[2]Sudhan Majhi

1]Department of Mathematics, Indian Institute of Technology, Patna, 801106,
Bihar, India

2]Department of Electrical Communication Engineering, Indian Institute of
Sciene, Bangalore, 560012, Karnataka, India

# A Direct Construction of Type-II $Z$ Complementary Code Set with Arbitrarily
Large Codes

Rajen Kumar<EMAIL_ADDRESS>Prashant Kumar Srivastava
<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [

###### Abstract

In this paper, we propose a construction of type-II $Z$-complementary code set
(ZCCS), using a multi-variable function with Hamiltonian paths and disjoint
vertices. For a type-I $(K,M,Z,N)$-ZCCS, $K$ is bounded by $K\leq
M\left\lfloor\frac{N}{Z}\right\rfloor$. However, the proposed type-II ZCCS
provides $K=M(N-Z+1)$. The proposed type-II ZCCS provides a larger number of
codes compared to that of type-I ZCCS. Further, the proposed construction can
generate the Kernel of complete complementary code (CCC) as $(p,p,p)$-CCC, for
any integral value of $p\geq 2$.

###### keywords:

$Z$-complementary code set (ZCCS), complete complementary code (CCC),
generalized Boolean function (GBF), aperiodic correlation, zero correlation
zone

##  1 Introduction

A collection of matrices is described as a complete complementary code (CCC)
set if the aperiodic auto-correlation sum (AACS) of each matrix/code is zero
for all non-zero time shifts and the aperiodic cross-correlation sum (ACCS)
between any two matrices is zero for all time shifts [1]. CCC has shown a
significant role over asynchronous multi-carrier code division multiple access
(MC-CDMA) for interference-free communication [2]. Rathinakumar and Chaturvedi
proposed $(2^{k},2^{k},2^{k+m})$-CCC using a generalized Boolean function
(GBF) associating it with a graph of a Hamiltonian path of $m$ vertices after
deleting the $k$ vertices from the graph [3]. To utilize more users in the MC-
CDMA system, a zero correlation zone (ZCZ) has been introduced in the code and
formed $Z$-complementary code set (ZCCS) in [4]. ZCCSs are capable of
supporting interference-free MC-CDMA in the quasi-synchronous environment
without the need of power control [2]. Depending on whether the ZCZ width
exists close to zero time shift or end time shift, the set is referred to as a
type-I ZCCS or type-II ZCCS, respectively. A $(K,M,Z,N)$-ZCCS is a collection
of $K$ matrices; each matrix has $M$ constituent sequences, each of length $N$
and the ZCZ width is $Z$. For a type-I $(K,M,Z,N)$-ZCCS, $K$ is bounded by
$K\leq M\left\lfloor\frac{N}{Z}\right\rfloor$ [4]. In other words, if ZCZ
width is considered to be $1/n$ of the complete time zone for some natural
number $n$, the number of users can be increased by $n$ times to the number of
constituent sequences. In [5, 6, 7, 8, 9, 10], construction of type-I ZCCS has
been proposed following the above bound. Recently, type-II ZCCS has been
proposed to improve the number of codes in the MC-CDMA system in [11].
However, the code in [11] is based on CCC and indirect construction. Best of
the Authors‘ knowledge, there is no direct construction of type-II ZCCS in the
existing literature.

In this paper, we propose a direct construction of type-II
$(p^{r+k},p^{k},p^{n+r}-p^{r}+1,p^{n+r})$-ZCCS, where $p\geq 2$, $0<k<n$ and
$r\geq 0$, i.e., we have $p^{r}$ times more codes compared to the flock size
by compromising $p^{r}-1$ shift of ZCZ width. Therefore, a large number of
codes can be generated compared to type-I ZCCS with the same number of
constituent sequences in a code. To construct type-II ZCCS, a multi-variable
function (MVF) is used, which is associated with a graph of a collection of
$k$ Hamiltonian paths and $r$ disjoint vertices. The proposed method can
generate the Kernel of CCC as $(p,p,p)$-CCC, for any integral value of $p\geq
2$, but in [12] the Kernel can be obtained only for the prime values of $p$.
As the proposed type-II ZCCS has larger ZCZ width and more codes compared to
type-I ZCCS, it is suitable for the asynchronous environment.

The rest of this paper is written as follows. The notations, fundamental
definitions, and relationship between a graph and an MVF are defined in
Section 2. In Section 3, we describe the construction of type-II ZCCS and
provide an example of the proposed construction. Section 4, contains the
conclusion of the paper.

##  2 Preliminaries

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

######  Definition 1.

Let $\mathbf{a}=(a_{0},a_{1},\ldots,a_{N-1})$ and
$\mathbf{a^{\prime}}=(a^{\prime}_{0},a^{\prime}_{1},$$\ldots,$$a^{\prime}_{N-1})$
be two complex-valued sequences of length $N$. The ACCS between $\mathbf{a}$
and $\mathbf{a^{\prime}}$ is defined as

$\mathcal{C}_{\mathbf{a},\mathbf{a^{\prime}}}(\tau)=\left\\{\begin{array}[]{ll}\sum_{i=0}^{N-1-\tau}a_{i+\tau}a_{i}^{{}^{\prime}*},&0\leq\tau<N,\\\
\sum_{i=0}^{N+\tau-1}a_{i}a_{i-\tau}^{{}^{\prime}*},&-N<\tau<0,\\\ 0,&\text{
otherwise, }\end{array}\right.$

where $a_{i}^{{}^{\prime}*}$ is complex conjugate of $a^{\prime}_{i}$. When
$\mathbf{a}=\mathbf{a^{\prime}}$,
$\mathcal{C}_{\mathbf{a},\mathbf{a^{\prime}}}(\tau)$ is called AACS of
$\mathbf{a}$ and is denoted as $\mathcal{C}_{\mathbf{a}}(\tau)$.

Let $C_{i}$ be a code and defined as

$C_{i}=\left[\begin{array}[]{c}\mathbf{a}_{0}^{i}~{}\mathbf{a}_{1}^{i}~{}\ldots~{}\mathbf{a}_{M-1}^{k}\end{array}\right]^{T}_{M\times
N}.$

######  Definition 2.

Let $C_{s}$ and $C_{t}$ be any two codes, then ACCS between $C_{s}$ and
$C_{t}$ is defined by

$\mathcal{C}\left({C}_{s},{C}_{t}\right)(\tau)=\sum_{\nu=0}^{M-1}\mathcal{C}\left(\mathbf{a}_{\nu}^{s},\mathbf{a}_{\nu}^{t}\right)(\tau).$
(1)

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

######  Definition 3.

Let $\mathbf{C}=\left\\{{C}_{0},{C}_{1},\ldots,{C}_{K-1}\right\\}$ be a set of
$K$ matrices (codes), each of having order $M\times N$. $\mathbf{C}$ is called
a type-I $(K,M,Z,N)$-ZCCS if it satisfies the following properties

$\mathcal{C}\left(C_{k_{1}},C_{k_{2}}\right)(\tau)=\left\\{\begin{array}[]{ll}NM,&\tau=0,k_{1}=k_{2},\\\
0,&\tau=0,k_{1}\neq k_{2},\\\ 0,&1\leq|\tau|<Z.\end{array}\right.$ (2)

######  Definition 4.

Let $\mathbf{C}=\left\\{{C}_{0},{C}_{1},\ldots,{C}_{K-1}\right\\}$ be a set of
$K$ matrices (codes), each of having order $M\times N$. $\mathbf{C}$ is called
a type-II $(K,M,Z,N)$-ZCCS if it satisfies the following properties

$\mathcal{C}\left(C_{k_{1}},C_{k_{2}}\right)(\tau)=\left\\{\begin{array}[]{ll}NM,&\tau=0,k_{1}=k_{2},\\\
0,&\tau=0,k_{1}\neq k_{2},\\\ 0,&N-Z<|\tau|<N.\end{array}\right.$ (3)

When $K=M$ and $Z=N$, type-II ZCCS is called a $(K,K,N)-$CCC.

###  2.1 Multi-variable Functions and Corresponding Sequences

Let $\mathbf{x}=\\{x_{0},x_{1},\ldots,x_{n-1}\\}$ such that
$\mathbf{x}\in\mathbb{Z}_{p}^{n}$, where
$\mathbb{Z}_{p}=\\{0,1,\ldots,p-1\\}$, for $p\geq 2$. We say $f$ is a MVF if
$f:\mathbb{Z}_{p}^{n}\to\mathbb{Z}_{q}$. We define
$\chi(f)=\left(\zeta_{q}^{f(0)},\zeta_{q}^{f(1)},\ldots,\zeta_{q}^{f(p^{n}-1)}\right)$,
where $f(I)=f(I_{0},I_{1},\ldots,I_{n-1})$ [13], such that

$I=\sum_{i=0}^{n-1}p^{n-i-1}I_{i}.$ (4)

###  2.2 Relation with graph and multi-variable function

Let $Q$$:$$\mathbb{Z}_{p}^{n}$$\to$$\mathbb{Z}_{q}$ be a MVF defined by

$Q(x_{0},x_{1},\ldots,x_{n-1})=\sum_{0\leq i<j<n}h_{ij}x_{i}x_{j},$ (5)

where $h_{ij}$$\in$$\mathbb{Z}_{q}$, so that $Q$ is a quadratic form in $n$
variables, $x_{0},x_{1},\ldots,x_{n-1}$ over $\mathbb{Z}_{p}$. Let
$\mathcal{G}(Q)$ be a labelled graph of $n$ vertices with $Q$ as given in (5).
Label the vertices by $x_{0},x_{1},\ldots,x_{n-1}$ and if $h_{ij}\neq 0$, join
vertices $x_{i}$ and $x_{j}$ by an edge with the label of $h_{ij}$. A graph
$\mathcal{G}(Q)$ of the type defined above is a path if $n\geq 1$ and
$\mathcal{G}(Q)$ has exactly $n-1$ edges, all are labelled by $\frac{q}{p}$
and form a Hamiltonian path in $\mathcal{G}(Q)$. For $n\geq 1$, a path on $n$
vertices corresponds to a quadratic form of the type

$Q(x_{0},x_{1},\ldots,x_{n-1})=\frac{q}{p}\sum_{i=1}^{n-1}x_{\pi(i-1)}x_{\pi(i)},$
(6)

where $\pi$ is a permutation of $\\{0,1,\ldots,n-1\\}$. This $\pi$ is
dependent on the Hamiltonian path. Now, we define MVF over Hamiltonian path
$\mathcal{G}(Q)$ by $H:\mathbb{Z}_{p}^{n}\to\mathbb{Z}_{p}^{h}$ as

$H(x_{0},x_{1},\ldots,x_{n-1})=Q+\sum_{i=0}^{n-1}\gamma_{i}x_{i}+\theta,$ (7)

where $\gamma_{i},\theta\in\mathbb{Z}_{p}$.

######  Definition 5.

Let
$f_{1}(x_{1,0},x_{1,1},\ldots,x_{1,n_{1}-1})$$:$$\mathbb{Z}_{p}^{n_{1}}$$\to$$\mathbb{Z}_{q}$
and
$f_{2}(x_{2,0},x_{2,1},\ldots,x_{2,n_{2}-1})$$:$$\mathbb{Z}_{p}^{n_{2}}$$\to$$\mathbb{Z}_{q}$
be two MVFs then we define a new MVF as
$f(x_{0},x_{1},\ldots,x_{n_{1}+n_{2}-1})$$:$$\mathbb{Z}_{p}^{n_{1}+n_{2}}$$\to$$\mathbb{Z}_{q}$
such that
$(x_{0},x_{1},\ldots,x_{n_{1}+n_{2}-1})$$=$$(x_{1,0},x_{1,1},\ldots,x_{1,n_{1}-1},x_{2,0},$
$x_{2,1},\ldots,x_{2,n_{2}-1})$ and represented by $f=f_{1}\oplus f_{2}$.

Let $\mathcal{G}(Q_{0}),\mathcal{G}(Q_{1}),\ldots,\mathcal{G}(Q_{k-1})$ are
$k$ many Hamiltonian paths with $n_{0},n_{1},\ldots,n_{k-1}$ vertices,
respectively, and $x_{\pi_{i}(0)}$ and $x_{\pi_{i}(n_{i}-1)}$ are the end
vertices of $\mathcal{G}(Q_{i})$. We define a MVF of
$n(=n_{0}+n_{1}+\cdots+n_{k-1})$ variable as follows

$f_{\beta}^{\alpha}=\sum_{i=0}^{k-1}\left(H(\mathbf{x}_{i})+\frac{q}{p}\left(\alpha_{i}x_{\pi_{i}(0)}+\beta_{i}x_{\pi_{i}(n_{i}-1)}\right)\right),$
(8)

where $\mathbf{x}_{i}$ are vertices of $\mathcal{G}(Q_{i})$, $H$ is as defined
in (7), $\alpha=\sum_{i=0}^{k-1}p^{k-i-1}\alpha_{i}$ and
$\beta=\sum_{i=0}^{k-1}p^{k-i-1}\beta_{i}$, i.e.,
$\boldsymbol{\alpha}=(\alpha_{0},\alpha_{1},\ldots,\alpha_{k-1})$ and
$\boldsymbol{\beta}=(\beta_{0},\beta_{1},\ldots,\beta_{k-1})$ are vector
representation of $\alpha$ and $\beta$ with the base $p$, respectively.

###  2.3 PMEPR

Let $\mathbf{a}=(a_{0},a_{1},\ldots,a_{M-1})$ be a complex valued sequence of
length $N$. For a multi-carrier system with $M$ subcarriers, the time domain
multi-carrier signal can be written as

$s_{\mathbf{a}}(t)=\sum_{j=0}^{M-1}a_{j}e^{2\pi\sqrt{-1}jt},$ (9)

where $0\leq t<1$ and the carrier spacing has been normalized to $1$ and
$\mathbf{a}$ is spreaded over $M$ subcarriers. The instantaneous envelope
power of the signal is the real-valued function
$P_{\mathbf{a}}(t)=|s_{\mathbf{a}}(t)|^{2}$. A ZCCS based MC-CDMA is given in
[14]. From [15], the PMEPR is bounded by the AACS value of $\mathbf{a}$ as
follows,

$PMEPR(\mathbf{a})\leq\sum_{\tau=-M+1}^{M+1}\mathcal{C}(\mathbf{a})(\tau).$
(10)

##  3 Proposed Construction

In this section first we are proposing two lemmas, which are nothing but
extenstion of two well known results. These lemmas are in support for complete
the proof of the proposed theorems. Lemma 1 is extension of sum of root of
unity and Lemma 2 is extension orthogonality of Walash function over prime to
any number bigger than two.

######  Lemma 1.

For $q(\geq 2)\in\mathbb{N}$, $k,r\in\mathbb{Z}$ such that $k\neq rq$. Let
$\zeta_{q}=\exp{2\pi\iota/q}$, then

$\sum_{i=0}^{q-1}\zeta_{q}^{k\cdot i}=0.$ (11)

###### Proof.

Since $k\neq rq$, let $k_{1}=k\ (\mathrm{mod}\ q)$, since $k\neq rq$ it
implies $k_{1}\neq 0$. Now $k_{1}\in\mathbb{Z}_{q}$. To complete the proof we
consider two case.

Case 1. $k_{1}$ is relatively prime with $q$. Then

$\displaystyle\sum_{i=0}^{q-1}\zeta_{q}^{k\cdot i}$
$\displaystyle=\sum_{i=0}^{q-1}\zeta_{q}^{k_{1}\cdot i}$ (12)
$\displaystyle=\sum_{i=0}^{q-1}\zeta_{q}^{i}=0,~{}~{}\text{sum of $q$-th root
of unity.}$

Case 2. When $k_{1}$ is not relatively prime with $q$. Let
$d\in\mathbb{Z}_{q}$, such that $k_{1}=k_{2}d$ and $q=q_{1}d$ and $k_{2}$ is
relatively prime with $q_{1}$. Now,

$\displaystyle\sum_{i=0}^{q-1}\zeta_{q}^{k\cdot i}$
$\displaystyle=\sum_{i=0}^{q-1}\zeta_{q}^{k_{1}\cdot i}$ (13)
$\displaystyle=\sum_{i=0}^{q-1}\zeta_{q}^{k_{2}d\cdot i}$
$\displaystyle=\sum_{i=0}^{q-1}\zeta_{q_{1}}^{k_{2}\cdot i}$
$\displaystyle=d\sum_{i=0}^{q-1}\zeta_{q_{1}}^{i}$
$\displaystyle=0,~{}~{}\text{sum of $q_{1}$-th root of unity.}$

Concluding above two cases, the proof completes. ∎

######  Lemma 2.

Let $\mathcal{G}(\Omega)$ be graph of $r$ isolated vertices then MVF
$\Omega^{\delta}:\mathbb{Z}_{p}^{r}\to\mathbb{Z}_{q}$ defined by

$\Omega^{\delta}(\mathbf{x})=\frac{q}{p}\boldsymbol{\delta}\cdot\mathbf{x},$
(14)

where $\boldsymbol{\delta}$ is vector representation of $0\leq\delta<p^{r}$,
with base $p$ and $q$ is integral multiple of $p$.
$\Omega^{\delta}(\mathbf{x})$ is known as Walsh function [16]. Therefore, dot
product between sequences $\chi(\Omega^{\delta^{1}}(\mathbf{x}))$ and
$\chi(\Omega^{\delta^{2}}(\mathbf{x}))$ is zero, for
$\delta^{1}\neq\delta^{2}$.

###### Proof.

Let $\delta^{1}\neq\delta^{2}$, therefore, $\boldsymbol{\delta}^{1}$ is differ
from $\boldsymbol{\delta}^{2}$ for at least one elements. To proof completely,
we consider two cases as follows:

Case 1. Let
$\boldsymbol{\delta}^{1}=(\delta_{0},\delta_{1},\ldots,\delta^{1}_{u},\ldots,\delta_{r-1})$
and
$\boldsymbol{\delta}^{2}=(\delta_{0},\delta_{1},\ldots,\delta^{2}_{u},\ldots,\delta_{r-1})$.

$\Omega^{\delta^{1}}(\mathbf{x})-\Omega^{\delta^{2}}(\mathbf{x})=\frac{q}{p}(\delta^{1}_{u}-\delta^{2}_{u})x_{u}.$
(15)

Let for and integer $I$, the $p$-ary vector representation is
$\mathbf{I}=(I_{0},I_{1},\ldots,I_{r-1})$.

$\Omega^{\delta^{1}}(\mathbf{I})-\Omega^{\delta^{2}}(\mathbf{I})=\frac{q}{p}(\delta^{1}_{u}-\delta^{2}_{u})I_{u}.$
(16)

Now, we use this in dot product of $\chi(\Omega^{\delta^{1}}(\mathbf{x}))$ and
$\chi(\Omega^{\delta^{2}}(\mathbf{x}))$.

$\displaystyle\chi(\Omega^{\delta^{1}}(\mathbf{x}))\cdot\chi(\Omega^{\delta^{2}}(\mathbf{x}))=$
$\displaystyle\sum_{I=0}^{p^{r}-1}\zeta_{q}^{\frac{q}{p}(\delta^{1}_{u}-\delta^{2}_{u})I_{u}}$
(17) $\displaystyle=$ $\displaystyle
p^{r-1}\sum_{I_{u}=0}^{p-1}\zeta_{q}^{\frac{q}{p}(\delta^{1}_{u}-\delta^{2}_{u})I_{u}}$
$\displaystyle=$ $\displaystyle
p^{r-1}\sum_{I_{u}=0}^{p-1}\zeta_{p}^{(\delta^{1}_{u}-\delta^{2}_{u})I_{u}}$
$\displaystyle=$ $\displaystyle 0,~{}~{}\text{from {Lemma}
\ref{Lem:sum_zero}.}$

Case 2. Let $\boldsymbol{\delta}^{1}$ is differ form $\boldsymbol{\delta}^{2}$
at more than one place. Let $0\leq j_{0}<j_{1}<\ldots<j_{k-1}\leq r-1$, for
$1<k\leq r$ such that $\boldsymbol{\delta}^{1}$ is differ form
$\boldsymbol{\delta}^{2}$ at $j_{0},j_{1},\ldots,j_{k}$ place. Now

$\Omega^{\delta^{1}}(\mathbf{x})-\Omega^{\delta^{2}}(\mathbf{x})=\frac{q}{p}\sum_{u=0}^{k-1}(\delta^{1}_{j_{u}}-\delta^{2}_{j_{u}})x_{j_{u}}.$
(18)

Now, we use this in dot product of $\chi(\Omega^{\delta^{1}}(\mathbf{x}))$ and
$\chi(\Omega^{\delta^{2}}(\mathbf{x}))$.

$\displaystyle\chi(\Omega^{\delta^{1}}(\mathbf{x}))\cdot\chi(\Omega^{\delta^{2}}(\mathbf{x}))=$
$\displaystyle\sum_{I=0}^{p^{r}-1}\zeta_{q}^{\sum_{u=0}^{k-1}\frac{q}{p}(\delta^{1}_{j_{u}}-\delta^{2}_{j_{u}})I_{j_{u}}}$
(19) $\displaystyle=$
$\displaystyle\sum_{I=0}^{p^{r}-1}\zeta_{p}^{\sum_{u=0}^{k-1}(\delta^{1}_{j_{u}}-\delta^{2}_{j_{u}})I_{j_{u}}}$
$\displaystyle=$
$\displaystyle\sum_{I=0}^{p^{r}-1}\left(\prod_{u=0}^{k-1}\zeta_{p}^{(\delta^{1}_{j_{u}}-\delta^{2}_{j_{u}})I_{j_{u}}}\right)$
$\displaystyle=$ $\displaystyle
p^{r-k}\sum_{I_{j_{u}}=0}^{p-1}\zeta_{p}^{(\delta^{1}_{j_{u}}-\delta^{2}_{j_{u}})I_{j_{u}}}$
$\displaystyle=$ $\displaystyle 0,~{}~{}\text{from {Lemma}
\ref{Lem:sum_zero}.}$

From the above two cases the proof is completed. ∎

We use two collections of graphs, one with Hamiltonian paths and the other is
with disjoint vertices, to construct type-II ZCCS.

######  Theorem 1.

Let $p\geq 2$, $\mathcal{G}(\Omega)$ be graph of $r$ isolated vertices and
$\mathcal{G}(Q)$ be a Hamiltonian path with $n$ vertices and $x_{\pi(0)}$ and
$x_{\pi(n-1)}$ be the end vertices of $\mathcal{G}(Q)$, since there is a
single Hamiltonian path, we use $\pi$ instead of $\pi_{0}$.
$f_{\beta}^{\alpha}$ and $\Omega^{\delta}$ are defined as (8) and (14),
respectively. Now, we define an MVF as

$F_{\beta}^{\alpha p^{r}+\delta}=f_{\beta}^{\alpha}\oplus\Omega^{\delta},$
(20)

for $0\leq\alpha,\beta<p$ and $0\leq\delta<p^{r}$. Now, we define the order
set as

$C_{\alpha p^{r}+\delta}=\left[\chi(F_{0}^{\alpha
p^{r}+\delta})~{}\chi(F_{1}^{\alpha
p^{r}+\delta})~{}\cdots~{}\chi(F_{p-1}^{\alpha p^{r}+\delta})\right]^{T},$
(21)

and

$\mathbf{C}=\left\\{C_{0},C_{1},\ldots,C_{p^{1+r}-1}\right\\}.$ (22)

Then $\mathbf{C}$ is a type-II
$\left(p^{1+r},p,p^{n+r}-p^{r}+1,p^{n+r}\right)$-ZCCS.

###### Proof.

Since, $0\leq\alpha,\beta\leq p-1$, then $\boldsymbol{\alpha}=\alpha_{0}$ and
$\boldsymbol{\beta}=\beta_{0}$, in this condition $\alpha=\alpha_{0}$ and
$\beta=\beta_{0}$. So we are using $\alpha,\beta$ and $\pi$ instead of
$\alpha_{0},\beta_{0}$ and $\pi_{0}$ in the proof. Now,
$f_{\beta}^{\alpha}:\mathbb{Z}_{p}^{n+r}\to\mathbb{Z}_{q}$ be an MVF, given by

$\displaystyle f_{\beta}^{\alpha}(x_{0},x_{1},\ldots,x_{n-1})=$ $\displaystyle
Q(x_{0},x_{1},\ldots,x_{n-1})+\sum_{i=0}^{n-1}\gamma_{i}x_{i}$ (23)
$\displaystyle+\frac{q}{p}\left(\alpha x_{\pi(0)}+\beta x_{\pi(n-1)}\right).$
(24)

and

$\displaystyle F_{\beta}^{\alpha
p^{r}+\delta}(x_{0},x_{1},\ldots,x_{n-1},x_{n},x_{n+1},\ldots,x_{n+r-1})=f_{\beta}^{\alpha}\oplus\Omega^{\delta}$
(25)
$\displaystyle=f_{\beta}^{\alpha}(x_{0},x_{1},\ldots,x_{n-1})+\Omega^{\delta}(x_{n},x_{n+1},\ldots,x_{n+r-1}).$
(26)

Consider two codes $C_{s}$ and $C_{t}$ such that

$C_{s}=\left[\chi(F_{0}^{s})~{}\chi(F_{1}^{s})~{}\cdots~{}\chi(F_{p-1}^{s})\right]^{T},$
(27)

and

$C_{t}=\left[\chi(F_{0}^{t})~{}\chi(F_{1}^{t})~{}\cdots~{}\chi(F_{p-1}^{t})\right]^{T}.$
(28)

Consider $I$ and $J$ are two integers such that ${I}={J}+\tau$ for some
integral value of $\tau$. Let $\mathbf{I}$ and $\mathbf{J}$ be the vector
representation of $I$ and $J$ with base $p$, respectively.

$\displaystyle F_{\beta}^{s}(\mathbf{I})-F_{\beta}^{t}(\mathbf{J})=$
$\displaystyle(H(I_{0},I_{1},\ldots,I_{n-1})-H(J_{0},J_{1},\ldots,J_{n-1}))$
(29)
$\displaystyle+\Omega^{\delta^{s}}(I_{n},I_{n+1},\ldots,I_{n+r-1})-\Omega^{\delta^{t}}(J_{n},J_{n+1},\ldots,J_{n+r-1})$
$\displaystyle+\frac{q}{p}\beta(I_{\pi(n-1)}-J_{\pi(n-1)})+\frac{q}{p}\alpha(I_{\pi(0)}-J_{\pi(0)}),$

where $\delta^{s}$ and $\delta^{t}$ are $\delta$ in $s=\alpha p^{r}+\delta$
and $t=\alpha p^{r}+\delta$, respectively. When $s=t$, $\alpha$ will not
appear in (29). We denote
$H(I_{0},I_{1},\ldots,I_{n-1})-H(J_{0},J_{1},\ldots,J_{n-1})$ by $H^{IJ}$,
$\Omega^{\delta^{s}}(I_{n},I_{n+1},\ldots,I_{n+r-1})-\Omega^{\delta^{t}}(J_{n},J_{n+1},\ldots,J_{n+r-1})$
by $\Omega_{st}^{IJ}$, $(I_{\pi(n-1)}-J_{\pi(n-1)})$ by $D^{IJ}_{n-1}$ and
$(I_{\pi(0)}-J_{\pi(0)})$ by $D^{IJ}_{0}$. Substitute these in (29), we have

$F_{\beta}^{s}(\mathbf{I})-F_{\beta}^{t}(\mathbf{J})=H^{IJ}+\Omega_{st}^{IJ}+\frac{q}{p}\alpha
D_{0}^{IJ}+\frac{q}{p}\beta D_{n-1}^{IJ}.$ (30)

When $|\tau|\geq p^{r}$, at least one of the components from the first $n$
component is not the same for any number representation (with the base of $p$)
of $\mathbf{I}$ and $\mathbf{J}$. We consider two cases to complete our proof.

Case 1. $I_{\pi(n-1)}\neq J_{\pi(n-1)}$,

$\zeta_{q}^{F_{\beta}^{s}(\mathbf{I})-{F_{\beta}^{t}(\mathbf{J})}}=\zeta_{q}^{H^{IJ}+\Omega_{st}^{IJ}+\frac{q}{p}\alpha
D^{IJ}_{0}}\zeta_{q}^{D^{IJ}_{n-1}\frac{q}{p}\beta}.$ (31)

Thus

$\displaystyle\sum_{\beta=0}^{p-1}\left(\zeta_{q}^{F_{\beta}^{s}(\mathbf{I})-F_{\beta}^{t}(\mathbf{J})}\right)$
$\displaystyle=\zeta_{q}^{H^{IJ}+\Omega_{st}^{IJ}+\frac{q}{p}\alpha
D^{IJ}_{0}}\sum_{\beta=0}^{p-1}\zeta_{q}^{D^{IJ}_{n-1}\frac{q}{p}\beta}$ (32)
$\displaystyle=\zeta_{q}^{H^{IJ}+\Omega_{st}^{IJ}+\frac{q}{p}\alpha
D^{IJ}_{0}}\sum_{\beta=0}^{p-1}\zeta_{p}^{D^{IJ}_{n-1}\beta}$
$\displaystyle=0,~{}\text{from {Lemma} \ref{Lem:sum_zero}.}$

Case 2. If $I_{\pi(n-1)}=J_{\pi(n-1)}$ then there exist a largest integer
$0\leq u\leq n-2$, such that $I_{\pi(u)}\neq J_{\pi(u)}$. Let us consider
$\mathbf{I}^{k}$, which is an integer having vector representation of $I^{k}$
with base $p$ differs from that of $\mathbf{I}$ only at the position $\pi(u)$,
i.e.,

$\mathbf{I}^{k}=(I_{0},I_{1},\ldots,[(I_{\pi(u+1)}-k)]\ (\mathrm{mod}\
p),\ldots,I_{n},\ldots,I_{n+r}),$ (33)

where $1\leq k\leq p-1$. Similarly, $\mathbf{J}^{k}$ can be defined by
assumption $I^{k}=J^{k}+\tau$. It can be observed that an invertible map from
the pair $(\mathbf{I},\mathbf{J})$ to the pair
$(\mathbf{I}^{k},\mathbf{J}^{k})$ can be made. Thus, both pairs
$(\mathbf{I},\mathbf{J})$ and $(\mathbf{I}^{k},\mathbf{J}^{k})$, for
$k=1,2,\ldots,p-1$ contribute to $\mathcal{C}(C_{s},C_{t})(\tau)$.

$F^{s}_{\beta}(\mathbf{I}^{k})-F^{s}_{\beta}(\mathbf{I})=-k\left(\frac{q}{p}\left(I_{\pi(u)}+I_{\pi(u+2)}\right)+\gamma_{\pi(u+1)}\right).$
(34)

If $u=n-2$, we just remove the term $I_{\pi(u+2)}$. Similarly, we can write

$F^{t}_{\beta}(\mathbf{J}^{k})-F^{t}_{\beta}(\mathbf{J})=-k\left(\frac{q}{p}\left(J_{\pi(u)}+J_{\pi(u+2)}\right)+\gamma_{\pi(u+1)}\right),$
(35)

and

$\displaystyle
F^{s}_{\beta}(\mathbf{I})-F^{t}_{\beta}(\mathbf{J})-F^{s}_{\beta}(\mathbf{I}^{k})+F^{t}_{\beta}(\mathbf{J}^{k})=$
$\displaystyle\frac{qk}{p}(I_{\pi(u)}-J_{\pi(u)})$ (36) $\displaystyle=$
$\displaystyle\frac{qk}{p}(I_{\pi(u)}-J_{\pi(u)})$ $\displaystyle=$
$\displaystyle\frac{qk}{p}D^{IJ}_{u}.$

Thus,

$\zeta_{q}^{F^{s}_{\beta}(\mathbf{I})-F^{t}_{\beta}(\mathbf{J})-(F^{s}_{\beta}(\mathbf{I}^{k})-F^{t}_{\beta}(\mathbf{J}^{k}))}=\zeta_{p}^{kD^{IJ}_{u}}.$
(37)

and summing over $\beta$, we have

$\sum_{\beta=0}^{p-1}\zeta_{q}^{F^{s}_{\beta}(\mathbf{I})-F^{t}_{\beta}(\mathbf{J})-F^{s}_{\beta}(\mathbf{I}^{k})+F^{t}_{\beta}(\mathbf{J}^{k})}$

$\displaystyle=\sum_{k=1}^{p-1}\zeta_{q}^{F^{s}_{\beta}(\mathbf{I})-F^{t}_{\beta}(\mathbf{J})-(F^{s}_{\beta}(\mathbf{I}^{k})-F^{t}_{\beta}(\mathbf{J}^{k}))}$
(38) $\displaystyle=\sum_{k=1}^{p-1}\zeta_{p}^{kD^{IJ}_{u}}$
$\displaystyle=0.$

Combining the above two cases, we can conclude that

$\mathcal{C}(C_{s},C_{t})(\tau)=0\text{~{}for~{}}|\tau|\geq p^{r}.$ (39)

Now, we have to show that for $s\neq t$, $\mathcal{C}(C_{s},C_{t})(0)=0$. To
prove this again, we consider two cases as follows.

Case 1. Let $s=\alpha p^{r}+\delta^{s}$ and $t=\alpha p^{r}+\delta^{t}$. Then
ACCS between $C_{s}$ and $C_{t}$ at zero shift can be written as

$\displaystyle\mathcal{C}(C_{s},C_{t})(0)=$
$\displaystyle\sum_{I=0}^{p^{n+r}-1}\sum_{\beta=0}^{p-1}\zeta_{q}^{F_{\beta}^{\mathbf{s}}(\mathbf{I})-F_{\beta}^{\mathbf{t}}(\mathbf{I})}$
(40) $\displaystyle=$
$\displaystyle\sum_{I=0}^{p^{n+r}-1}\sum_{\beta=0}^{p-1}\zeta_{q}^{\frac{q}{p}(\Omega^{\delta^{s}}(\mathbf{I})-\Omega^{\delta^{t}}(\mathbf{I}))}$
$\displaystyle=$ $\displaystyle
p^{n}\sum_{I=0}^{p^{r}-1}\sum_{\beta=0}^{p-1}\zeta_{p}^{\Omega^{\delta^{s}}(\mathbf{I})-\Omega^{\delta^{t}}(\mathbf{I})}$
$\displaystyle=$ $\displaystyle
p^{n}\sum_{\beta=0}^{p-1}\chi(\Omega^{\delta^{s}})\cdot\chi(\Omega^{\delta^{t}})$
$\displaystyle=$ $\displaystyle 0.$

Case 2. Let $s=\alpha^{s}p^{r}+\delta$ and $t=\alpha^{t}p^{r}+\delta$. Then

$F_{\beta}^{s}(\mathbf{I})-F_{\beta}^{t}(\mathbf{I})=\frac{q}{p}(\alpha^{s}-\alpha^{t})I_{\pi(0)}.$
(41)

Thus

$\zeta_{q}^{F_{\beta}^{s}(\mathbf{I})-F_{\beta}^{t}(\mathbf{I})}=\zeta_{q}^{\frac{q}{p}(\alpha^{s}-\alpha^{t})I_{\pi(0)}},$
(42)

and

$\displaystyle\sum_{I=0}^{p^{n+r}-1}\zeta_{q}^{F_{\beta}^{s}(\mathbf{I})-F_{\beta}^{t}(\mathbf{I})}=$
$\displaystyle\sum_{I=0}^{p^{n+r}-1}\zeta_{p}^{(\alpha^{s}-\alpha^{t})I_{\pi(0)}}$
(43) $\displaystyle=$ $\displaystyle
p^{n+r-1}\sum_{I=0}^{p}\zeta_{p}^{(\alpha^{s}-\alpha^{t})I_{\pi(0)}}$
$\displaystyle=$ $\displaystyle
p^{n+r-1}\sum_{I=0}^{p}\zeta_{p}^{(\alpha^{s}-\alpha^{t})I}$ $\displaystyle=$
$\displaystyle 0.$

Thus

$\sum_{\beta=0}^{p-1}\sum_{I=0}^{p^{n+r}-1}\zeta_{q}^{F_{\beta}^{s}(\mathbf{I})-F_{\beta}^{t}(\mathbf{I})}=0.$
(44)

From the above cases, we can conclude that

$\mathcal{C}(C_{s},C_{t})(0)=0.$ (45)

Equations (39) and (45) lead us to conclude the Theorem 1. ∎

Now, we provide construction for type-II
$(p^{k+r},p^{k},p^{n+r}$$-$$p^{r}$$+$$1,p^{n+r})$-ZCCS.

######  Theorem 2.

Let $\mathcal{G}(\Omega)$ be set of $r$ disjoint vertices and
$\mathcal{G}(Q_{0}),\mathcal{G}(Q_{1}),\ldots,\mathcal{G}(Q_{k-1})$ be $k$
many Hamiltonian paths, with $n_{0},n_{1},\ldots,n_{k-1}$ vertices such that
$x_{\pi_{i}(0)}$ and $x_{\pi_{i}(n_{i}-1)}$ are end vertices of
$\mathcal{G}(Q_{i})$. $f_{\beta}^{\alpha}$ and $\Omega^{\delta}$ are defined
in (8) and (14), respectively. Now we define an MVF as

$F_{\beta}^{\alpha p^{r}+\delta}=f_{\beta}^{\alpha}\oplus\Omega^{\delta},$
(46)

for $0\leq\alpha,\beta<p^{k}$ and $0\leq\delta<p^{r}$. Now, we define the
order set as

$C_{\alpha p^{r}+\delta}=\left[\chi(F_{0}^{\alpha
p^{r}+\delta})~{}\chi(F_{1}^{\alpha
p^{r}+\delta})~{}\cdots~{}\chi(F_{p^{k}-1}^{\alpha p^{r}+\delta})\right]^{T},$
(47)

and

$\mathbf{C}=\left\\{C_{0},C_{1},\ldots,C_{p^{k+r}-1}\right\\}.$ (48)

Then $\mathbf{C}$ is a type-II
$\left(p^{k+r},p^{k},p^{n+r}-p^{r}+1,p^{n+r}\right)$-ZCCS.

###### Proof.

To prove $\mathcal{C}(C_{s},C_{t})=0$ for $|\tau|\geq p^{r}$, first we prove
it for $p^{r}\geq|\tau|\geq p^{r+n_{k}}-1$, which will be same proof as
Theorem 1, and after that the same process can be followed for the interval
$p^{r+\sum_{j=0}^{i}n_{k-j}}\geq|\tau|\geq p^{r+\sum_{j=0}^{i+1}n_{k-j}}-1$,
for $i\in\\{1,2,\ldots,k-2\\}$. ∎

We found that some codes are completely uncorrelated, i.e., ACCS sums are zero
for the entire time shift, which can be obtained by following Remark 1.

######  Remark 1.

In the proposed type-II ZCCS, for a fix $\delta$ and
$\alpha^{s}\neq\alpha^{t}$, we have $\mathcal{C}(C_{s},C_{t})(\tau)=0$ for
each integral value of $\tau$, where $s=\alpha^{s}p^{r}+\delta$ and
$t=\alpha^{t}p^{r}+\delta$.

######  Remark 2.

For Theorem 1 and 2, when number of isolated vertices is zero, we get
$(p,p,p^{n})$-CCC and $(p^{k},p^{k},p^{n})$-CCC.

Theorem $1$ of [12] appears as a special case of Remark 2. Remark 2 can also
generate Kernel of CCC $(p,p,p)$ for any integral value of $p\geq 2$, however,
[12] can generate Kernel of CCC $(p,p,p)$ for only prime number $p$. We know
that aperiodic correlation values are unchanged by cycling shifts of elements
in the sequence. In Theorem 2, path has been considered as first $n_{1}$
vertices should construct $1^{\text{st}}$ path, now first $n_{2}$ vertices
should construct $2^{\text{nd}}$ path and so on. Since aperiodic correlation
is unchanged under cyclic shifts in the elements of a sequence. The
restriction of vertices to a path in the same order is necessary. We are
providing one more remark considering the graph for which Theorem 2 is valid.

######  Remark 3.

Let $\mathcal{G}(Q)$ be a collection of Hamiltonian paths with
$n_{0},n_{1},\ldots,n_{k-1}$ vertices without the restriction that each path
is made of consecutive vertices only Theorem 2 is true.

In Theorem 2, column sequence PMEPR upper bound is $p^{r}$, which can be
reduced by adding a suitable function. In [7], there is an explanation and
proof is given that how to find a suitable function. We know that if a
sequence is part of GCS with $p$ sequences, its PMEPR upper bound is $p$.
Every column of a code obtained from Theorem 2 can be represented by a linear
function of $(\beta_{0},\beta_{1},\ldots,\beta_{k-1})$, and if we add some
function of $(\beta_{0},\beta_{1},\ldots,\beta_{k-1})$ in (46) it will act as
constant multiplication in the sequence which does not affect AACS/ACCS
property of codes. Therefore, we are proposing a remark that ensures low
column PMEPR.

######  Remark 4.

In Theorem 2 replace (46) it by

$F_{\beta}^{\alpha
p^{r}+\delta}=f_{\beta}^{\alpha}\oplus\Omega^{\delta}+\sum_{\nu=1}^{k-1}\beta_{\nu-1}\beta_{\nu},$
(49)

for $0\leq\alpha,\beta<p^{k}$ and $0\leq\delta<p^{r}$ and
$(\beta_{0},\beta_{1},\ldots,\beta_{k-1})$ is $p$-ary vector representation of
$\beta$. Then the obtained type-II ZCCS have column sequence PMEPR $p$.

######  Remark 5.

As $p=2$ and $q=2$, we get type-II binary
$(2^{k+r},2^{k},2^{n+r}-2^{r}+1,2^{n+r})$-ZCCS. The direct construction for
binary type-II ZCCS is not reported in the existing literature.

######  Example 1.

Let $\mathcal{G}(Q_{0})$ and $\mathcal{G}(Q_{1})$ be two Hamiltonian paths
each of having two vertices denoted by $x_{0},x_{1}$ and $x_{2},x_{3}$,
respectively and $\mathcal{G}(\Omega)$ be an isolated vertex denoted by
$x_{4}$, as shown in Fig. 1.

$x_{0}$$x_{1}$$x_{3}$$x_{2}$$x_{4}$ Figure 1: Graph of two Hamiltonian paths
and one disjoint vertices.

Now, defining $f:\mathbb{Z}_{2}^{5}\to\mathbb{Z}_{2}$ as follows,

$F_{\beta}^{2\alpha+\delta}=x_{0}x_{1}+x_{2}x_{3}+\alpha_{0}x_{0}+\alpha_{1}x_{2}+\beta_{0}x_{1}+\beta_{1}x_{3}+\delta_{0}x_{4}+\beta_{0}\beta_{1}.$
(50)

Using (50), in Theorem 2, we get

$C_{2\alpha+\delta}=\left[\chi(F_{0}^{2\alpha+\delta})~{}\chi(F_{1}^{2\alpha+\delta})~{}\chi(F_{2}^{2\alpha+\delta})~{}\chi(F_{3}^{2\alpha+\delta})\right]^{T},$
(51)

for $0\leq\alpha,\beta<8$ and $0\leq\delta<2$. Then
$\mathbf{C}=\\{C_{0},C_{1},\ldots,C_{8}\\}$ are given below ($+$ represents
$1$ and $-$ represents $-1$) is a binary type-II $(8,4,31,32)$-ZCCS. Using
(10), we get column PMEPR upper bound is $2$ for the obtained codes.

$\displaystyle C_{0}=$
$\displaystyle\begin{bmatrix}++++++--++++++--++++++--------++\\\
++++++--++++++--------++++++++--\\\ ++++--++++++--++++++--++----++--\\\
\----++------++--++++--++----++--\end{bmatrix},$ $\displaystyle C_{1}=$
$\displaystyle\begin{bmatrix}++++++--------++++++++--++++++--\\\
++++++--------++------++------++\\\ ++++--++----++--++++--++++++--++\\\
\----++--++++--++++++--++++++--++\end{bmatrix},$ $\displaystyle C_{2}=$
$\displaystyle\begin{bmatrix}++--++++++--++++++--++++--++----\\\
++--++++++--++++--++----++--++++\\\ ++------++------++--------++++++\\\
\--++++++--++++++++--------++++++\end{bmatrix},$ $\displaystyle C_{3}=$
$\displaystyle\begin{bmatrix}++--++++--++----++--++++++--++++\\\
++--++++--++------++------++----\\\ ++--------++++++++------++------\\\
\--++++++++------++------++------\end{bmatrix},$ $\displaystyle C_{4}=$
$\displaystyle\begin{bmatrix}+-+-+--++-+-+--++-+-+--+-+-+-++-\\\
+-+-+--++-+-+--+-+-+-++-+-+-+--+\\\ +-+--++-+-+--++-+-+--++--+-++--+\\\
-+-++--+-+-++--++-+--++--+-++--+\end{bmatrix},$ $\displaystyle C_{5}=$
$\displaystyle\begin{bmatrix}+-+-+--+-+-+-++-+-+-+--++-+-+--+\\\
+-+-+--+-+-+-++--+-+-++--+-+-++-\\\ +-+--++--+-++--++-+--++-+-+--++-\\\
-+-++--++-+--++-+-+--++-+-+--++-\end{bmatrix},$ $\displaystyle C_{6}=$
$\displaystyle\begin{bmatrix}+--++-+-+--++-+-+--++-+--++--+-+\\\
+--++-+-+--++-+--++--+-++--++-+-\\\ +--+-+-++--+-+-++--+-+-+-++-+-+-\\\
-++-+-+--++-+-+-+--+-+-+-++-+-+-\end{bmatrix},$ $\displaystyle C_{7}=$
$\displaystyle\begin{bmatrix}+--++-+--++--+-++--++-+-+--++-+-\\\
+--++-+--++--+-+-++--+-+-++--+-+\\\ +--+-+-+-++-+-+-+--+-+-++--+-+-+\\\
-++-+-+-+--+-+-++--+-+-++--+-+-+\end{bmatrix}.$

In type-I ZCCS with $4$ sequences of length $32$, the number of codes can be
$8$ only if ZCZ width is less than $16$ due to the bound $K\leq
M\left\lfloor\frac{N}{Z}\right\rfloor$. However, Example 1 provides $8$ codes
with the ZCZ width $31$.

In Table 1, we provide some available parameters for type-I ZCCS and type-II
ZCCS, where it can be observed that the type-II ZCCS provides a larger number
of codes compared to type-I ZCCS for a specific ZCZ width.

Table 1: Comparison with some available ZCCS

Source | Method | ZCCS | $\left(K,M,Z,N\right)$
---|---|---|---
[7] | Direct | Type-I |  $\left(2^{k+p},2^{k},2^{m-p},2^{m}\right)$111 $p\leq m$, $k+1\geq m$, $k\geq 0$
[8] | Direct | Type-I |  $\left(p2^{k+1},2^{k+1},2^{m},p2^{m}\right)$222$p$ is prime, $k+1\leq m$
[9] | Direct | Type-I |  $\left(P2^{k+1},2^{k+1},2^{m},P2^{m}\right)$333$P\geq 2$, $k+1\leq m$
[11] | Indirect | Type-II |  $\left(rK,K,NP-P+1,NP\right)$555$P\geq 2$, $k\leq n$, $r,k\geq 0$
Th. 2 | Direct | Type-II |  $\left(P^{k+r},P^{k},P^{n+r}-P^{r}+1,P^{n+r}\right)$66footnotemark: 6
|  |  |

44footnotetext: $r\leq P$ & $K\times N$ code size of a CCC.

##  4 Conclusion

In this work, we connected MVF to a graph, which is mainly composed of
disjoint vertices and Hamiltonian paths, in order to construct type-II ZCCS. A
type-II $(p^{k+r},p^{k},p^{r+n}-p^{r}+1,p^{r+n})$-ZCCS is for $p,k,r$ and $n$
non-negative integer such that $p\geq 2$, $r\geq 0$ and $0<k<n$. Furthermore,
the proposed construction can generate the Kernel of CCC as $(p,p,p)$-CCC, for
any integral value of $p$. The proposed construction follows the code bound
$K=M(N-Z+1)$, given in [13], for type-II $(K,M,Z,N)$-ZCCS.

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