it complete complementary-ZCZ (CC-
ZCZ) code set. The code set is constructed by using multivariable functions.
The proposed construction also provides Golay-ZCZ codes with new lengths,
i.e., prime-power lengths. The proposed Golay-ZCZ codes are optimal and
asymptotically optimal for binary and non-binary cases, respectively, by
_Tang-Fan-Matsufuzi_ bound. Furthermore, the proposed direct construction
provides novel ZCZ sequences of length $p^{k}$, where $k$ is an integer $\geq
2$. We establish a relationship between the proposed CC-ZCZ code set and the
first-order generalized Reed-Muller (GRM) code, and proved that both have the
same Hamming distance. We also counted the number of CC-ZCZ code set in first-
order GRM codes. The column sequence peak-to-mean envelope power ratio (PMEPR)
of the proposed CC-ZCZ construction is derived and compared with existing
works. The proposed construction is also deduced to Golay-ZCZ and ZCZ
sequences which are compared to the existing work. The proposed construction
generalizes many of the existing work.

###### Index Terms:

Golay complementary sets (GCSs), complete complementary code (CCC),
multivariable functions, zero correlation zone (ZCZ) sequence set, Golay-ZCZ
sequence set, CC-ZCZ code set, prime-power length, peak-to-mean envelope power
ratio (PMEPR), orthogonal frequency division multiplexing (OFDM), multi-
carrier code division multiple access (MC-CDMA), generalized Reed-Muller (GRM)
codes.

##  I Introduction

GOLAY was the first to introduce Golay complementary pair (GCP) in 1949 but
defined formally in his article “Complementary Series” in 1960 [1]. GCPs show
their applications in many areas such as infrared multislit spectrometry,
orthogonal frequency-division multiplexing (OFDM) and many others. The idea of
GCP was further extended to Golay complementary sets (GCSs) [2]. A
relationship between GCP of length in the form $2^{m}$ and generalized Boolean
function (GBF) has been established in [3]. Later in [4], Paterson _et al._
obtained GCSs of polyphase sequences from cosets of the first-order
generalized Reed-Muller (GRM) code. Further, an upper bound on peak-to-mean
envelope power ratio (PMEPR) for second-order cosets of this code was
obtained. In [5], the idea of GCSs was further extended to complete
complementary codes (CCCs) which is a collection of GCSs with some extra
aperiodic cross-correlation property. Additionally, CCC has found application
in multi-carrier code division multiple access (MC-CDMA) systems to eliminate
multiple-access interference (MAI) for multiple users over asynchronous
environment [6, 7]. CCCs are also utilized in optimal channel estimation for
multiple-input multiple-output (MIMO) frequency-selective fading channels [8],
MIMO radar [9, 10], cell search in OFDM systems [11], and data hiding[12]. A
detailed study on GCSs and CCCs can be carried out in [3, 4, 13, 14, 15, 16,
17, 18, 19].

In OFDM systems, GCSs are widely used to control PMEPR [3], an upper bound on
PMEPR was obtained in [4]. By using the good aperiodic correlation properties
of GCP and GCSs, these can be used for asynchronous CDMA system, channel
estimation [20], synchronization [21], signal detection etc. Quasi-synchronous
CDMA (QS-CDMA) system is proposed to reduce the impact of synchronization
accuracy in communication systems [22]. QS-CDMA system can eliminate co-
channel interference for multiuser environment when synchronization error is
controlled within a specific range [22].

Zero correlation zone (ZCZ) sequences have ideal periodic correlation property
inside a zone about the origin. Therefore, they can reduce MAI and multipath
interference (MPI) [23], and hence they are used to take benefit of quasi
synchronization in the QS-CDMA system. They are also used in radar [24],
channel estimation [25, 26, 27], pilot design [28, 29], multicode MIMO system
[30], and as a training sequence in MIMO-OFDM systems[28]. The ZCZ sequences
can achieve optimal channel estimation performance under the condition that
all the received signals are quasi-synchronous within the ZCZ [31, 32].

The relation between GCSs and ZCZ sequences was first established in [33].
Later, numerous ZCZ sequences are constructed by using mutually orthogonal
GCSs (MOGCS) [34, 35, 36]. While most of the ZCZ sequence constructions need
predefined sequences like perfect sequences and use unitary matrices[36],
interleaving technique [37] and Hadamard matrices [38]. The direct
constructions of ZCZ sequences based on GBFs are provided in [39, 40, 41]. The
connection between GRM codes and ZCZ sequences is also provided in [40, 41].
The GCSs and ZCZ sets are characterized by their aperiodic auto-correlation
and periodic correlation properties, respectively. The authors in [42, 43],
investigated the periodic auto-correlation behaviour of individual Golay
sequences in 2013. Specifically, they presented two constructions of Golay
sequences of length $2^{m}$ each displaying a periodic zero auto-correlation
zone (ZACZ) of $2^{m-2}$, and $2^{m-3}$, respectively, around the in-phase
position. Motivated by the work in [42, 43], the authors in [44] studied the
periodic zero cross-correlation zone (ZCCZ) of the GCSs presented in [18] and
provided a relation between ZCZ sets and GCSs and proposed a construction of a
new class of sequence sets and named it Golay-ZCZ sequence sets. Golay-ZCZ
sequence set consists of sequences having their aperiodic auto-correlation sum
zero except at the zero shift and ideal periodic cross-correlations and auto-
correlations within the ZCZ. Precisely, a Golay-ZCZ sequence set is a ZCZ
sequence set which is also a GCS. Therefore, from the application point of
view, Golay-ZCZ sequence sets are used to obtain both the advantages in a
single code. For an instant, ZCZ sequences are used as pilots or training
sequences in OFDM systems [29, 28] but it is not utilized for PMEPR reduction.
Since Golay-ZCZ sequence sets are themselves GCSs, they can be utilized as
training sequences as well as for PMEPR reduction. Therefore, the proposed
code provides low PMEPR when employed in OFDM systems due to the good auto-
correlation properties of GCSs. Besides, the Golay-ZCZ sets possess zero auto-
correlations and cross-correlations within their ZCZ and hence have the
potential application to synchronization. Inspired by the Golay-ZCZ work given
in [44], the authors in [22] given construction of Golay-ZCZ sequence sets
which have large ZCZ width. The length of proposed Golay-ZCZ sequence sets in
[22] and [44] is limited to power-of-two. Continuing in this chain, recently
in 2021, the authors in [45] provided a construction of Golay-ZCZ sequences of
length $M^{2}N$ using an CCC of length $N$ and flock size $M$. Furthermore, in
[46], they also constructed Golay-ZCZ complementary pair of length $4N$, where
$N$ is a positive integer and provided an open problem to construct Golay-ZCZ
sequence set of non-power-of-two length as _“An interesting future work will
be to design Golay-ZCZ sequence sets, consisting sequences of length non-
power-of-two, which have periodic ZACZ ***zero auto-correlation zone and
ZCCZ†††zero cross-correlation zone around the in-phase position”_.
Furthermore, they also extended this work to two-dimension [47].

Motivated by the above works, and open problem provided in [46], in this
paper, we provide a construction of novel type of code set called CC-ZCZ code
set of prime-power length in which each code is Golay-ZCZ sequence set having
non-power-of-two length sequences which settles the open problem provided by
in [46]. The proposed $p$-ary Golay-ZCZ sequence sets are asymptotically
optimal with respect to _Tang-Fan-Matsufuji_ bound and optimal for binary
case. The proposed CC-ZCZ code sets are also seen as second-order cosets of
first-order GRM codes. Since first-order GRM codes enjoy high Hamming
distance, we prove that the proposed Golay-ZCZ sequence set in CC-ZCZ code set
has the same Hamming distance as GRM codes. Additionally, we also count the
number of cosets of GRM codes, which is further used to count the total number
of CC-ZCZ code sets and Golay-ZCZ sequence sets. The column sequence PMEPR of
the proposed CC-ZCZ construction is reduced. The row and column sequence PMEPR
for Golay-ZCZ and CCC are also been reduced over the existing work [44] and
[14], respectively. The proposed construction can generate new ZCZ sequences
of length $p^{k}$, where $k$ is a positive integer $\geq 2$. The proposed
construction also contributes Golay-ZCZ sequence sets and ZCZ sequences which
are again compared to the existing state-of-the-art.

The remainder of the paper is structured as follows. We provide some basic
notations and definitions in Section II. In Section III, we provide the
construction of CC-ZCZ code sets and provide examples in support of our
construction. In Section IV, a relation between proposed CC-ZCZ code set and
GRM codes is established. Moreover, the number of CC-ZCZ code set in a second-
order coset of GRM code and Hamming distance are provided. In Section V, we
discuss column sequence PMEPR of constructed CC-ZCZ code based MC-CDMA system.
Section VI provides another result that increases the number of CC-ZCZ code
sets. In section VII, we defend the work’s novelty by comparing it to existing
works in the literature. Lastly, we provide conclusion in Section VIII.

##  II Preliminary

###  II-A Definition and Correlation Functions

Let $\mathbf{a}=(a_{0},a_{1},\ldots,a_{L-1})$ and
$\mathbf{b}=(b_{0},b_{1},\ldots,b_{L-1})$ be two $L$-length complex-valued
sequences. For an integer $\tau$, define

$\mathcal{A}(\mathbf{a},\mathbf{b})(\tau)=\begin{cases}\sum_{i=0}^{L-1-\tau}a_{i}b^{*}_{(i+\tau)},&0\leq\tau<L,\\\
\sum_{i=0}^{L+\tau-1}a_{(i-\tau)}b^{*}_{i},&-L<\tau<0.\end{cases}$ (1)

The function $\mathcal{A}(\mathbf{a},\mathbf{b})$ is referred to as aperiodic
cross-correlation function (ACCF) of $\mathbf{a}$ and $\mathbf{b}$. If
$\mathbf{a}=\mathbf{b}$ then this function is called aperiodic auto-
correlation function (AACF) and denoted as $\mathcal{A}(\mathbf{a})$.

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

$\mathcal{P}(\mathbf{a},\mathbf{b})(\tau)=\begin{cases}\sum_{i=0}^{L-1}a_{i}b^{*}_{(i+\tau)\mod
L},&0\leq\tau<L,\\\ \sum_{i=0}^{L-1}a_{i}b^{*}_{(i-\tau)\mod
L},&-L<\tau<0.\end{cases}$ (2)

When $\mathbf{a}=\mathbf{b}$, then this function is called periodic auto-
correlation function (PACF) and denoted as $\mathcal{P}(\mathbf{a})$ [48].

###### 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

$\displaystyle\mathbf{C_{\xi}}=\begin{bmatrix}\mathbf{a}_{0}^{\xi}\\\
\mathbf{a}_{1}^{\xi}\\\ \vdots\\\ \mathbf{a}_{M-1}^{\xi}\end{bmatrix},$ (3)

where $\mathbf{a}_{\upsilon}^{\xi}$ ($0\leq\upsilon\leq M-1,0\leq\xi\leq K-1$)
is the $\upsilon$th row sequence or $\upsilon$th constituent sequence. Then
the ACCF of two codes $\mathbf{C_{\xi_{1}}},\mathbf{C_{\xi_{2}}}\in\textbf{C}$
is defined as

$\gamma(\mathbf{C_{\xi_{1}}},\mathbf{C_{\xi_{2}}})(u)=\sum_{\upsilon=0}^{M-1}{\gamma(\mathbf{a}_{\upsilon}^{\xi_{1}},\mathbf{a}_{\upsilon}^{\xi_{2}})(u)}.$
(4)

###### Definition 2

Let C be a code set satisfying the following correlation properties

$\gamma(\mathbf{C_{\xi_{1}}},\mathbf{C_{\xi_{2}}})(u)=\begin{cases}LM,&\xi_{1}=\xi_{2}\
\text{and}\ u=0,\\\ 0,&\xi_{1}=\xi_{2}\ \text{and}\ 0<|u|<L,\\\
0,&\xi_{1}\neq\xi_{2}\ \text{and}\ |u|<L.\end{cases}$ (5)

Then C is known as $(K,M,L)$-MOGCS. If $K=M$, it is referred to be CCC set and
we write it as $(K,K,L)$-CCC. Additionally, each code of C is known as GCS
[49]. When $M=2$ it is called GCP [19].

###### Definition 3

Let
$\bm{\mathcal{Z}}=\\{\mathbf{z}_{0},\mathbf{z}_{1},\ldots,\mathbf{z}_{K-1}\\}$
be a set of $K$ sequences each of length $L$, i.e.,

$\mathbf{z}_{i}=(\mathbf{z}_{i0},\mathbf{z}_{i1},\ldots,\mathbf{z}_{iL-1}),\ \
0\leq i\leq K-1.$

Then, $\bm{\mathcal{Z}}$ is referred to be $(K,L,Z)$-ZCZ sequence set for
$0\leq i,j\leq K-1$, if $\bm{\mathcal{Z}}$ satisfies the following,

$\mathcal{P}(\mathbf{z}_{i},\mathbf{z}_{j})(u)=\begin{cases}0,&i=j\
\text{and}\ 1\leq|u|<Z,\\\ 0,&i\neq j\ \text{and}\ 0\leq|u|<Z,\\\ L,&i=j\
\text{and}\ u=0,\end{cases}$ (6)

where $Z$ is termed as ZCZ width.

###### Definition 4

_(Tang-Fan-Matsufuji Bound[50])_ Let $\bm{\mathcal{Z}}$ be any ZCZ sequence
set with parameter $(K,L,Z)$. Then, $KZ\leq L$ and if $KZ=L$, then
$\bm{\mathcal{Z}}$ is said to be optimal. If for the larger value of $K$,
$KZ\approx L$, $\bm{\mathcal{Z}}$ is said to be asymptotically optimal. For
the binary case, it is widely accepted that the bound is reduced to $2KZ\leq
L$ [51].

###### Definition 5

Let $\bm{\mathcal{Z}}$ be a $(K,L,Z)$-ZCZ sequence set then $\bm{\mathcal{Z}}$
is called $(K,L,Z)$-Golay-ZCZ sequence set if it additionally satisfies the
GCS properties.

###### Definition 6

Let $\mathbf{C}=\\{\mathbf{C}_{0},\mathbf{C}_{1},\ldots,\mathbf{C}_{K-1}\\}$
be a $(K,K,L)$-CCC. Then $\mathbf{C}$ is called $(K,K,L,Z)$-CC-ZCZ code set if
each $\mathbf{C}_{i},~{}\forall~{}i=0,1,\ldots,k-1$ is $(K,L,Z)$-Golay-ZCZ
sequence set.

###  II-B Multivariable Functions and Corresponding Sequences [14]

Let $\mathbf{x}=(x_{1},x_{2},\ldots,x_{k-1})\in\mathbb{Z}_{p}^{k}$, be a
vector of finite length, where $p$ is a prime number and $k$ is a positive
integer. It is clear that $x_{i}\in\mathbb{Z}_{p}$ for $1\leq i\leq k.$ Now,
we take a specific collection of monomials of degree at most $r$ over the
variables $x_{1},x_{2},\ldots,x_{k}$ as follows

$\mathscr{M}(\mathbf{x},r)=\\{x_{1}^{j_{1}}x_{2}^{j_{2}}\ldots
x_{k}^{j_{k}}:j_{i}\in\\{0,1\\}\ \text{for}\ 1\leq i\leq
k,j_{1}+j_{2}+\cdots+j_{k}\leq r\\},$ (7)

where $0\leq r\leq k$. A monomial’s degree in $\mathscr{M}$ is determined by
the number of independent variables associated with it and this can be
verified by examining the expressions of the monomials in $\mathscr{M}$. A
linear combination of monomials in $\mathscr{M}$ with $\mathbb{Z}_{q}$-valued
coefficients, where $q$ is a positive integer, leads to a multivariable
function
$f(x_{1},x_{2},\ldots,x_{k}):\mathbb{Z}_{p}^{k}\rightarrow\mathbb{Z}_{q}$
which has maximum degree $r$. Let $\mathscr{F}(\mathscr{M}(\mathbf{x},r),q)$
be the set of all multivariable functions
$f(x_{1},x_{2},\ldots,x_{k}):\mathbb{Z}_{p}^{k}\rightarrow\mathbb{Z}_{q}$,
over the monomials in $\mathscr{M}$ and defined as

$\mathscr{F}(\mathscr{M}(\mathbf{x},r),q)=\bigg{\\{}\sum_{(j_{1},j_{2},\ldots,j_{k})\in\\{0,1\\}^{k}}{c_{j_{1},j_{2},\ldots,j_{k}}x_{1}^{j_{1}}x_{2}^{j_{2}}\ldots
x_{k}^{j_{k}}}:c_{j_{1},j_{2},\ldots,j_{k}}\in\mathbb{Z}_{q}\bigg{\\}},$ (8)

where $r$ is the maximum degree of a multivariable function $f$ in
$\mathscr{F}(\mathscr{M}(\mathbf{x},r),q)$. When $p=2$, the set of
multivariable functions $\mathscr{F}(\mathscr{M}(\mathbf{x},r),q)$ is reduced
to a set of $\mathbb{Z}_{q}$-valued GBFs of maximum degree $r$ over the vector
variable $\mathbf{x}$ .

For a multivariable function, $f\in\mathscr{F}(\mathscr{M}(\mathbf{x},r),q)$,
the $\mathbb{Z}_{q}$-valued sequence $\theta(f)$ and the complex-valued
sequence $\Theta(f)$ having length $p^{k}$ are defined as

$\theta(f)=({f_{0}},{f_{1}},\ldots,{f_{p^{k}-1}}),$ (9)
$\Theta(f)=(\omega^{f_{0}},\omega^{f_{1}},\ldots,\omega^{f_{p^{k}-1}}).$ (10)

where $f_{i}=f(i_{0},i_{1},\ldots,i_{k-1})$, $\omega=\exp(2\pi\sqrt{-1}/q)$,
and $(i_{0},i_{1},\ldots,i_{k-1})$ is the $p$-ary vector representation of
$i$.

###  II-C Generalized Reed-Muller (GRM) Code

For a prime $p$, let $\lambda=p^{n}$ and $n\geq 1$, then codewords of
$\lambda$-ary GRM codes consist of the sequence corresponding to multivariable
polynomials over the field $\mathbb{F}_{\lambda}$, where $\lambda=p^{n}$ and
$n\geq 1$. Let us consider a polynomial ring
$\mathbb{F}_{\lambda}[x_{1},x_{2},\ldots,x_{m}]$ with $m$ variables. The GRM
code with parameters $m$ and $r$ consists of all the sequences corresponding
to polynomials with $m$ variables and degree no larger than $r$.

###### Definition 7

The $r$-th order $r\leq\lambda$, $\lambda$-ary GRM code denoted by
$GRM_{\lambda}(m,r)$, is defined by the set of $\lambda$-ary vectors as

$GRM_{\lambda}(m,r)=\\{\theta(f):f\in\mathbb{F}_{\lambda}[x_{1},x_{2},\ldots,x_{m}],~{}deg(f)\leq
r\\}.$ (11)

It is to be noted that we always have $x^{\lambda}=1$ in
$\mathbb{F}_{\lambda}$, so we only need to consider polynomials in which the
degree of each $x_{i}$ is no larger than $\lambda-1$. All such polynomials
with degree no larger than $r$ are linear combinations of the following set of
monomials

$\mathcal{N}(r)=\Big{\\{}x_{1}^{j_{1}}x_{2}^{j_{2}},\ldots
x_{m}^{j_{m}}:j_{i}=0,1,2,\ldots,\lambda-1~{}\text{and}~{}\sum_{i=1}^{m}{j_{i}}\leq
r\Big{\\}}.$ (12)

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

######  Lemma 1 ([52])

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 $p-1$.

###  II-D Peak-to-Mean Envelope Power Ratio (PMEPR)

In spite of several benefits arising from the OFDM technique, its wide
acceptance has been hindered by its high PMEPR in uncoded signals. In this
subsection, we define PMEPR for OFDM signals.

One can model the OFDM signal for a complex valued word
$\mathbf{B}=(B_{1},B_{2},\ldots,B_{N})$ of length $N$ as the real part of the

$\mathcal{S}_{\mathbf{B}}(\tau)=\sum_{i=1}^{N}{B_{i}e^{2\pi(i+\zeta)(\tau)\sqrt{-1}}},$
(13)

where $0\leq\tau<1$ and $\zeta$ is a positive constant. The sequence
$\mathbf{B}=(B_{1},B_{2},\ldots,B_{N})$ is termed as modulating codeword of
the OFDM symbol. The PMEPR of word $\mathbf{B}$ is defined as

$PMEPR(\mathbf{B})=\frac{1}{N}\sup_{0\leq\tau<1}{|\mathcal{S}_{\mathbf{B}}(\tau)|^{2}}.$
(14)

It is to be observed that the largest value that PMEPR can have is $N$.
However, it is preferable to use codewords with a lower PMEPR than $N$.
Additionally, the PMEPR of GCS of size $M$ is bounded above by $M$ [4].

##  III Proposed Construction of CC-ZCZ Code Set

In this section, we provide a construction of CC-ZCZ code set of prime-power
length. For that we start with a result which provides the construction of CCC
of prime-power length [14]. For the ease of representation, we present this
result in a modified way without loosing the essence of the result.

######  Lemma 2 ([14])

For an integer $m\geq 2,$ let $\\{1,2,\ldots,m\\}$ is divided into $k$
partitions namely, $E_{1},E_{2},\ldots,E_{k}$ where $1\leq k\leq m-1$.
Further, let us also assume that $\lambda=p^{n}$ and $n\geq 1$ and
$\pi_{\beta}$ be bijection mapping from $\\{1,2,\ldots,n_{\beta}\\}$ to
$E_{\beta}$ where $n_{\beta}=|E_{\beta}|\geq 1,~{}\forall{\beta=2,3,\ldots,k}$
and $n_{1}=|E_{1}|\geq 2$. Now, we define a multivariable function
$f:\mathbb{Z}_{p}^{m}\rightarrow\mathbb{Z}_{\lambda}$ as

$f(x_{1},x_{2},\ldots,x_{m})=\frac{\lambda}{p}\sum_{\beta=1}^{k}{\sum_{\gamma=1}^{n_{\beta}-1}{x_{\pi_{\beta}(\gamma)}x_{\pi_{\beta}(\gamma+1)}}}+\sum_{\alpha=1}^{m}{g_{\alpha}x_{\alpha}},$
(15)

where $g_{\alpha}\in\mathbb{Z}_{\lambda}$ for $\alpha=1,2,\ldots,m$. Further,
let
$\mathbf{u_{k}}=[u_{1},u_{2},\ldots,u_{k}],~{}\mathbf{v_{k}}=[v_{1},v_{2},\ldots,v_{k}]$
be $p$-ary representation for $0\leq\mathbf{u_{k},v_{k}}\leq p^{k}-1$. Define

$a_{\mathbf{u_{k}}}^{\mathbf{v_{k}}}=a_{u_{1}}^{v_{1}}(E_{1})+a_{u_{2}}^{v_{2}}(E_{2})+\cdots+a_{u_{k}}^{v_{k}}(E_{k}),$
(16)

where for any $1\leq\beta\leq k$, multivariable function
$a_{u_{\beta}}^{v_{\beta}}(E_{\beta})$ is defined as

$a_{u_{\beta}}^{v_{\beta}}(E_{\beta})=\frac{\lambda}{p}\sum_{\gamma=1}^{n_{\beta}-1}{x_{\pi_{\beta}(\gamma)}x_{\pi_{\beta}(\gamma+1)}}+\sum_{\gamma=1}^{n_{\beta}}{g_{\pi_{\beta}{(\gamma)}}x_{\pi_{\beta}{(\gamma)}}}+\frac{\lambda}{p}x_{\pi_{\beta}(1)}u_{\beta}+\frac{\lambda}{p}x_{\pi_{\beta}(n_{\beta})}v_{\beta}.$
(17)

Let us also define

$\mathbf{C_{u_{k}}}=[\Theta(a_{\mathbf{u_{k}}}^{\mathbf{v_{k}}})]_{0\leq\mathbf{v_{k}}\leq
p^{k}-1}=\begin{bmatrix}\Theta(a_{\mathbf{u_{k}}}^{0})\\\
\Theta(a_{\mathbf{u_{k}}}^{1})\\\ \vdots\\\
\Theta(a_{\mathbf{u_{k}}}^{p^{k}-1})\end{bmatrix}.$ (18)

Then $\mathbf{C}=\\{\mathbf{C_{u_{k}}}:0\leq\mathbf{u_{k}}\leq p^{k}-1\\}$ is
$(p^{k},p^{k},p^{m})$-CCC.

###### Theorem 1

For a fixed $\mathbf{u_{k}}$, $\mathbf{C_{u_{k}}}$ is GCS as defined in _Lemma
2_. Let $\mathbf{C^{\prime}_{u_{k}}}$ be $\mathbf{C_{u_{k}}}$ together with
the permutation $\pi_{\beta}(1)=m-\beta+1$ for $\beta=1,2,\ldots,k$. Then
$\mathbf{C^{\prime}_{u_{k}}}$ is $(p^{k},p^{m},(p-1)p^{\pi_{1}(2)-1})$-Golay-
ZCZ sequence set and hence
$\mathbf{C^{\prime}}=\\{\mathbf{C^{\prime}_{u_{k}}}:0\leq\mathbf{u_{k}}\leq
p^{k}-1\\}$ is $(p^{k},p^{k},p^{m},(p-1)p^{\pi_{1}(2)-1})$-CC-ZCZ code set.

###### Proof:

Please see Appendix A. ∎

###### Remark 1

If we have $\pi_{1}(2)=m-k$, then from _Theorem 1_,
$\mathbf{C^{\prime}_{u_{k}}}$, is $(p^{k},p^{m},(p-1)p^{m-k-1})$-ZCZ
$\forall~{}0\leq\mathbf{u_{k}}\leq p^{k}-1$, which implies $KZ=(p-1)p^{m-1}$
and $L=p^{m}$. Now, for $\lambda=2$, $KZ=2L$ and hence
$\mathbf{C^{\prime}_{u_{k}}}$ is optimal ZCZ sequence set. But if $p\neq 2$
then for larger value of $p$, $(p-1)\approx p$ and hence $KZ\approx L$,
therefore, in this case, $\mathbf{C^{\prime}_{u_{k}}}$ is asymptotically
optimal.

According to _Theorem 1_, we can provide a construction of
$(p^{k},p^{k},p^{m},(p-1)p^{\pi_{1}(2)-1})$-CC-ZCZ code sets having variable
ZCZ width $\pi_{1}(2)$. It is desirable to have a larger ZCZ width, we take
$\pi_{1}(2)=m-k$ to propose a family of
$(p^{k},p^{k},p^{m},(p-1)p^{m-k-1})$-CC-ZCZ code sets having larger ZCZ width.
Furthermore, each code of CC-ZCZ is $(p^{k},p^{m},(p-1)p^{m-k-1})$-Golay-ZCZ
sequence set and it is optimal _(Tang-Fan-Matsufuji Bound)_ for $\lambda=2$
and asymptotically optimal for $\lambda\neq 2$. To support our construction,
we present examples below.

###### Example 1

(_Binary Case_) Taking $\lambda=2,p=2,k=2,$ and $m=5$, we let
$E_{1}=\\{1,3,5\\}$ and $E_{2}=\\{2,4\\}$ be a partition of $\\{1,2,3,4,5\\}$.
Also let $\pi_{1}(1)=5,\pi_{1}(2)=3,\pi_{1}(3)=1,\pi_{2}(1)=4$ and
$\pi_{2}(2)=2$. Let $f:\mathbb{Z}_{2}^{5}\rightarrow\mathbb{Z}_{2}$ be defined
as

$f(x_{1},x_{2},\ldots,x_{5})=x_{5}x_{3}+x_{3}x_{1}+x_{4}x_{2}+x_{1}+x_{3}.$

Further, let $\mathbf{u_{k}}=0$, Then

$\displaystyle\mathbf{C^{\prime}_{0}}=\begin{bmatrix}\theta(a_{\mathbf{0}}^{\mathbf{0}})\\\
\theta(a_{\mathbf{0}}^{\mathbf{1}})\\\ \theta(a_{\mathbf{0}}^{\mathbf{2}})\\\
\theta(a_{\mathbf{0}}^{\mathbf{3}})\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\theta(f+x_{5}.0+x_{4}.0+x_{1}.0+x_{2}.0)\\\
\theta(f+x_{5}.0+x_{4}.0+x_{1}.1+x_{2}.0)\\\
\theta(f+x_{5}.0+x_{4}.0+x_{1}.0+x_{2}.1)\\\
\theta(f+x_{5}.0+x_{4}.0+x_{1}.1+x_{2}.1)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}01011111011011000101000001100011\\\
00001010001110010000010100110110\\\ 01101100010111110110001101010000\\\
00111001000010100011011000000101\end{bmatrix},$

where the above matrix is phase matrix and $\mathbf{C^{\prime}_{0}}$ is a
$(4,32,4)$-Golay-ZCZ sequence set. Moreover, we can also derive
$\mathbf{C^{\prime}_{1}},\mathbf{C^{\prime}_{2}}$ and
$\mathbf{C^{\prime}_{3}}$ and these are also $(4,32,4)$-Golay-ZCZ sequence
sets. Hence
$\mathbf{C^{\prime}}=\\{\mathbf{C^{\prime}_{0}},\mathbf{C^{\prime}_{1}},\mathbf{C^{\prime}_{2}},\mathbf{C^{\prime}_{3}}\\}$
is a $(4,4,32,4)$-CC-ZCZ code set. Since $K=4,~{}L=32,~{}Z=4$ which implies
$KZ=16$. Hence, $2KZ=L$. Therefore, $\mathbf{C^{\prime}}_{i}$ is optimal by
_Tang-Fan-Matsufuji_ Bound. A graphical representation of periodic correlation
of sequences in $\mathbf{C^{\prime}_{0}}$ is shown in Fig. 1

###### Example 2

(_Non-binary case_) Here let $\lambda=3,k=2,m=3$ and $p=3$. Also let
$E_{1}=\\{2,3\\},~{}E_{2}=\\{1\\}$ be a partition of $\\{1,2,3\\}$. Further,
let $\pi_{1}(1)=3,\pi_{1}(2)=2,$ and $\pi_{2}(1)=1$. Define
$f:\mathbb{Z}_{3}^{3}\rightarrow\mathbb{Z}_{3}$ as
$f(x_{1},x_{2},x_{3})=x_{3}x_{2}$. Now, let us fix $\mathbf{u_{k}=0}$ then

$\displaystyle\mathbf{C^{\prime}_{0}}$
$\displaystyle=\begin{bmatrix}a_{\mathbf{0}}^{\mathbf{0}}\\\
a_{\mathbf{0}}^{\mathbf{1}}\\\ a_{\mathbf{0}}^{\mathbf{2}}\\\
a_{\mathbf{0}}^{\mathbf{3}}\\\ a_{\mathbf{0}}^{\mathbf{4}}\\\
a_{\mathbf{0}}^{\mathbf{5}}\\\ a_{\mathbf{0}}^{\mathbf{6}}\\\
a_{\mathbf{0}}^{\mathbf{7}}\\\
a_{\mathbf{0}}^{\mathbf{8}}\end{bmatrix}=\begin{bmatrix}f+x_{3}.0+x_{1}.0+x_{2}.0+x_{1}.0\\\
f+x_{3}.0+x_{1}.0+x_{2}.1+x_{1}.0\\\ f+x_{3}.0+x_{1}.0+x_{2}.2+x_{1}.0\\\
f+x_{3}.0+x_{1}.0+x_{2}.0+x_{1}.1\\\ f+x_{3}.0+x_{1}.0+x_{2}.1+x_{1}.1\\\
f+x_{3}.0+x_{1}.0+x_{2}.2+x_{1}.1\\\ f+x_{3}.0+x_{1}.0+x_{2}.0+x_{1}.2\\\
f+x_{3}.0+x_{1}.0+x_{2}.1+x_{1}.2\\\
f+x_{3}.0+x_{1}.0+x_{2}.2+x_{1}.2\end{bmatrix}=\begin{bmatrix}000000000012012012021021021\\\
012012012021021021000000000\\\ 021021021000000000012012012\\\
000111222012120201021102210\\\ 012120201021102210000111222\\\
021102210000111222012120201\\\ 000222111012201120021210102\\\
012201120021210102000222111\\\ 021210102000222111012201120\end{bmatrix},$ (19)

$\mathbf{C^{\prime}_{0}}$ is a $(9,27,2)$-Golay-ZCZ sequence set. Similarly,
$\mathbf{C^{\prime}_{u_{k}}}$ for $1\leq\mathbf{u_{k}}\leq 8$ are also
$(9,27,2)$-Golay-ZCZ sequence set. Hence
$\mathbf{C^{\prime}}=\\{\mathbf{C^{\prime}_{0}},\mathbf{C^{\prime}_{1}},\ldots,\mathbf{C^{\prime}_{8}}\\}$
is a $(9,9,27,2)$-CC-ZCZ code set.

Figure 1: $\mathbf{(a)}$ Auto-correlation plot for
{$\theta(a_{\mathbf{0}}^{\mathbf{0}})$, $\theta(a_{\mathbf{0}}^{\mathbf{1}})$,
$\theta(a_{\mathbf{0}}^{\mathbf{2}})$, $\theta(a_{\mathbf{0}}^{\mathbf{3}})$},
$\mathbf{(b)}$ Cross-correlation plot for
$\\{(\theta(a_{\mathbf{0}}^{\mathbf{1}}),\theta(a_{\mathbf{0}}^{\mathbf{2}})),(\theta(a_{\mathbf{0}}^{\mathbf{3}}),\theta(a_{\mathbf{0}}^{\mathbf{4}}))\\}$,
$\mathbf{(c)}$ Cross-correlation plot for
$\\{(\theta(a_{\mathbf{0}}^{\mathbf{1}}),\theta(a_{\mathbf{0}}^{\mathbf{3}})),(\theta(a_{\mathbf{0}}^{\mathbf{2}}),\theta(a_{\mathbf{0}}^{\mathbf{4}}))\\}$
$\mathbf{(d)}$ Cross-correlation plot for
$\\{(\theta(a_{\mathbf{0}}^{\mathbf{1}}),\theta(a_{\mathbf{0}}^{\mathbf{4}})),\allowbreak(\theta(a_{\mathbf{0}}^{\mathbf{2}}),\theta(a_{\mathbf{0}}^{\mathbf{3}}))\\}.$

##  IV CC-ZCZ Code Sets from GRM Codes

In this section, a relation between CC-ZCZ code set and elements of second-
order coset of $GRM_{p}(m,1)$ is established for $\lambda=p$. The precise
number of cosets and number of CC-ZCZ code set in each coset is also given.
Hamming distance of Golay-ZCZ sequence is also calculated.

It can be obtained that if we have a sequence in
$\mathbf{C^{\prime}_{u_{k}}}$, then it is a codeword belonging to second-order
coset $\mathcal{Q}_{1}+GRM_{p}(m,1)$ where $\mathcal{Q}_{1}$ is given by

$\mathcal{Q}_{1}=\sum_{\beta=1}^{k}{\sum_{\gamma=1}^{n_{\beta}-1}{x_{\pi_{\beta}(\gamma)}x_{\pi_{\beta}(\gamma+1)}}}.$
(20)

Conversely, if any sequence is in the coset $\mathcal{Q}_{1}+GRM_{p}(m,1)$
then it belongs to a certain $(p^{k},p^{m},(p-1)p^{\pi_{1}(2)-1})$-Golay-ZCZ
sequence set. Now, let us consider the constructed
$(p^{k},p^{m},(p-1)p^{m-k-1})$-Golay-ZCZ sequence family by taking
$\pi_{1}(2)=m-k$.

###### Corollary 1

The number of coset representative $\mathcal{Q}_{1}$ in the second-order coset
of the form $\mathcal{Q}_{1}+GRM_{p}(m,1)$ is given by

$\sum_{n_{1}+n_{2}+\cdots+n_{k}=m}{{\binom{m-k-1}{n_{1}-2,n_{2}-1,\ldots,n_{k}-1}}(n_{1}-2)!\prod_{\beta=2}^{k}{(n_{\beta}-1)!}}.$
(21)

###### Proof:

Since, we have $\pi_{1}(1)=m$ and $\pi_{1}(2)=m-k$, so we can get $(n_{1}-2)!$
different quadratic forms of the type

$x_{\pi_{1}(1)}x_{\pi_{1}(2)}+x_{\pi_{1}(2)}x_{\pi_{1}(3)}+\cdots+x_{\pi_{1}(n_{1}-1)}x_{\pi_{1}(n_{1})},$
(22)

where $\pi_{1}(i)$ is a permutation of $n_{1}-2$ integers where
$i=3,4,\ldots,n_{1}$. Similarly, there are $(n_{\beta}-1)!$ different
quadratic forms of the type
$\sum_{\beta=1}^{n_{\beta}-1}{x_{\pi_{\beta}(\gamma)}x_{\pi_{\beta}(\gamma+1)}}$
for $\beta=2,3,\ldots,k$ as $\pi_{\beta}(1)$ is restricted to $m-\beta+1$. In
addition, we have the condition $n_{1}+n_{2}+\cdots+n_{k}=m$ where
$n_{\beta}=|E_{\beta}|$ and non-empty sets $E_{1},E_{2},\ldots,E_{k}$ form a
partition of $\\{1,2,\ldots,m\\}$. Therefore, we can explicitly determine the
number of distinct coset representatives $\mathcal{Q}_{1}$ as

$\sum_{n_{1}+n_{2}+\cdots+n_{k}=m}{{\binom{m-k-1}{n_{1}-2,n_{2}-1,\ldots,n_{k}-1}}(n_{1}-2)!\prod_{\beta=2}^{k}{(n_{\beta}-1)!}}.$

∎

###### Corollary 2

Let $\mathcal{Q}_{1}+GRM_{p}(m,1)$ be any second-order coset with coset
representative $\mathcal{Q}_{1}$ and let $\pi_{1}(2)=m-k$. Then the coset
$\mathcal{Q}_{1}+GRM_{p}(m,1)$ consists of $p^{m-k+1}$ and $p^{m+1}$ distinct
$p$-ary $(p^{k},p^{k},p^{m},(p-1)p^{m-k-1})$-CC-ZCZ code set and
$(p^{k},p^{m},(p-1)p^{m-k-1})$-Golay-ZCZ sequence set respectively.

###### Proof:

We have from _Theorem 1_, that a codeword $\mathbf{c}$ in
$\mathcal{Q}_{1}+GRM_{p}(m,1)$ lies in the
$(p^{k},p^{m},(p-1)p^{m-k-1})$-Golay-ZCZ sequence set

$\mathbf{C^{\prime}_{u_{k}}}=\bigg{\\{}f+\sum_{\beta=0}^{k-1}{x_{\pi_{\beta+1}{(1)}}u_{\beta+1}}+\sum_{\beta=1}^{k}{x_{\pi_{\beta}{(n_{\beta})}}v_{\beta}}+b_{0}:v_{\beta}\in\mathbb{Z}_{p},~{}1\leq\beta\leq
k\bigg{\\}}.$ (23)

So it can easily be obtained that for a fixed value of $\mathbf{u_{k}}$,
$\mathcal{Q}_{1}+GRM_{p}(m,1)$ consists of $p^{m-k+1}$ distinct $p$-ary
$(p^{k},p^{m},(p-1)p^{m-k-1})$-Golay-ZCZ sequence set. Hence, we have
$p^{m-k+1}$ distinct $p$-ary $(p^{k},p^{k},p^{m},(p-1)p^{m-k-1})$-CC-ZCZ code
set. Since $\mathbf{u_{k}}$ can vary in $p^{k}$ ways, therefore,
$\mathcal{Q}_{1}+GRM_{p}(m,1)$ consists of $p^{m-k+1}\cdot p^{k}=p^{m+1}$
distinct $p$-ary $(p^{k},p^{m},(p-1)p^{m-k-1})$-Golay-ZCZ sequence set. ∎

Since the codes in our construction lie inside the second-order cosets of
first-order $GRM$ codes, therefore, they have the high Hamming distance.
Hamming distance of a code measures how efficient a code is to detect and
correct errors. If the Hamming distance of a code is $h$ then we can correct
the errors of Hamming weight less than $h/2$. Therefore, in the next
corollary, we prove that minimum Hamming distance of the constructed CC-ZCZ
code set is equal to minimum Hamming distance of $GRM_{p}(m,1)$.

###### Corollary 3

The minimum Hamming distance of $(p^{k},p^{m},(p-1)p^{m-k-1})$-Golay-ZCZ
sequence sets which are constructed in _Theorem 1_ is $(p-1)p^{m-1}$.

###### Proof:

Since, $\mathbf{C^{\prime}_{u_{k}}}$ is contained in the second-order coset

$\mathcal{Q}_{1}+GRM_{p}(m,1)=\\{\mathcal{Q}_{1}+c:c\in GRM_{p}(m,1)\\}.$ (24)

Hence, according to _Lemma 1_, $\mathbf{C^{\prime}_{u_{k}}}$ has the minimum
Hamming distance $(p-1)p^{m-1}$ . ∎

Considering $p$-ary sequences, let $N(\mathcal{Q}_{1})$ be the number of coset
representatives $\mathcal{Q}_{1}$, $N(GZ)$ be the number of distinct
$(p^{k},p^{m},(p-1)p^{m-k-1})$-Golay-ZCZ sequence set in a coset
$\mathcal{Q}_{1}+GRM_{p}(m,1)$, $N(CZ)$ be the number of distinct
$(p^{k},p^{k},p^{m},(p-1)p^{m-k-1})$-CC-ZCZ code set in a coset
$\mathcal{Q}_{1}+GRM_{p}(m,1)$, and $d(\mathcal{Q}_{1}+GRM_{p}(m,1))$ be the
minimum Hamming distance of $\mathcal{Q}_{1}+GRM_{p}(m,1)$. Taking $p=3$ in
_corollary_ 21, _corollary_ 2, and _corollary_ 3, Table I shows the values of
$N(\mathcal{Q}_{1}),N(GZ),N(CZ)$ and $d(\mathcal{Q}_{1}+GRM_{3}(m,1))$ for
lengths $27$, $81$, and $243$. Taking $m=5$ and $k=1$, it can be observed from
Table I that there are $6$ second-order cosets $\mathcal{Q}_{1}+GRM_{3}(5,1)$
and each coset consists of $243$ distinct $(3,3,243,54)$-CC-ZCZ code set. If
we want to increase the set size from $3$ to $9$, we need to take $k=2$. Then,
Table I gives that we can obtain $81$, $(9,9,243,18)$-CC-ZCZ code set with
reduced ZCZ width from one of $6$ different second-order cosets
$\mathcal{Q}_{1}+GRM_{3}(5,2)$.

TABLE I: Calculated Values of $N(\mathcal{Q}_{1})$, $N(GZ)$, $(K,L,Z)$ and
$d(\mathcal{Q}_{1}+GRM_{3}(m,1))$.

$\mathbf{m}$ | $\mathbf{k}$ | $\mathbf{(K,K,L,Z)}$ | $\mathbf{N(\mathcal{Q}_{1})}$ | $\mathbf{N(CZ)}$ | $\mathbf{N(GZ)}$ | $\mathbf{d(\mathcal{Q}_{1}+GRM_{3}(m,1))}$
---|---|---|---|---|---|---
$3$ | $1$ | $(3,3,27,6)$ | $1$ | $27$ | $81$ | $18$
$2$ | $(9,9,27,2)$ | $1$ | $9$ | $18$
$4$ | $1$ | $(3,3,81,18)$ | $2$ | $81$ | $243$ | $54$
$2$ | $(9,9,81,6)$ | $2$ | $27$ | $54$
$3$ | $(27,27,81,2)$ | $1$ | $9$ | $54$
$5$ | $1$ | $(3,3,243,54)$ | $6$ | $243$ | $729$ | $162$
$2$ | $(9,9,243,18)$ | $6$ | $81$ | $162$
$3$ | $(27,27,243,6)$ | $3$ | $27$ | $162$
$4$ | $(81,81,243,2)$ | $1$ | $9$ | $162$

##  V PMEPR of Proposed CC-ZCZ Code Set

In this section, a method has been discussed to reduce column sequence PMEPR
of proposed CC-ZCZ code set. Since in the proposed CC-ZCZ code set, each code
is Golay-ZCZ sequence set of size $p^{k}$ and hence its row sequence PMEPR is
bounded by $p^{k}$. But column sequence PMEPR of CC-ZCZ can further be
reduced. If we want to generate $j$th column of $\mathbf{C^{\prime}_{u_{k}}}$,
$0\leq j\leq p^{m}-1$ given in (23) then only variable terms in (23) is
$\sum_{\beta=1}^{k}{x_{\pi_{\beta}{(n_{\beta})}}v_{\beta}}$. Now, we add some
constant to it and we can get,

$\displaystyle\phi_{j}(v)$
$\displaystyle=\frac{\lambda}{p}\sum_{\beta=1}^{k-1}{v_{\pi^{\prime}(\beta)}v_{\pi^{\prime}(\beta+1)}}+\frac{\lambda}{p}\sum_{\beta=1}^{k}{j_{\pi_{\beta}{(n_{\beta})}}v_{\beta}}+\frac{\lambda}{p}v_{\pi^{\prime}(1)}t+\frac{\lambda}{p}v_{\pi^{\prime}(k)}l$
(25)

where $l,t,v\in\mathbb{Z}_{p}$, $\pi^{\prime}$ be the permutation of symbols
$1,2,\ldots,k$, $[v_{1},v_{2},\ldots,v_{k}]$ be the $p$-ary representation of
$v$. By putting $m=k$ and $k=1$ in _Lemma 2_ it can easily be seen that
$\Theta(\phi_{j}(v))$ is a member of a GCS which has $p$ sequences each of
length $p^{k}$. Hence its column sequence PMEPR is bounded by $p$. Now, let us
redefine $\mathbf{C^{\prime}_{u_{k}}}$ as

$\mathbf{C^{\prime}_{u_{k}}}=\bigg{\\{}f+\frac{\lambda}{p}\sum_{\beta=0}^{k-1}{x_{\pi_{\beta+1}{(1)}}u_{\beta+1}}+\phi_{x}(v)+b_{0}:v_{\beta}\in\mathbb{Z}_{p},~{}1\leq\beta\leq
k\bigg{\\}},$ (26)

where

$\phi_{x}(v)=\frac{\lambda}{p}\sum_{\beta=1}^{k-1}{v_{\pi^{\prime}(\beta)}v_{\pi^{\prime}(\beta+1)}}+\sum_{\beta=1}^{k}{x_{\pi_{\beta}{(n_{\beta})}}v_{\beta}}+\frac{\lambda}{p}v_{\pi^{\prime}(1)}t+\frac{\lambda}{p}v_{\pi^{\prime}(k)}l.$
(27)

For some multivariable functions $g_{1},g_{2}$ and constants $c_{1},c_{2}$, it
can easily be verified that

$\displaystyle\mathcal{A}(g_{1}+c_{1})$ $\displaystyle=\mathcal{A}(g_{1}),$
$\displaystyle\mathcal{A}(g_{1}+c_{1},g_{2}+c_{2})$
$\displaystyle=\mathcal{A}(g_{1},g_{2})\omega^{c_{1}-c_{2}},$
$\displaystyle\mathcal{P}(g_{1}+c_{1})$ $\displaystyle=\mathcal{P}(g_{1}),$
(28) $\displaystyle\mathcal{P}(g_{1}+c_{1},g_{2}+c_{2})$
$\displaystyle=\mathcal{P}(g_{1},g_{2})\omega^{c_{1}-c_{2}}.$

Using (V), we can assure that redefining of $\mathbf{C^{\prime}_{u_{k}}}$ as
in (26) affect its correlation properties by a multiplication of constant,
i.e., $\mathbf{C^{\prime}_{u_{k}}}$ appearing in (26) is still a Golay-ZCZ
sequence set whose column sequence PMEPR is bounded by $p$.

###### Remark 2

In [44], the PMEPR of Golay-ZCZ sequence set is upper bounded by the number of
sub-carriers, i.e, $2^{k}$. The PMEPR of Golay-ZCZ sequence set increases with
the increase in the value of $k$. On the other hand, in the proposed Golay-ZCZ
sequence set, the number of sub-carriers is $p^{k}$ and upper bound for its
column sequence PMEPR is $p$. For $p=2$, we have $2^{k}$ sub-carriers and
maximum column sequence PMEPR $2$, which remains same with increase in the
value of $k$. So, the proposed construction have advantage over the
construction presented in [44] as maximum column sequence PMEPR of the
proposed construction is bounded above by $2$ for $2^{k}$ sub-carriers.

###### Remark 3

Although, we have taken CCCs from [14], but authors in [14] didn’t bound the
column sequence PMEPR in proposed construction. In the proposed construction,
we bound the maximum column sequence PMEPR of CCCs having flock size $p^{k}$
by $p$.

##  VI More CC-ZCZ Code Sets

In this section, another construction of CC-ZCZ code set is provided which are
different from those generated from _Theorem 1_. Similar to CC-ZCZ constructed
in _Theorem 1_, we also provide number of cosets of GRM codes, number of CC-
ZCZ code set in each coset, and Hamming distance but omit the proof as they
are same as proof given in section IV.

###### Theorem 2

For a fixed $\mathbf{u_{k}}$, $\mathbf{C_{u_{k}}}$ is GCS as defined in _Lemma
2_. Let $\mathbf{C^{\prime\prime}_{u_{k}}}$ be $\mathbf{C_{u_{k}}}$ together
with the permutation $\pi_{\beta}(n_{\beta})=m-k+\beta$, for
$\beta=1,2,\ldots,k$. Then $\mathbf{C^{\prime\prime}_{u_{k}}}$ is
$(p^{k},p^{m},(p-1)p^{\pi_{k}(n_{k}-1)-1})$-Golay-ZCZ sequence set and hence
$\mathbf{C^{\prime\prime}}=\\{\mathbf{C^{\prime\prime}_{u_{k}}}:0\leq\mathbf{u_{k}}\leq
p^{k}-1\\}$ is $(p^{k},p^{k},p^{m},(p-1)p^{\pi_{k}(n_{k}-1)-1})$-CC-ZCZ code
set.

###### Corollary 4

The number of coset representative $\mathcal{Q}_{2}$ in the second-order coset
of the form $\mathcal{Q}_{2}+GRM_{p}(m,1)$ is given by

$\sum_{n_{1}+n_{2}+\cdots+n_{k}=m}{{{m-k-1}\choose{n_{1}-1,n_{2}-1,\ldots,n_{k}-2}}(n_{k}-2)!\prod_{\beta=1}^{k-1}{(n_{\beta}-1)!}},$
(29)

where

$\mathcal{Q}_{2}=\sum_{\beta=1}^{k}{\sum_{\gamma=1}^{n_{\beta}-1}{x_{\pi_{\beta}(\gamma)}x_{\pi_{\beta}(\gamma+1)}}}.$
(30)

###### Corollary 5

Let $\mathcal{Q}_{2}+GRM_{p}(m,1)$ be any second-order coset with coset
representative $\mathcal{Q}_{2}$ as defined in (30) and let
$\pi_{k}(n_{k}-1)=m-k$. Then coset $\mathcal{Q}_{2}+GRM_{p}(m,1)$ consists of
$p^{m-k+1}$ and $p^{m+1}$ distinct $p$-ary
$(p^{k},p^{k},p^{m},(p-1)p^{m-k-1})$-CC-ZCZ code set and
$(p^{k},p^{m},(p-1)p^{m-k-1})$-Golay-ZCZ sequence set respectively.

###### Corollary 6

The minimum Hamming distance of $(p^{k},p^{m},(p-1)p^{m-k-1})$-Golay-ZCZ
sequence sets which are constructed in _Theorem 2_ is $(p-1)p^{m-1}$.

##  VII Comparison with Existing Works

Since the construction of CC-ZCZ is not available in the literature therefore
in this section, the contributed Golay-ZCZ sequences, ZCZ sequences, and PMEPR
of Golay-ZCZ sequences and CCCs are compared with existing works along with
comparison tables.

###  VII-A Comparison with existing constructions of Golay-ZCZ Sequence Set

####  VII-A1 Comparison with [22, 44]

In [44], authors proposed the construction of
$(2^{k},2^{m},2^{\pi_{1}(2)-1})$-Golay-ZCZ sequence sets. In our construction
(_Theorem_ 1), if we take $p=2$ then the parameters of Golay-ZCZ sequence sets
in [44] appear as special case of our parameters of Golay-ZCZ sequence sets.
Similarly, in [22], parameters of proposed
$(2^{k},2^{m},2^{\pi_{k}(n_{k})-1})$-Golay-ZCZ sequence sets appear as special
case of ours for $p=2$ in _Theorem_ 2.

####  VII-A2 Comparison with [45, 46]

In [45], authors proposed an indirect construction of
$(M,(M-1)N,M^{2}N)$-Golay-ZCZ sequence set with the help of $(M,M,N)$-CCC.
This construction requires CCC as a seed code to construct Golay-ZCZ sequence
set which makes it indirect. Furthermore, in the proposed construction, we are
getting Golay-ZCZ sequence set through a multivariable function which makes
our construction direct and advantageous over [45]. In [46], authors proposed
an indirect construction of Golay-ZCZ complementary pair of length $4N$ using
GCPs of length $N$, having ZCZ width $N+1$. This construction can produce only
GCPs but not GCSs. Moreover, length of sequence is also dependent on length of
GCP because the construction is indirect. On the other hand, the proposed
construction is direct which is based on multivariable functions. Furthermore,
the proposed construction can produce GCPs (for $p=2$) as well as GCSs (for
$p\neq 2$).

Author, Reference | Based on |  Parameter (K,L,Z) & Constraints | Direct/Indirect |  Optimality
---|---|---|---|---
Chen _et. al_ , [44] |  GBF of order $2$ |  $(2^{k},2^{m},2^{\pi_{1}{(2)}-1}),~{}\pi_{\alpha}(1)=m-\alpha+1$, $1\leq\beta\leq k$ | Direct |  Optimal for binary case
Wang _et. al_ , [22] |  GBF of order $2$ |  $(2^{k},2^{m},2^{\pi_{k}{(n_{k}-1)}-1}),~{}\pi_{\alpha}(n_{\alpha})=m-k+\alpha$, $1\leq\beta\leq k$ | Direct |  Optimal for binary case
Gu _et. al_ , [45] |  $(M,M,N)$-CCC |  $(M,(M-1)N,M^{2}N)$ | Indirect |  Optimal for binary case
Gu _et. al_ , [46] | Concatenation of GCP |  $(2,4N,N),~{}N$ is a positive integer | Indirect |  Asymptotically optimal
Theorem 1 | Second-order multivariable functions |  $(p^{k},p^{m},(p-1)p^{\pi_{1}(2)-1}),~{}p$ is prime and $\pi_{\beta}(1)=m-\beta+1,$ $1\leq\beta\leq k$ | Direct |  Optimal for binary case else asymptotically optimal
Theorem 2 | Second-order multivariable functions |  $(p^{k},p^{m},(p-1)p^{\pi_{k}(n_{k}-1)-1}),~{}p$ is prime and $\pi_{\beta}(n_{\beta})=m-k+\beta,$ $1\leq\beta\leq k$ | Direct |  Optimal for binary case else asymptotically optimal

TABLE II: Comparision with [22, 44, 45, 46]

A detailed comparison of contributed Golay-ZCZ sequences with existing works
is provided in Table II.

###  VII-B Comparison with existing constructions of ZCZ Sequences

The constructions of ZCZ sequences based on RM codes or Boolean functions [39,
40, 44], multivariable functions [41], bent functions[53], and perfect non-
linear function[54] are direct constructions of ZCZ sequences available in the
literature while many constructions need kernel sequences such as perfect
sequences [36, 55, 38, 37]. Although, constructions presented in [39, 40, 44]
are direct but length of ZCZ sequences is limited to power-of-two only. In
[53] and[54], the length of ZCZ sequences is limited to $N^{2}$ and $p^{2}$
respectively, where $p$ is an odd prime, and $N$ is a positive integer. Hence
flexibility in length parameter is not adequate here. Furthermore, in [41],
authors provided a direct construction of ZCZ sequences having flexible
length, i.e., $p^{n}$, where $n$ is a positive integer $\geq 3$ but it is
unable to produce ZCZ sequences of length $p^{2}$. The proposed construction
can generate ZCZ sequences of length $p^{n}$, where $n$ is a positive integer
$\geq 2$. A detailed comparison is provided in Table III.

TABLE III: Comparison with [35],[44],[55], [40], [54], [53], and [39]*

Author, Reference | Based on |  Parameter ($K$,$L,Z$) | Direct/Indirect |  Optimality
---|---|---|---|---
Tang _et. al_ , [35] | MOGCS |  $(2^{n},2^{n+1}Z_{cz},Z_{cz}+1),\ n\geq 1$ | Indirect |  Optimal
Chen _et. al_ , [44] | Generalised Boolean function |  $(2^{k},2^{m},2^{m-k-1}),$ $m\geq 2,$ and $k\leq m-1$. | Direct |  Optimal for binary case
Hayashi _et. al_ , [55] | Perfect sequences |  $(2(2n+1),4(2k+1)(2n+1),4k+1),\ n\geq 1,\ k\geq 1$ | Indirect |  Neither optimal nor almost optimal
Liu _et. al_ , [40] | Second-order Reed-Muller codes |  $(2^{k+1},2^{n+k+2},2^{n}),k\geq 0,n\geq 1$ | Direct |  Optimal for binary case
Zhou _et. al_ , [54] | Perfect non-linear function |  $(p,p^{2},p)$, $p$ is an odd prime | Direct |  Optimal
Zhang _et. al_ , [53] | Generalised Bent function |  $(N,N^{2},N)$, $N$ is positive integer. | Direct |  Optimal.
Tang _et. al_ , [39] | Generalised Boolean function |  $(2^{k},2^{m},2^{m-k}-2^{m-k-z})$ $,k\geq 0,n\geq 1$ | Direct |  optimal if $m=k+z$ and almost optimal if $m=k+z+1$
Kumar _et. al_ , [41] | Multivariable functions |  ${(p^{t},p^{n+t+1},(p-1)p^{n})},\ n,t\geq 1,p$ is prime and $t\leq n$ | Direct |  Optimal for binary case else asymptotically optimal
Theorem 1 and Theorem 2 | Multivariable functions |  $(p^{k},p^{m},(p-1)p^{m-k-1}),~{}p$ is prime | Direct |  Optimal for binary case else asymptotically optimal

  * *

The parameters presented in table are according to the definition of ZCZ in
the corresponding paper. However, changing the definition of ZCZ also changes
the bound and hence it doesn’t affect the optimality of ZCZ sequence sets.

###  VII-C Comparison with PMEPR of Existing Works.

In this section, the proposed PMEPR of CC-ZCZ code set is compared with the
PMEPR of existing constructions of CCCs, detailed comparison is provided in
Table IV. It can be observed from the Table IV that if we put $p=2$ in our
construction then, the parameter of CCCs presented in [13, 18, 56, 57] and
Golay-ZCZ sequence sets presented in [44, 22] can be obtained from our
construction. Furthermore, since PMEPR of our CC-ZCZ code set is bounded by 2
hence it has advantage over [13, 18, 57, 44, 22] and PMEPR of our CC-ZCZ code
set coincide with PMEPR of CCC in [56]. Additionally, the authors in [14]
presented a construction of CCCs but did not bound the column sequence PMEPR.
In Section V, we tackled this problem and provided a bound for column sequence
PMEPR of CCC.

TABLE IV: Comparison of the PMEPR of construction with [13, 18, 22, 44, 45,
46, 56, 57]

Author, Ref. | Parameters | Based on | Constraints |  Column sequence PMEPR is upper bounded by
---|---|---|---|---
Rathinakumar _et. al_ , [13] |  $(2^{k+1},2^{k+1},2^{m})$-CCC | GBF of order $2$ | $m,k\in\mathbb{Z}^{+},~{}m>1$ | $2^{k+1}$
Chen _et. al_ , [18] |  $(2^{k},2^{k},2^{m})$-CCC | GBF of order $>2$ |  $k,m\in\mathbb{Z}^{+}$, $m\geq 1,k\leq m$ | $2^{k}$
Wang _et. al_ [22] |  $(2^{k},2^{m},2^{m-k-1})$-Golay-ZCZ | GBF | $m,k\in\mathbb{Z}^{+},1\leq k\leq m-1$ | $2^{k}$
Chen _et. al_ [44] |  $(2^{k},2^{m},2^{m-k-1})$-Golay-ZCZ | GBF | $m,k\in\mathbb{Z}^{+},1\leq k\leq m-1$ | $2^{k}$
Gu _et. al_ , [45] |  $(M,(M-1)N,M^{2}N)$-Golay-ZCZ | CCC | There exist a $(M,M,N)$-CCC | $M$
Gu _et. al_ [46] |  $(2,4N,N)$-Golay-ZCZ | Concatenation of GCP |  $N$ is a positive integer | $2$
Z. Liu, [56] |  $(2^{k+1},2^{k+1},2^{m})$-CCC | GBF of order $>2$ | $m,k\in\mathbb{Z}^{+},~{}m>1$ | $2$
Wu _et. al_ , [57] |  $(2^{k},2^{k},2^{m})$-CCC | GBF of order $2$ |  $m,k\in\mathbb{Z}^{+}$, $m\geq 3,1\leq k\leq m$ | $2^{k}$
This paper* |  $(p^{k},p^{k},(p-1)p^{m-k-1},p^{m})$-CC-ZCZ | MVF of order $2$ |  $k,m\in\mathbb{Z}^{+}$, $p$ is prime | $p$

  * *

In this paper CCCs are directly taken from [14] but we have bounded the column
sequence PMEPR of these CCCs.

##  VIII Conclusion

In this paper, we settled open problem provided by Gong. _et al._ by providing
a direct construction of $(p^{k},p^{k},p^{m},(p-1)p^{\pi_{1}(2)-1})$-CC-ZCZ
code set and $(p^{k},p^{m},(p-1)p^{\pi_{1}(2)-1})$-Golay-ZCZ sequence set
using multivariable functions. The proposed construction provides the non-
power-of-two sequence length and larger ZCZ width. We are interested in larger
ZCZ width therefore by substituting $\pi_{1}(2)=m-k$,
$(p^{k},p^{m},(p-1)p^{m-k-1})$-CC-ZCZ code set is achieved, in which each
Golay-ZCZ sequence set is asymptotically optimal for non-binary case else
optimal, with respect to _Tang-Fan-Matsufuji_ Bound. We also established a
relation between proposed CC-ZCZ code set and GRM codes and provided minimum
Hamming distance of such codes. Moreover, the proposed construction provides
tighter upper bound for column sequence PMEPR of proposed CC-ZCZ code set. The
comparison of the proposed construction in the context of Golay-ZCZ sequences,
ZCZ sequences, and PMEPR is provided. Further, relying on the existing
literature, we propose two open problems as __

  1. 1.

_Construction of CC-ZCZ code set and Golay-ZCZ sequence set having length in
the form of product of different primes._

  2. 2.

_Since each code in CC-ZCZ code set is ZCZ sequence set so we can treat CC-ZCZ
code set as multiple ZCZ sequence set. Hence, it would be interesting to find
inter-set ZCZ which is beneficial in multiuser environment to resist inter-
cell interference caused by users from different cells in CDMA systems._

##  Appendix A Proof of Theorem 1

We first state and prove a lemma which will later be used to prove the
_Theorem_ 1.

###### Lemma 3

For two non-negative integers $i$ and $j$, let $(i_{1},i_{2},\ldots,i_{m})$
and $(j_{1},j_{2},\ldots,j_{m})$ be $p$-ary representation of $i$ and $j$
respectively. Let $f:\mathbb{Z}_{p}^{m}\rightarrow\mathbb{Z}_{\lambda}$ be a
function as defined in (15). Now, we can write
$a_{\mathbf{u_{k}}}^{\mathbf{v_{k}}}$ by using (16) and (17) as

$a_{\mathbf{u_{k}}}^{\mathbf{v_{k}}}=f+\frac{\lambda}{p}\sum_{\beta=1}^{k}{x_{\pi_{\beta}{(1)}}u_{\beta}}+\frac{\lambda}{p}\sum_{\beta=1}^{k}{x_{\pi_{\beta}{(n_{\beta})}}v_{\beta}}.$
(31)

Further, suppose we have
$i_{\pi_{\beta}(1)}=j_{\pi_{\beta}(1)}~{}\forall~{}\beta=1,2,\ldots,k$. For a
certain $\beta^{\prime}\leq k$, let us assume $t$ be the smallest integer such
that $i_{\pi_{\beta^{\prime}}(t)}\neq j_{\pi_{\beta^{\prime}}(t)}$. Let us
define $i^{\eta}$ to be an integer whose vector representation with base $p$
is

$(i_{1},i_{2},\ldots,i_{\pi_{\beta^{\prime}}(t-1)}-\eta,\ldots,i_{m}),$ (32)

which differs from that of $i$ only at the position
$\pi_{\beta^{\prime}}(t-1)$ and $\eta=1,2,\ldots,(p-1)$. Similarly, we define
$j^{\eta}$ such that its vector representation with base $p$ is

$(j_{1},j_{2},\ldots,j_{\pi_{\beta^{\prime}}(t-1)}-\eta,\ldots,j_{m}).$ (33)

Then we have

$\sum_{\eta=1}^{p-1}{\omega_{\lambda}^{(a_{\mathbf{u_{k}}}^{\mathbf{v_{k}}})_{j^{\eta}}-(a_{\mathbf{u_{k}}}^{\mathbf{v_{k}}})_{i^{\eta}}}}+\omega_{\lambda}^{(a_{\mathbf{u_{k}}}^{\mathbf{v_{k}}})_{j}-(a_{\mathbf{u_{k}}}^{\mathbf{v_{k}}})_{i}}=0.$
(34)

###### Proof:

For notational convenience, we write
$a_{\mathbf{u_{k}}}^{\mathbf{v_{k}}}=\mathbf{c}$. Now using (15) and (31), we
can write

$\displaystyle\mathbf{c}_{i^{\eta}}-\mathbf{c}_{i}=$ $\displaystyle
f_{i^{\eta}}-f_{i}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{p}[i_{\pi_{\beta^{\prime}}(t-2)}i_{\pi_{\beta^{\prime}}(t-1)}^{\eta}+i_{\pi_{\beta^{\prime}}(t-1)}^{\eta}i_{\pi_{\beta^{\prime}}(t)}-i_{\pi_{\beta^{\prime}}(t-2)}i_{\pi_{\beta^{\prime}}(t-1)}^{\eta}-i_{\pi_{\beta^{\prime}}(t-1)}i_{\pi_{\beta^{\prime}}(t)}]$
$\displaystyle+g_{\pi_{\beta^{\prime}}(t-1)}[i_{\pi_{\beta^{\prime}}(t-1)}^{\eta}-i_{\pi_{\beta^{\prime}}(t-1)}]$
$\displaystyle=$
$\displaystyle[-\eta\frac{\lambda}{p}i_{\pi_{\beta^{\prime}}(t)}-\eta\frac{\lambda}{p}i_{\pi_{\beta^{\prime}}(t-2)}-\eta
g_{\pi_{\beta^{\prime}}(t-1)}]$ $\displaystyle=$
$\displaystyle-\eta[\frac{\lambda}{p}i_{\pi_{\beta^{\prime}}(t)}+\frac{\lambda}{p}i_{\pi_{\beta^{\prime}}(t-2)}+g_{\pi_{\beta^{\prime}}(t-1)}].$
(35)

Similarly,

$\mathbf{c}_{j^{\eta}}-\mathbf{c}_{j}=f_{j^{\eta}}-f_{j}=-\eta[\frac{\lambda}{p}j_{\pi_{\beta^{\prime}}(t)}+\frac{\lambda}{p}j_{\pi_{\beta^{\prime}}(t-2)}+g_{\pi_{\beta^{\prime}}(t-1)}].$
(36)

From (35) and (36), we have

$(\mathbf{c}_{i^{\eta}}-\mathbf{c}_{j^{\eta}})-(\mathbf{c}_{i}-\mathbf{c}_{j})=-\eta\frac{\lambda}{p}(i_{\pi_{\beta^{\prime}}(t)}-j_{\pi_{\beta^{\prime}}(t)}).$
(37)

Now, taking sum of
$\omega_{\lambda}^{(\mathbf{c}_{i^{\eta}}-\mathbf{c}_{j^{\eta}})-(\mathbf{c}_{i}-\mathbf{c}_{j})}$
over $\eta$, we have

$\sum_{\eta=1}^{p-1}{\omega_{\lambda}^{\mathbf{c}_{i^{\eta}}-\mathbf{c}_{j^{\eta}}-(\mathbf{c}_{i}-\mathbf{c}_{j})}=\sum_{\eta=1}^{p-1}{\omega_{\lambda}^{\eta\frac{\lambda}{p}(j_{\pi_{\beta^{\prime}}(t)-i_{\pi_{\beta^{\prime}}(t)}})}}}=\sum_{\eta=1}^{p-1}{\omega_{p}^{\eta(j_{\pi_{\beta^{\prime}}(t)-i_{\pi_{\beta^{\prime}}(t)}})}}.$
(38)

It is given that $i_{\pi_{\beta^{\prime}}(t)}\neq
j_{\pi_{\beta^{\prime}}(t)}$, implies that
$i_{\pi_{\beta^{\prime}}(t)}-j_{\pi_{\beta^{\prime}}(t)}\neq 0$ and hence RHS
of (38) is the sum of roots of polynomial $z^{p}-1=0$ except the root $z=1$.
Hence

$\displaystyle\sum_{\eta=1}^{p-1}{\omega_{p}^{\eta(j_{\pi_{\beta^{\prime}}(t)-i_{\pi_{\beta^{\prime}}(t)}})}}=-1.$
(39)

Therefore, from (38) and (39), we have

$\sum_{\eta=1}^{p-1}{\omega_{\lambda}^{\mathbf{c}_{i^{\eta}}-\mathbf{c}_{j^{\eta}}-(\mathbf{c}_{i}-\mathbf{c}_{j})}}=-1,$
(40)

which further implies that

$\sum_{\eta=1}^{p-1}{\omega_{\lambda}^{\mathbf{c}_{i^{\eta}}-\mathbf{c}_{j^{\eta}}}+\omega_{\lambda}^{(\mathbf{c}_{i}-\mathbf{c}_{j})}}=0.$
(41)

∎

###### Proof:

We need to show that for a fixed $\mathbf{u_{k}}$, $\mathbf{C_{u_{k}}}$ is a
$(p^{k},(p-1)p^{\pi_{1}(2)-1},p^{m})$-ZCZ. Except the ZCZ width all the
parameters are directly inherited from _Lemma 2_. So, we only need to show
that the ZCZ width is $(p-1)p^{\pi_{1}(2)-1}$. Let
$\mathbf{c}\in\mathbf{C_{u_{k}}}$, by (18),
$\mathbf{c}=\psi(a_{\mathbf{u_{k}}}^{\mathbf{v_{k_{1}}}})$ for some
$0\leq\mathbf{v_{k_{1}}}\leq p^{k}-1$. First, we find the PACF of $\mathbf{c}$
and show that for $0<\tau\leq(p-1)p^{\pi_{1}(2)}-1$,

$\mathcal{P}(\mathbf{c})(\tau)=\sum_{i=0}^{L-1}{\omega_{\lambda}^{\mathbf{c}_{(i+\tau)mod~{}L}-\mathbf{c}_{i}}}=0,$
(42)

where $L$ is the length of the sequence, i.e., $L=p^{m}$. For any integer $i$,
let us denote another integer $j=(i+\tau)mod~{}L$. Then we consider two cases
and demonstrate that for each pair $(i,j)$ there exist other $(p-1)$ pairs
$(i^{\eta},j^{\eta}),~{}\eta=1,2,\ldots,p-1$ such that

$\sum_{\eta=1}^{p-1}{\omega_{\lambda}^{\mathbf{c}_{i^{\eta}}-\mathbf{c}_{j^{\eta}}}+\omega_{\lambda}^{(\mathbf{c}_{i}-\mathbf{c}_{j})}}=0,$
(43)

in each case.

######  Case 1 ($i_{\pi_{1}(2)}=j_{\pi_{1}(2)}$)

In this case, we have
$i_{\pi_{\beta}(1)}=j_{\pi_{\beta}(1)},~{}\forall\beta=1,2,\ldots,k$. For the
contrary, suppose this is not true. Then, assume that $\bar{\beta}$ is the
largest integer such that $i_{\pi_{\bar{\beta}}(1)}\neq
j_{\pi_{\bar{\beta}}(1)}$. For ease of presentation, let
$d=\pi_{\bar{\beta}}(1)$, now if $j_{d}>i_{d}$, we have

$\displaystyle\tau=j-i=$ $\displaystyle\sum_{s=1}^{d}{(j_{s}-i_{s})p^{s-1}}$
$\displaystyle=$
$\displaystyle(j_{d}-i_{d})p^{d-1}+\sum_{s=1,s\neq\pi_{1}(2)}^{d-1}{(j_{s}-i_{s})p^{s-1}}$
$\displaystyle\geq$
$\displaystyle(j_{d}-i_{d})p^{d-1}-(p-1)\sum_{s=1}^{d-1}{p^{s-1}}+(p-1)p^{\pi_{1}(2)-1}$
$\displaystyle=$
$\displaystyle(j_{d}-i_{d})p^{d-1}-(p-1)\Big{[}\frac{p^{d-1}-1}{p-1}\Big{]}+(p-1)p^{\pi_{1}(2)-1}$
$\displaystyle=$
$\displaystyle(j_{d}-i_{d}-1)p^{d-1}+1+(p-1)p^{\pi_{1}(2)-1}>(p-1)p^{\pi_{1}(2)-1}.$
(44)

Hence (44) implies that $\tau>(p-1)p^{\pi_{1}(2)-1}$ which is a contradiction.
Similarly, if $j_{d}<i_{d}$, then

$\displaystyle\tau$
$\displaystyle=j-i+p^{m}=\sum_{s=1}^{d}{(j_{s}-i_{s})p^{s-1}}+p^{m}$
$\displaystyle=(j_{d}-i_{d})p^{m-\bar{\beta}}+p^{m}+\sum_{s=1,s\neq\pi_{1}(2)}^{d-1}{(j_{s}-i_{s})p^{s-1}}$
$\displaystyle=(j_{d}-i_{d}+p^{\bar{\beta}})p^{m-\bar{\beta}}+\sum_{s=1,s\neq\pi_{1}(2)}^{d-1}{(j_{s}-i_{s})p^{s-1}}$
$\displaystyle\geq(j_{d}-i_{d}+p^{\bar{\beta}})p^{m-\bar{\beta}}-(p-1)\sum_{s=1}^{d-1}{p^{s-1}}+(p-1)p^{\pi_{1}(2)-1}$
$\displaystyle=$
$\displaystyle(j_{d}-i_{d}+p^{\bar{\beta}}-1)p^{m-\bar{\beta}}+1+(p-1)p^{\pi_{1}(2)-1}>(p-1)p^{\pi_{1}(2)-1}.$
(45)

Again we got a contradiction. Hence
$i_{\pi_{\beta}(1)}=j_{\pi_{\beta}(1)}~{}\forall\beta=1,2,\ldots,k$. Now
without loss of generality, we assume that there exist a positive integer
$\beta^{\prime}\leq k$ such that
$i_{\pi_{\beta}(r)}=j_{\pi_{\beta}(r)},~{}\forall\beta=1,2,\ldots,\beta^{\prime}-1$
and $r=1,2,\ldots,n_{\beta}$. Assume $t$ be the smallest integer with
$i_{\pi_{\beta^{\prime}}(t)}\neq j_{\pi_{\beta^{\prime}}(t)}$. Now, let us
define $i^{\eta}$ and $j^{\eta}$ same as in (32) and (33) respectively. Then
it can easily be obtained that $j^{\eta}=(i^{\eta}+\tau)mod~{}L$ and hence
using _Lemma_ 3, we get (43).

######  Case 2 ($i_{\pi_{1}(2)}\neq j_{\pi_{1}(2)}$)

In this case, let $i^{\eta}$ and $j^{\eta}$ are modified from $i$ and $j$ by
changing only last bit of $i$ and $j$ as $i_{m}^{\eta}=i_{m}-\eta$ and
$j_{m}^{\eta}=j_{m}-\eta$. Then

$\displaystyle\mathbf{c}_{i^{\eta}}-\mathbf{c}_{i}$
$\displaystyle=f_{i^{\eta}}-f_{j}+\frac{\lambda}{p}(i_{m}-\eta-i_{m})u_{1}$
$\displaystyle=\frac{\lambda}{p}\left[i_{m}^{\eta}i_{\pi_{1}(2)}-i_{m}i_{\pi_{1}(2)}\right]+g_{m}(i_{m}-\eta-
i_{m})+\frac{\lambda}{p}(i_{m}-\eta-i_{m})u_{1}$
$\displaystyle=-\eta\left[\frac{\lambda}{p}i_{\pi_{1}(2)}+g_{m}+\frac{\lambda}{p}u_{1}\right].$
(46)

Similarly,

$\mathbf{c}_{j^{\eta}}-\mathbf{c}_{j}=-\eta\left[\frac{\lambda}{p}j_{\pi_{1}(2)}+g_{m}+\frac{\lambda}{p}u_{1}\right].$
(47)

By subtracting (47) from (46), we get

$(\mathbf{c}_{i^{\eta}}-\mathbf{c}_{j^{\eta}})-(\mathbf{c}_{i}-\mathbf{c}_{j})=-\eta\frac{\lambda}{p}(i_{\pi_{1}(2)}-j_{\pi_{1}(2)}).$
(48)

Now following the same steps as in (37), (38), (39), and (40), we get

$\sum_{\eta=1}^{p-1}{\omega_{\lambda}^{\mathbf{c}_{i^{\eta}}-\mathbf{c}_{j^{\eta}}}+\omega_{\lambda}^{(\mathbf{c}_{i}-\mathbf{c}_{j})}}=0.$
(49)

Till now we have proved that for $0<\tau\leq(p-1)p^{\pi_{1}(2)-1}$, the value
of $\mathcal{P}(\mathbf{c})(\tau)=0$. Now in the rest of the proof, we will
prove that for any $0\leq\tau\leq(p-1)p^{\pi_{1}(2)-1}$, the PCCF of any two
different sequences in $\mathbf{C_{u_{k}}}$ is zero. For that let
$0\leq\bm{\gamma_{k}},\bm{\delta_{k}}\leq p^{k}-1$ such that
$\psi(a_{\mathbf{u_{k}}}^{\bm{\gamma_{k}}}),\psi(a_{\mathbf{u_{k}}}^{\bm{\delta_{k}}})\in\mathbf{C_{u_{k}}}$.
Again for notational convenience, we denote
$\psi(a_{\mathbf{u_{k}}}^{\bm{\gamma_{k}}})$ and
$\psi(a_{\mathbf{u_{k}}}^{\bm{\delta_{k}}})$ by $\mathbf{b}$ and $\mathbf{c}$
respectively. Then, we need to prove that for
$0\leq\tau\leq(p-1)p^{\pi_{1}(2)-1}$,

$\mathcal{P}(\mathbf{c},\mathbf{b})(\tau)=\sum_{i=0}^{L-1}{\omega_{\lambda}^{\mathbf{c}_{(i+\tau)mod~{}L}-\mathbf{b}_{i}}}=0.$
(50)

Let $j=(i+\tau)mod~{}L$. Following similar arguments as in _Case_ $1$ and
_Case_ $2$, for any pair $(i,j)$, we can find other pairs
$(i^{\eta},j^{\eta}),~{}\eta=1,2,\ldots,k-1$ such that

$\sum_{\eta=1}^{p-1}{\omega_{\lambda}^{\mathbf{c}_{i^{\eta}}-\mathbf{b}_{j^{\eta}}}+\omega_{\lambda}^{(\mathbf{c}_{i}-\mathbf{b}_{j})}}=0,$
(51)

for $\tau\neq 0$, where $j^{\eta}=(i^{\eta}+\tau)mod~{}L$. Therefore, we can
obtain that (50) holds for $\tau\neq 0$. Now, it remains to prove that,

$\mathcal{P}(\mathbf{c},\mathbf{b})(0)=\sum_{i=0}^{L-1}{\omega_{\lambda}^{\mathbf{c}_{i}-\mathbf{b}_{i}}}=0.$
(52)

Taking (31) into consideration, let
$\mathbf{c}-\mathbf{b}=\frac{\lambda}{p}.\mathbf{d}$. Then $\mathbf{d}$ is a
non-zero codeword in $GRM_{p}(1,m)$. Now let
$\mathbf{d}=(d_{1},d_{2},\ldots,d_{p^{m}})$ and hence $\mathbf{d}$ can be
written as linear combination of $\theta(x_{1}),\theta(x_{2}),$
$\ldots,\theta(x_{m})$ as
$\mathbf{d}=c_{1}\cdot\theta(x_{1})+c_{2}\cdot\theta(x_{2})+\cdots+c_{m}\cdot\theta(x_{m})$,
where $c_{i}\in\mathbb{Z}_{p},~{}1\leq i\leq m$. For each $i$, $\theta(x_{i})$
contains each element of the set $\\{0,1,\ldots,p-1\\}$, $p^{m-1}$ times.
Hence $\mathbf{d}$ will also contains each element of the set
$\\{0,1,\ldots,p-1\\}$, $p^{m-1}$ times. Now (52) can be written as

$\mathcal{P}(\mathbf{c},\mathbf{b})(0)=\sum_{i=0}^{L-1}{\omega_{\lambda}^{\frac{\lambda}{p}\mathbf{d}_{i}}}=\sum_{i=0}^{L-1}{\omega_{\lambda}^{\frac{\lambda}{p}\mathbf{d}_{i}}}=\sum_{i=0}^{L-1}{\omega_{p}^{\mathbf{d}_{i}}}=0.$
(53)

Hence the _Theorem_ 1 is proved.

∎

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