# A Direct Construction of Complete Complementary Codes With Arbitrary Lengths

Palash Sarkar, Zilong Liu, and Sudhan Majhi Palash Sarkar and Sudhan Majhi is
with the Department of Electrical Engineering, Indian Institute of Technology
Patna, India, e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Liu is
with the School of Computer Science and Electronics Engineering, University of
Essex, UK<EMAIL_ADDRESS>
###### Abstract

This paper is focused on efficient design of complete complementary codes
(CCCs) which have found wide applications in coding, signal processing and
wireless communication due to their zero auto- and cross-correlation sum
properties. A major motivation of this research is that the existing state-of-
the-art can generate CCCs with certain lengths only and therefore may not meet
the diverse requirements in practice. We introduce a new tool called
multivariable functions and propose a direct construction of CCCs with any
arbitrary lengths in the form $\prod_{i=1}^{k}p_{i}^{m_{i}}$, where $k$ is a
positive integer, $p_{1},p_{2},\ldots,p_{k}$ are prime numbers and
$m_{1},m_{2},\ldots,m_{k}$ are positive integers whose sum is not less than 1.
For $k=1$ and $p_{1}=2$, our proposed generator reduces to the exact Golay-
Davis-Jedwab (GDJ) sequence generator as a special case. For $k>1$ and
$p_{1}=p_{2}=\cdots=p_{k}=2$, it gives rise to the conventional CCCs with
power-of-two lengths which are obtained from generalized Boolean functions.

###### Index Terms:

Complementary set (CS), Golay complementary pair (GCP), complete complementary
codes (CCCs), generalized Boolean functions (GBF), multivariable functions.

##  I Introduction

###  I-A Background

The study of complementary sequences has been attracting increasing research
attention in recent years due to their interesting correlation properties as
well as their wide applications in engineering. Formally, a collection of
sequences with identical length and zero aperiodic auto-correlation sum (AACS)
for all non-zero time shifts are called a complementary set (CS) and each
constituent sequence is called a complementary sequence [1, 2]. In particular,
a CS of size two is called a Golay complementary pair (GCP), a concept
proposed by Michael J. E Golay in his study of infrared spectrometry [3] and
in which the two constituent sequences are called Golay sequences.
Furthermore, by rearranging every CS to an ordered matrix, a group of CSs with
zero aperiodic cross-correlation sum (ACCS) for all time shifts is called
mutually orthogonal Golay complementary sets (MOGCSs) [4]. It is noted that
the set size (denoted by $K$) of MOGCSs is upper bounded by the number of
constituent sequences (denoted by $M$), i.e., $K\leq M$. When $K=M$, the
resultant MOGCSs are called a set of complete complementary codes (CCCs) [5].

So far, complementary sequences and CCCs have been applied in a number of
applications in coding, signal processing, and wireless communications. For
example, they can be used for peak-to-average power ratio (PAPR) reduction in
code-keying orthogonal frequency division multiplexing (OFDM) [6], signature
sequences for interference-free multicarrier code-division multiple-access
(MC-CDMA) over asynchronous wireless channels [7, 8], training/sensing
sequences in multiple-input multiple-output (MIMO) communication/radar systems
[9, 10, 11], information hiding in image/audio/video signals [12], and zero-
correlation zone (ZCZ) sequences [13, 14, 15, 16, 17] in quasi-synchronous
direct-sequence CDMA communications.

Significant research attempts have been made on efficient constructions of
complementary sequences (including GCPs, MOGCSs and CCCs) with flexible
lengths. With interleaving and concatenation, Golay first constructed binary
GCPs of lengths $2MN$ from two shorter GCPs with lengths $M$ and $N$,
respectively [18]. Budišin developed recursive algorithms for more polyphase
and multi-level complementary sequences through a series of sequence
operations (e.g., phase rotations, shifting) associated with permutation
vectors [19, 20]. In 1999, Davis and Jedwab proposed a systematic construction
of $2^{h}$-ary ($h\in\mathbb{Z}^{+}$) GCPs with power-of-two lengths using
second-order generalized Boolean functions (GBFs) [6]. Such Golay sequences
are called Golay-Davis-Jedwab (GDJ) sequences in this paper. Paterson extended
the idea of $2^{h}$-ary GCPs to $q$-ary (for even $q$) CSs with power-of-two
constituent sequences having power-of-two lengths [2]. His work was further
generalized in 2007 by Schmidt by moving to the higher-order GBFs [21].
Recently, Palash _et al._ proposed a novel construction in [22] using higher-
order GBFs for complementary sequences with lower PMEPR upper bound. For
complementary sequences with non-power-of-two lengths, Chen made a
contribution with the aid of certain truncated Reed-Muller codes [23].

The constructions of MOGCSs especially CCCs may be more stringent owing to the
additional requirement of zero aperiodic cross-correlation sums [4, 8, 24].
Shing _et al._ introduced an construction of MOGCSs with non-power-of-two
lengths based on GBFs in [25] where the set size is exactly half of the number
of constituent sequences in a CS, i.e., $K=M/2$. Another powerful tool is
paraunitary (PU) generator with which a set of CCC can be efficiently obtained
by sending a pipeline of shifted impulses to multistage filterbanks whose
coefficients are extracted from a group of PU matrices [26, 27, 28, 29, 30].
The inherent connections between the GBF generator and the PU generator in CCC
construction have been uncovered lately by Wang _et al._ [31].

###  I-B Contributions

Despite numerous constructions of CCCs in the literature, to the best of our
knowledge, there is no direct construction of CCCs for any arbitrary lengths.
Addressing such a design problem will bring more flexibility to practical
applications of CCCs, whilst pushing forward the knowledge boundaries of this
topic. We observe that a GBF is a $\mathbb{Z}_{n}$-valued function where the
domain is restricted to several binary variables $\mathbb{Z}_{2}^{m}$ (for
positive integer $m$). To proceed, we study a new tool called “multi-variable
function” where the domain is extended to
$\mathbb{Z}_{p_{1}}^{m_{1}}\times\mathbb{Z}_{p_{2}}^{m_{2}}\times\cdots\times\mathbb{Z}_{p_{k}}^{m_{k}}$
(where $k$ is a positive integer, $p_{1},p_{2},\ldots,p_{k}$ are prime numbers
and $m_{1},m_{2},\ldots,m_{k}$ are chosen as positive integers whose sum is no
less than $1$). The key idea is to first construct CCCs of lengths
$p_{i}^{m_{i}}$ and then expand them using an implicit Kronecker product for
generic lengths in the form of $\prod_{i=1}^{k}p_{i}^{m_{i}}$. As every
integer greater than $1$ can be factored uniquely as the product of prime
numbers (by unique factorization theorem), our proposed multivariable
generator is capable of producing CCCs of all possible lengths. The set size
and the number of constituent sequences of the proposed CCCs are given by
$\prod_{i=1}^{k}p_{i}$ . For $k=1$ and $p_{1}=2$, the GDJ sequences are
obtained as a special case of our construction. For $k>1$ and
$p_{1}=p_{2}=\cdots=p_{k}=2$, our proposed multivariable generator is able to
produce CSs and CCCs with the same parameters as that reported in the works by
Paterson [2], Schmidt [21], and Palash et al. [22].

##  II Preliminary

###  II-A Notations

  * •

$\mathbb{Z}^{+}$ denotes the set of all nonnegative integers.

  * •

$p_{1},p_{2},\ldots,p_{k}$, denote $k$ prime numbers where
$k\in\mathbb{Z}^{+}\setminus\\{0\\}$.

  * •

$v_{p_{\alpha},1},v_{p_{\alpha},2},\ldots,v_{p_{\alpha},m_{\alpha}}$ denote
$m_{\alpha}$ variables which take values from $\mathbb{Z}_{p_{\alpha}}$, where
$\alpha=1,2,\ldots,k$.

  * •

$\lambda=l.c.m(p_{1},p_{2},\ldots,p_{k},q)$,
$q\in\mathbb{Z}^{+}\setminus\\{0\\}$.

  * •

$\omega_{\lambda}=\exp(2\pi\sqrt{-1}/\lambda)$.

  * •

$V_{\alpha}=\left(v_{p_{\alpha},1},v_{p_{\alpha},2},\ldots,v_{p_{\alpha},m_{\alpha}}\right)$,
where $\alpha=1,2,\ldots,k$.

  * •

$\mathbf{u}_{p_{\alpha},i_{\alpha}}\in\mathbb{Z}_{p_{\alpha}}^{m_{\alpha}}$,
denotes the vector representation of $i_{\alpha}$ with base $p_{\alpha}$.

  * •

$\otimes$ denotes Kronecker product operator.

###  II-B Definitions of Correlations and Sequences

Let $\textbf{a}=(a_{\alpha}:\alpha=0,1,\ldots,L-1)$ and
$\textbf{b}=(b_{\alpha}:\alpha=0,1,\ldots,L-1)$ be two complex-valued
sequences of equal length $L$ and let $u$ be an integer. Define

$\begin{split}C(\textbf{a},\textbf{b})(u)=\begin{cases}\sum_{i=0}^{L-1-u}a_{i+u}b^{*}_{i},&0\leq
u<L,\\\ \sum_{i=0}^{L+u-1}a_{i}b^{*}_{i-u},&-L<u<0,\\\
0,&\textnormal{otherwise},\end{cases}\end{split}$ (1)

and $A(\textbf{a})(u)=C(\textbf{a},\textbf{a})(u)$. The above mentioned
functions are called the aperiodic cross-correlation function between a and b
and the aperiodic auto-correlation function (AACF) of a, respectively.

Let $\textbf{C}=\\{C_{0},C_{1},\cdots,C_{K-1}\\}$ be a set of $K$ matrices
(codes), each having order $M\times L$. Let us define $C_{\mu}$ as

$\begin{split}C_{\mu}=\begin{bmatrix}\textbf{a}_{0}^{\mu}\\\
\textbf{a}_{1}^{\mu}\\\ \vdots\\\ \textbf{a}_{M-1}^{\mu}\end{bmatrix}_{M\times
L},\end{split}$ (2)

where $\textbf{a}_{\nu}^{\mu}$ $(0\leq\nu\leq M-1,0\leq\mu\leq K-1)$ is the
$\nu$-th row sequence or $\nu$-th constituent sequence of $C_{\mu}$. Let
$C_{\mu_{1}}$, $C_{\mu_{2}}\in\textbf{C}$ $(0\leq\mu_{1},\mu_{2}\leq K-1)$ be
any two matrices in C. The ACCF of $C_{\mu_{1}}$ and $C_{\mu_{2}}$ is defined
by

$C(C_{\mu_{1}},C_{\mu_{2}})(u)=\displaystyle\sum_{\nu=0}^{M-1}C(\textbf{a}_{\nu}^{\mu_{1}},\textbf{a}_{\nu}^{\mu_{2}})(u).$
(3)

###### Definition 1

Code set C is said to be MOGCSs or $(K,M,L)$-MOGCSs when $K\leq M$ and it
satisfy the following properties:

$C(C_{\mu_{1}},C_{\mu_{2}})(u)=\begin{cases}LK,&u=0,\mu_{1}=\mu_{2},\\\
0,&0<|u|<L,\mu_{1}=\mu_{2},\\\ 0,&|u|<L,\mu_{1}\neq\mu_{2}.\end{cases}$ (4)

We call C is a CCC if $K=M$ and we denote it by $(K,K,L)$-CCC. In the above
definition, each of the code $C_{\mu}$ $(0\leq\mu\leq K-1)$, is said to be a
CS. We denote a CS with the parameters $M$ and $L$ by $(M,L)$-CS. A $(2,L)$-CS
is known as a GCP where either sequence of the pair is called a Golay
sequence. A GCP of length $L$ is denoted by $(2,L)$-GCP in this paper.

###  II-C Multivariable Functions and Corresponding Sequences

We consider the following subset set $M$ of the set of monomials of degree at
most $r$ over the variables
$v_{p_{\alpha},1},v_{p_{\alpha},2},\ldots,v_{p_{\alpha},m_{\alpha}}$:

$M=\left\\{\prod_{\alpha=1}^{k}v_{p_{\alpha},1}^{j_{1}^{\alpha}}v_{p_{\alpha},2}^{j_{2}^{\alpha}}\cdots
v_{p_{\alpha},m_{\alpha}}^{j_{m_{\alpha}}^{\alpha}}:0\leq\sum_{\alpha=1}^{k}\sum_{\beta=1}^{m_{\alpha}}j_{\beta}^{\alpha}\leq
r\right\\},$ (5)

where $j_{\beta}^{\alpha}\in\\{0,1\\}$, $0\leq
r\leq\sum_{\alpha=0}^{k}m_{\alpha}$. A multivariable function
$f:\mathbb{Z}_{p_{1}}^{m_{1}}\times\mathbb{Z}_{p_{2}}^{m_{2}}\times\cdots\mathbb{Z}_{p_{k}}^{m_{k}}\rightarrow\mathbb{Z}_{\lambda}$
over the monomials from the set $M$ is the linear combination of those
monomials with $\mathbb{Z}_{\lambda}$-valued coefficients. Order of $f$ is
given by the degree of highest degree monomials with non-zero coefficients.
The function $f$ is called a GBF when $k=1$ and $p_{1}=2$. Now, let us define
the ordered set $U$ as

$\begin{split}U=\left(\left(\mathbf{u}_{p_{1},i_{1}},\mathbf{u}_{p_{2},i_{2}},\ldots,\mathbf{u}_{p_{k},i_{k}}\right):i_{1}=0,1,\ldots,p_{1}^{m_{1}}-1,i_{2}=0,1,\ldots,p_{2}^{m_{2}}-1,\ldots,\right.\\\
\left.i_{k}=0,1,\ldots,p_{k}^{m_{k}}-1\right).\end{split}$ (6)

Corresponding to
$f:\mathbb{Z}_{p_{1}}^{m_{1}}\times\mathbb{Z}_{p_{2}}^{m_{2}}\times\cdots\mathbb{Z}_{p_{k}}^{m_{k}}\rightarrow\mathbb{Z}_{\lambda}$,
we define the complex-valued sequence $\psi(f)$ of length
$p_{1}^{m_{1}}p_{2}^{m_{2}}\cdots p_{k}^{m_{k}}$, as

$\begin{split}\psi(f)=\left(\omega_{\lambda}^{f\left(\mathbf{u}_{p_{1},i_{1}},\mathbf{u}_{p_{2},i_{2}},\ldots,\mathbf{u}_{p_{k},i_{k}}\right)}:i_{1}=0,1,\ldots,p_{1}^{m_{1}}-1,i_{2}=0,1,\ldots,p_{2}^{m_{2}}-1,\ldots,\right.\\\
\left.i_{k}=0,1,\ldots,p_{k}^{m_{k}}-1\right).\end{split}$ (7)

###### Example 1

Let us assume that $k=2,p_{1}=2,p_{2}=3,m_{1}=2,m_{2}=2,r=2,q=3$. Then
$\lambda=6$. From (5), we have

$\begin{split}M&=\left\\{\prod_{\alpha=1}^{2}v_{p_{\alpha},1}^{j_{1}^{\alpha}}v_{p_{\alpha},2}^{j_{2}^{\alpha}}\cdots
v_{p_{\alpha},m_{\alpha}}^{j_{m_{\alpha}}^{\alpha}}:0\leq\sum_{\alpha=1}^{k}\sum_{\beta=1}^{m_{\alpha}}j_{\beta}^{\alpha}\leq
2\right\\}\\\
&=\left\\{v_{2,1}^{j_{1}^{1}}v_{2,2}^{j_{2}^{1}}v_{3,1}^{j_{1}^{2}}v_{3,2}^{j_{2}^{2}}:0\leq
j_{1}^{1}+j_{2}^{1}+j_{1}^{2}+j_{2}^{2}\leq
2,j_{1}^{1},j_{2}^{1},j_{1}^{2},j_{2}^{2}\in\\{0,1\\}\right\\}\\\
&=\left\\{1,v_{2,1},v_{2,2},v_{3,1},v_{3,2},v_{2,1}v_{2,2},v_{2,1}v_{3,1},v_{2,1}v_{3,2},v_{2,2}v_{3,1},v_{2,2}v_{3,2},v_{3,1}v_{3,2}\right\\}\end{split}$
(8)

The set of all multivariable functions,
$f:\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{3}^{2}\rightarrow\mathbb{Z}_{6}$, over
the monomials in $M$ is given by

$\begin{split}\left\\{c_{0}+c_{1}v_{2,1}+c_{2}v_{2,2}+c_{3}v_{3,1}+c_{4}v_{3,2}+c_{5}v_{2,1}v_{2,2}+c_{6}v_{2,1}v_{3,1}+c_{7}v_{2,1}v_{3,2}+c_{8}v_{2,2}v_{3,1}+c_{9}v_{2,2}v_{3,2}\right.\\\
\left.+c_{10}v_{3,1}v_{3,2}:c_{0},c_{1},\ldots,c_{10}\in\mathbb{Z}_{6}\right\\}.\end{split}$
(9)

Let us assume $c_{4}=1,c_{5}=2,c_{6}=1$ and rest of the coefficients as zero.
Then we obtain a multivariable function $f$ as

$f(v_{2,1},v_{2,2},v_{3,1},v_{3,2})=2v_{2,1}v_{2,2}+v_{2,1},v_{3,1}+v_{3,2}.$

From (6), we have

$U=\left(\left(\mathbf{u}_{2,i_{1}},\mathbf{u}_{3,i_{2}}\right):i_{1}=0,1,2,3,i_{2}=0,1,\ldots,8\right),$

where $(\mathbf{u}_{2,i_{1}}:i_{1}=0,1,2,3)=((0,0),(1,0),(0,1),(1,1))$ and
$(\mathbf{u}_{3,i_{2}}:i_{2}=0,1,\ldots,8)=((0,0),(1,0),(2,0),(0,1),(1,1),$
$(2,1),(0,2),(1,2),(2,2))$.

We choose $q=6$, then $\lambda=6$. From (7), we have

$\begin{split}\psi(f)&=\left(\omega_{6}^{f\left(\mathbf{u}_{p_{1},i_{1}},\mathbf{u}_{p_{2},i_{2}}\right)}:i_{1}=0,1,2,3,i_{2}=0,1,\ldots,8\right)\\\
&=(000111222012123234000111222234345450)\end{split}$ (10)

In the above complex-valued sequence, $i$ denotes $\omega_{6}^{i}$, where
$i=0,1,\ldots,5$.

######  Lemma 1 (Unique Factorization Theorem [32])

Let $L$ be a positive integer greater than $1$. Then $L$ can be expressed as

$L=p_{1}^{m_{1}}p_{2}^{m_{2}}\cdots P_{k}^{m_{k}},$ (11)

where $p_{1},p_{2},\cdots,p_{k}$ are distinct prime numbers.

######  Lemma 2 ([6, Th. 3])

Let us assume that $k=1,p_{1}=2,q=2^{h}$, where
$h\in\mathbb{Z}^{+}\setminus\\{0\\}$. Then $\lambda=2^{h}$ and
$f:\mathbb{Z}_{2}^{m_{1}}\rightarrow\mathbb{Z}_{2^{h}}$ be a GBF which is
given by

$f(v_{2,1},v_{2,2},\ldots,v_{2,m_{1}})=\displaystyle
2^{h-1}\sum_{\alpha=1}^{m_{1}-1}v_{2,\pi(\alpha)}v_{2,\pi(\alpha+1)}+\sum_{\alpha=1}^{m_{1}}c_{\alpha}v_{2,\alpha},$
(12)

where $\pi$ is a permutation of the set $\\{1,2,\ldots,m_{1}\\}$ and
$c_{\alpha}\in\mathbb{Z}_{2^{h}}$, $\alpha=1,2,\ldots,m$. Then
$\left(\psi(f+c),\psi(f+2^{h-1}v_{2,\pi(1)}+c^{\prime})\right)$ forms
$(2,2^{m_{1}})$-GCP, where $c,c^{\prime}\in\mathbb{Z}_{2^{h}}$.

######  Lemma 3 ([33, Th. 1])

Let $\textbf{C}=\\{C_{\mu}:\mu=0,1,\ldots,K-1\\}$ be $(K,M,L)$-MOGCss and
$\textbf{D}=\\{D_{\nu}:\nu=0,1,\ldots,K_{1}-1\\}$ be
$(K_{1},M_{1},L_{1})$-MOGCSs. Then $\\{C_{\mu}\otimes
D_{\nu}:\mu=0,1,\ldots,K-1,\nu=0,1,\ldots,K_{1}-1\\}$ forms
$(KK_{1},MM_{1},LL_{1})$-CCC.

##  III Proposed Construction of Complete Complementary Codes

In this section, we present our proposed construction of CCCs. We start with
$k=1$, i.e., $f:\mathbb{Z}_{p_{1}}^{m_{1}}\rightarrow\mathbb{Z}_{\lambda}$ be
a multivariable variable function over the variables
$v_{p_{1},1},v_{p_{1},2},\ldots,v_{p_{1},m_{1}}$. Since, we assume $k=1$, for
a better presentation, we replace the variables
$v_{p_{1},1},v_{p_{1},2},\ldots,v_{p_{1},m_{1}}$ by
$v_{1},v_{2},\ldots,v_{m_{1}}$, respectively. We also replace $p_{1}$ by $p$
and $m_{1}$ by $m$. For the later presentation, specially, when $k>1$, we
shall follow the notations as defined in the begining of Section II. Let us
define the multivariable function $f$ as

$\begin{split}f(v_{1},v_{2},\ldots,v_{m})=\displaystyle\frac{\lambda}{p}\sum_{\beta=1}^{m-1}v_{\pi(\beta)}v_{\pi(\beta+1)}+\sum_{\beta=1}^{m}c_{\beta}v_{\beta},\end{split}$
(13)

where $\pi$ is a permutation of the set $\\{1,2,\ldots,m\\}$,
$c_{\beta}\in\mathbb{Z}_{\lambda}$, $\beta=1,2,\ldots,m$ and $m\geq 1$. We
define

$a_{t}^{\alpha,p}(v_{1},v_{2},\ldots,v_{m})=f(v_{1},v_{2},\ldots,v_{m})+\frac{\lambda}{p}x_{\pi(1)}\alpha+\frac{\lambda}{p}x_{\pi(m)}t+c,$
(14)

where $\alpha,t\in\\{0,1,\ldots,p-1\\}$ and $c\in\mathbb{Z}_{q}$. Again, we
define $C_{t}$ to be the ordered set or code given by

$\begin{split}C_{t}^{p}=\left[\psi(a_{t}^{\alpha,p})\right]_{\alpha\in(0,1,\ldots,p-1)}=\begin{bmatrix}\psi(a_{t}^{0,p})\\\
\psi(a_{t}^{1,p})\\\ \vdots\\\ \psi(a_{t}^{p-1,p})\end{bmatrix},\end{split}$
(15)

where $\psi(a_{t}^{\alpha,p})$ denotes the complex-valued sequence
corresponding to the polynomial $a_{t}^{\alpha,p}$.

###### Theorem 1

The collection of codes $\\{C_{t}^{p}:t=0,1,\ldots,p-1\\}$ forms
$(p,p,p^{m})$-CCC over $\mathbb{Z}_{\lambda}$.

###### Proof:

Let $(\gamma_{1},\gamma_{2},\ldots,\gamma_{m})$ and
$(\delta_{1},\delta_{2},\ldots,\delta_{m})$ be the vector representations of
$\gamma$ and $\delta$ with base $p$, respectively. Therefore,
$\gamma=\displaystyle\sum_{\alpha_{1}=0}^{m-1}\gamma_{\alpha_{1}+1}p^{\alpha_{1}}$
and
$\delta=\displaystyle\sum_{\alpha_{1}=0}^{m-1}\delta_{\alpha_{1}+1}p^{\alpha_{1}}$,
where $0\leq\gamma,\delta\leq p^{m}-1$. We also define
$(a_{t}^{\alpha,p})_{\gamma}=a_{t}^{\alpha,p}(\gamma_{1},\gamma_{2},\ldots,\gamma_{m})$
and
$(a_{t}^{\alpha,p})_{\delta}=a_{t}^{\alpha,p}(\delta_{1},\delta_{2},\ldots,\delta_{m})$.
We choose an integer $u$ such that $\delta=\gamma+u$. The aperiodic AACF
between $C_{t}^{p}$ and $C_{t^{\prime}}^{p}$ can be expressed as

$\begin{split}C(C_{t}^{p},C_{t^{\prime}}^{p})(u)=\displaystyle\sum_{\alpha=0}^{p-1}C(\psi(a_{t}^{\alpha,p}),\psi(a_{t^{\prime}}^{\alpha,p}))(u)=\displaystyle\sum_{\alpha=0}^{p-1}\sum_{\gamma=0}^{p^{m}-u-1}\omega_{\lambda}^{(a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\delta}}.\end{split}$
(16)

In (16), the expression
$(a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\delta}$ can be
expressed as

$(a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\delta}=f_{\gamma}-f_{\delta}+\frac{\lambda}{p}\alpha\left(\gamma_{\pi(1)}-\delta_{\pi(1)}\right)+\frac{\lambda}{p}t\gamma_{\pi(m)}-\frac{\lambda}{p}t^{\prime}\delta_{\pi(m)},$
(17)

where $f_{\gamma}=f(\gamma_{1},\gamma_{2},\ldots,\gamma_{m})$ and
$f_{\delta}=f(\delta_{1},\delta_{2},\ldots,\delta_{m})$. Taking sum of
$\omega^{(a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\delta}}$
over $\alpha$, from (17), we have

$\displaystyle\sum_{\alpha=0}^{p-1}\omega_{\lambda}^{(a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\delta}}=\omega_{\lambda}^{f_{\gamma}-f_{\delta}}\omega_{p}^{t\gamma_{\pi(m)}-t^{\prime}\delta_{\pi(m)}}\sum_{\alpha=0}^{p-1}\omega_{p}^{\alpha\left(\gamma_{\pi(1)}-\delta_{\pi(1)}\right)}.$
(18)

We have the following cases:

######  Case 1 ($u\neq 0,\gamma_{\pi(1)}\neq\delta_{\pi(1)}$)

Since, $\gamma_{\pi(1)}\neq\delta_{\pi(1)}$,
$\\{\omega_{p}^{\alpha\left(\gamma_{\pi(1)}-\delta_{\pi(1)}\right)}:\alpha=0,1,\ldots,p-1\\}$
becomes set of all roots of the single-variable polynomial: $z^{p}-1$. Using
the results from (17) and (18) in (16), we have

$\displaystyle\sum_{\alpha=0}^{p-1}\omega_{\lambda}^{(a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\delta}}=0.$
(19)

######  Case 2 ($u\neq 0,\gamma_{\pi(1)}=\delta_{\pi(1)}$)

We assume $v$ be the smallest positive integer for which
$\gamma_{\pi(v)}\neq\delta_{\pi(v)}$. Let us define $\gamma^{k}$ to be the
integer whose vector representation with base $s$ is given by

$(\gamma_{1},\gamma_{2},\ldots,\gamma_{\pi(v-1)}-k,\ldots,\gamma_{m}),$ (20)

which differs from that of $\gamma$ only at the position $\pi(v-1)$ and
$k\in\\{1,\ldots,p-1\\}$. Similarly, $\delta^{k}$ have the following vector
representation with base $p$:

$(\delta_{1},\delta_{2},\ldots,\delta_{\pi(v-1)}-k,\ldots,\delta_{m}).$ (21)

By assumption $\delta=\gamma+u$ and so, $\delta^{k}=\gamma^{k}+u$. It can be
observed that we can draw invertible maps from the pair ($\gamma,\delta$) to
the pair ($\gamma^{k},\delta^{k}$). Therefore, the pairs ($\gamma,\delta$) and
($\gamma^{k},\delta^{k}$), $k=1,2,\ldots,p-1$ contribute to
$C(C_{t},C_{t^{\prime}})(u)$.

From (20) and (21), the expression
$(a_{t}^{\alpha,p})_{\gamma^{k}}-(a_{t^{\prime}}^{\alpha,p})_{\delta^{k}}$ can
be expressed as

$(a_{t}^{\alpha,p})_{\gamma^{k}}-(a_{t^{\prime}}^{\alpha,p})_{\delta^{k}}=f_{\gamma^{k}}-f_{\delta^{k}}+\frac{\lambda}{p}t\gamma_{\pi(m)}-\frac{\lambda}{p}t^{\prime}\delta_{\pi(m)}.$
(22)

From (17) and (22), we have

$\begin{split}(&a_{t}^{\alpha,p})_{\gamma^{k}}-(a_{t^{\prime}}^{\alpha,p})_{\delta^{k}}-\left((a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\delta}\right)=f_{\gamma^{k}}-f_{\delta^{k}}-\left(f_{\gamma}-f_{\delta}\right).\end{split}$
(23)

From (13) and (20), we have

$\begin{split}f_{\gamma^{k}}-f_{\gamma}=&\frac{\lambda}{p}\gamma_{\pi(v-2)}\left(\gamma_{\pi(v-1)}-k\right)-\frac{\lambda}{p}\gamma_{\pi(v-2)}\gamma_{\pi(v-1)}+\frac{\lambda}{p}\gamma_{\pi(v)}\left(\gamma_{\pi(v-1)}-k\right)\\\
&-\frac{\lambda}{p}\gamma_{\pi(v-1)}\gamma_{\pi(v)}-kc_{v-1}=-k\left(\frac{\lambda}{p}\gamma_{\pi(v-2)}+\frac{\lambda}{p}\gamma_{\pi(v)}+c_{v-1}\right).\end{split}$
(24)

Similarly,

$\begin{split}f_{\delta^{k}}-f_{\delta}=-k\left(\frac{\lambda}{p}\delta_{\pi(v-2)}+\frac{\lambda}{p}\delta_{\pi(v)}+c_{v-1}\right).\end{split}$
(25)

Since, $\gamma_{\pi(v-2)}=\delta_{\pi(v-2)}$, we have

$\begin{split}f_{\gamma^{k}}-f_{\delta^{k}}-\left(f_{\gamma}-f_{\delta}\right)=f_{\gamma^{k}}-f_{\gamma}-\left(f_{\delta^{k}}-f_{\delta}\right)=&\frac{\lambda}{p}k\left(\delta_{\pi(v-2)}-\gamma_{\pi(v-2)}\right)+\frac{\lambda}{p}k\left(\delta_{\pi(v)}-\gamma_{\pi(v)}\right)\\\
=&\frac{\lambda}{p}k\left(\delta_{\pi(v)}-\gamma_{\pi(v)}\right).\end{split}$
(26)

From (23) and (26), we have

$\begin{split}\displaystyle\sum_{k=1}^{p-1}\omega_{\lambda}^{(a_{t}^{\alpha,p})_{\gamma^{k}}-(a_{t^{\prime}}^{\alpha,p})_{\delta^{k}}-\left((a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\delta}\right)}&=\displaystyle\sum_{k=1}^{p-1}\omega_{p}^{k\left(\delta_{\pi(v)}-\gamma_{\pi(v)}\right)}=-1.\end{split}$
(27)

Therefore,

$\omega_{\lambda}^{(a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\delta}}+\displaystyle\sum_{k=1}^{p-1}\omega_{\lambda}^{(a_{t}^{\alpha,p})_{\gamma^{k}}-(a_{t^{\prime}}^{\alpha,p})_{\delta^{k}}}=0,\leavevmode\nobreak\
\forall\alpha.$ (28)

Taking sum over $\alpha$, from (28), we have

$\begin{split}\displaystyle\sum_{\alpha=0}^{p-1}\omega_{\lambda}^{(a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\delta}}+\displaystyle\sum_{k=1}^{p-1}\sum_{\alpha=0}^{p-1}\omega_{\lambda}^{(a_{t}^{\alpha,p})_{\gamma^{k}}-(a_{t^{\prime}}^{\alpha,p})_{\delta^{k}}}=0.\end{split}$
(29)

######  Case 3 ($u=0$)

For $u=0$, from (16), we have

$\begin{split}C(C_{t}^{p},C_{t^{\prime}}^{p})(0)=\displaystyle\sum_{\alpha=0}^{p-1}\sum_{i=0}^{p^{m}-1}\omega_{\lambda}^{(a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\gamma}}.\end{split}$
(30)

From (18), we have

$\displaystyle\sum_{\alpha=0}^{s-1}\omega_{\lambda}^{(a_{t}^{\alpha,p})_{\gamma}-(a_{t^{\prime}}^{\alpha,p})_{\delta}}=p\omega_{p}^{(t-t^{\prime})\gamma_{\pi(m)}}.$
(31)

From (30) and (31), we have

$\begin{split}C(C_{t}^{p},C_{t^{\prime}}^{p})(0)&=p\displaystyle\sum_{i=0}^{p^{m}-1}\omega_{p}^{(t-t^{\prime})\gamma_{\pi(m)}}=p^{m}\sum_{\gamma_{\pi(m)}=0}^{p-1}\omega_{p}^{(t-t^{\prime})\gamma_{\pi(m)}}=\begin{cases}p^{m+1},&t=t^{\prime},\\\
0,&t\neq t^{\prime}.\end{cases}\end{split}$ (32)

Combining all the above cases and from (16), we have

$\begin{split}C(C_{t}^{p},C_{t^{\prime}}^{p})(u)=\begin{cases}p^{m+1},&u=0,\leavevmode\nobreak\
t=t^{\prime},\\\ 0,&0<|u|<s^{m},\leavevmode\nobreak\ t=t^{\prime},\\\
0,&0\leq|u|<2^{m},\leavevmode\nobreak\ t\neq
t^{\prime}.\end{cases}\end{split}$ (33)

From (33), we can see that the collection $\\{C_{t}^{p}:t=0,1,\ldots,p-1\\}$
forms $(p,p,p^{m})$-CCC. ∎

In Theorem 1, $\lambda=l.c.m(p,q)$, where $q$ can be chosen as any positive
integer. We introduce $q$ to extend the possibilities of Theorem 1. Below, we
introduce a corollary which shows us the importance of $q$.

###### Corollary 1

In Theorem 1, let us assume that $p=2$ and $q=2^{h}$, where
$h\in\mathbb{Z}^{+}\setminus\\{0\\}$. Then $\lambda=l.c.m(2,2^{h})=2^{h}$ and
the multivariable function
$f:\mathbb{Z}_{p}^{m}\rightarrow\mathbb{Z}_{\lambda}$ in Theorem 1 reduces to
$f:\mathbb{Z}_{2}^{m}\rightarrow\mathbb{Z}_{2^{h}}$ which is a GBF and given
by

$f(v_{1},v_{2},\ldots,v_{m})=\displaystyle
2^{h-1}\sum_{\beta=1}^{m-1}v_{\pi(\beta)}v_{\pi(\beta+1)}+\sum_{\beta=1}^{m}c_{\beta}v_{\beta}.$
(34)

Using the above described parameters in Theorem 1, we obtain
($2,2,2^{m}$)-CCC. Hence $\psi(f)$ is Golay sequence and the GBF in (34) is
same as reported in[6, Th. 3]. Therefore, [6, Th. 3] appears as special case
of Theorem 1.

Below we present an example to illustrate Theorem 1.

###### Example 2

Let us choose $m=2$, $p=3$, $\pi(1)=1$, and $\pi(2)=2$ in Theorem 1. We assume
that $q=3$ and hence $\lambda=3$. From (13), we have

$f(v_{1},v_{2})=v_{1}v_{2}+v_{1}+2v_{2},$ (35)

and from (14), we have

$a_{t}^{\alpha,3}=f+\alpha v_{1}+tv_{2}+1,$ (36)

where $\alpha,t\in\\{0,1,2\\}$. From (15), we have

$\begin{split}C_{t}^{3}&=\begin{bmatrix}\psi(a_{t}^{0,3})\\\
\psi(a_{t}^{1,3})\\\
\psi(a_{t}^{2,3})\end{bmatrix}=\begin{bmatrix}&\psi(v_{1}v_{2}+v_{1}+2v_{2}+0\cdot
v_{1}+tv_{2}+1)\\\ &\psi(v_{1}v_{2}+v_{1}+2v_{2}+1\cdot v_{1}+tv_{2}+1)\\\
&\psi(v_{1}v_{2}+v_{1}+2v_{2}+2\cdot v_{1}+tv_{2}+1)\end{bmatrix}.\end{split}$
(37)

Therefore,

$\begin{split}C_{0}^{3}&=\begin{bmatrix}&\psi(v_{1}v_{2}+v_{1}+2v_{2}+0\cdot
v_{1}+0\cdot v_{2}+1)\\\ &\psi(v_{1}v_{2}+v_{1}+2v_{2}+1\cdot v_{1}+0\cdot
v_{2}+1)\\\ &\psi(v_{1}v_{2}+v_{1}+2v_{2}+2\cdot v_{1}+0\cdot
v_{2}+1)\end{bmatrix}=\begin{bmatrix}120021222\\\ 102000201\\\
111012210\end{bmatrix},\\\
C_{1}^{3}&=\begin{bmatrix}&\psi(v_{1}v_{2}+v_{1}+2v_{2}+0\cdot v_{1}+1\cdot
v_{2}+1)\\\ &\psi(v_{1}v_{2}+v_{1}+2v_{2}+1\cdot v_{1}+1\cdot v_{2}+1)\\\
&\psi(v_{1}v_{2}+v_{1}+2v_{2}+2\cdot v_{1}+1\cdot
v_{2}+1)\end{bmatrix}=\begin{bmatrix}120102111\\\ 102111120\\\
111120102\end{bmatrix},\\\
C_{2}^{3}&=\begin{bmatrix}&\psi(v_{1}v_{2}+v_{1}+2v_{2}+0\cdot v_{1}+2\cdot
v_{2}+1)\\\ &\psi(v_{1}v_{2}+v_{1}+2v_{2}+1\cdot v_{1}+2\cdot v_{2}+1)\\\
&\psi(v_{1}v_{2}+v_{1}+2v_{2}+2\cdot v_{1}+2\cdot
v_{2}+1)\end{bmatrix}=\begin{bmatrix}120210000\\\ 102222012\\\
111201021\end{bmatrix}.\end{split}$ (38)

Figure 1: Correlation plots of $(3,3,9)$-CCC.

Fig. 1(a) represents aperiodic auto-correlation sum of any code from the set
of codes $\\{C_{0}^{3},C_{1}^{3},C_{2}^{3}\\}$ and Fig. 1(b) represents
aperiodic cross-correlation sum between any two distinct codes from the same
set of codes. Therefore, $\\{C_{0}^{3},C_{1}^{3},C_{2}^{3}\\}$ forms
$(3,3,9)$-CCC.

Now, we assume $k>1$ and we define the multivariable function
$f:\mathbb{Z}_{p_{1}}^{m_{1}}\times\mathbb{Z}_{p_{2}}^{m_{2}}\times\cdots\mathbb{Z}_{p_{k}}^{m_{k}}\rightarrow\mathbb{Z}_{\lambda}$
as

$f=f_{1}+f_{2}+\ldots+f_{k},$ (39)

where
$f_{\alpha}:\mathbb{Z}_{p_{\alpha}}^{m_{\alpha}}\rightarrow\mathbb{Z}_{\lambda_{\alpha}}$
is a multivariable function over the variables
$v_{p_{\alpha},1},v_{p_{\alpha},2},\ldots,v_{p_{\alpha},m_{\alpha}}$, and
defined as

$\begin{split}f_{\alpha}(V_{\alpha})=\displaystyle\frac{\lambda_{\alpha}}{p_{\alpha}}\sum_{j=1}^{m_{\alpha}-1}v_{p_{\alpha},\pi_{\alpha}(j)}v_{p_{\alpha},\pi_{\alpha}(j+1)}+\sum_{j=1}^{m_{\alpha}}c_{\alpha,j}v_{p_{\alpha},j},\end{split}$
(40)

where $\pi_{\alpha}$ is a permutation of the set
$\\{1,2,\ldots,m_{\alpha}\\}$, $m_{\alpha}\geq 1$,
$\lambda_{\alpha}=l.c.m(p_{\alpha},q)$ and
$c_{\alpha,j}\in\mathbb{Z}_{\lambda_{\alpha}}$, $j=1,2,\ldots,m_{\alpha}$,
$\alpha=1,2,\ldots,k$. Again, we define the following multivariable function
of $m_{1}+m_{2}+\cdots+m_{k}$ variables:

$a_{\mathbf{t}_{k}}^{\bm{\alpha}_{k},\mathbf{p}_{k}}=a_{t_{1}}^{\alpha_{1},p_{1}}(V_{1})+a_{t_{2}}^{\alpha_{2},p_{2}}(V_{2})+\cdots+a_{t_{k}}^{\alpha_{k},p_{k}}(V_{k}),$
(41)

where $\bm{\alpha}_{k}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})$,
$\mathbf{p}_{k}=(p_{1},p_{2},\ldots,p_{k})$,
$\mathbf{t}_{k}=(t_{1},t_{2},\ldots,t_{k})$ and
$\alpha_{i},t_{i}\in\\{0,1,\ldots,p_{i}-1\\}$, $i=1,2,\ldots,k$. The function
$a_{t_{i}}^{\alpha_{i},p_{i}}(V_{i})$ is obtained by substituting $c=0$,
$f=f_{i}$, $\pi=\pi_{i}$, $\alpha=\alpha_{i}$, $t=t_{i}$, and $p=p_{i}$ in
(14), $i=1,2,\ldots,k$. Let us also define the following ordered set:

$I_{k}=\left(\bm{\alpha}_{k}:\alpha_{1}=0,1,\ldots,p_{1}-1,\alpha_{2}=0,1,\ldots,p_{2}-1,\ldots,\alpha_{k}=0,1,\ldots,p_{k}-1\right),$
(42)

and the ordered set $C_{\mathbf{t}_{k}}^{\mathbf{p}_{k}}$ as

$C_{\mathbf{t}_{k}}^{\mathbf{p}_{k}}=\left[\psi(a_{\mathbf{t}_{k}}^{\bm{\alpha}_{k},\mathbf{p}_{k}})\right]_{\bm{\alpha}_{k}\in
I_{k}}$ (43)

###### Lemma 4

Let us assume that $p_{\alpha}|q$, $\alpha=1,2,\ldots,k$. Then the complex-
valued sequence $\psi(f)$ can be expressed as

$\psi(f)=\psi(f_{1})\otimes\psi(f_{2})\otimes\cdots\otimes\psi(f_{k}).$ (44)

###### Proof:

We prove the statement given in (44) using mathematical induction. For $k=1$,
(44) is satisfied trivially. For $k=2$, from (7), we have

$\begin{split}\psi(f)&=\left(\omega_{\lambda}^{f\left(\mathbf{u}_{p_{1},i_{1}},\mathbf{u}_{p_{2},i_{2}}\right)}:i_{1}=0,1,\ldots,p_{1}^{m_{1}}-1,i_{2}=0,1,\ldots,p_{2}^{m_{2}}-1\right)\\\
&=\left(\omega_{\lambda}^{f_{1}\left(\mathbf{u}_{p_{1},i_{1}}\right)}\omega_{\lambda}^{f_{2}\left(\mathbf{u}_{p_{2},i_{2}}\right)}:i_{1}=0,1,\ldots,p_{1}^{m_{1}}-1,i_{2}=0,1,\ldots,p_{2}^{m_{2}}-1\right).\end{split}$
(45)

Since $p_{1},p_{2}$ are prime numbers and $p_{1}|q$, $p_{2}|q$, we have

$l.c.m(p_{1},p_{2},q)=l.c.m(p_{1},q)=l.c.m(p_{2},q)=\lambda$

and

$\begin{split}\psi(f_{1})&=\left(\omega_{\lambda}^{f_{1}\left(\mathbf{u}_{p_{1},i_{1}}\right)}:i_{1}=0,1,\ldots,p_{1}^{m_{1}}-1\right),\\\
\psi(f_{2})&=\left(\omega_{\lambda}^{f_{2}\left(\mathbf{u}_{p_{2},i_{2}}\right)}:i_{2}=0,1,\ldots,p_{2}^{m_{2}}-1\right).\end{split}$
(46)

From (45), we have

$\begin{split}\psi(f)&=\left(\omega_{\lambda}^{f_{1}\left(\mathbf{u}_{p_{1},i_{1}}\right)}\psi(f_{2}):i_{1}=0,1,\ldots,p_{1}^{m_{1}}-1\right)\\\
&=\left(\omega_{\lambda}^{f_{1}\left(\mathbf{u}_{p_{1},0}\right)}\psi(f_{2}),\omega_{\lambda}^{f_{1}\left(\mathbf{u}_{p_{1},1}\right)}\psi(f_{2}),\ldots,\omega_{\lambda}^{f_{1}\left(\mathbf{u}_{p_{1},p_{1}^{m_{1}}-1}\right)}\psi(f_{2})\right)\\\
&=\psi(f_{1})\otimes\psi(f_{2}).\end{split}$ (47)

Let us assume that the statement is true for $k=n$. Therefore,

$\psi(f)=\psi(f_{1})\otimes\psi(f_{2})\otimes\cdots\otimes\psi(f_{n}).$ (48)

We have to show that the statement is true for $k=n+1$ also and for it we have

$\begin{split}\psi(f)&=\psi(f_{1}+f_{2}+\cdots+f_{n}+f_{n+1})=\psi(F+f_{n+1}),\end{split}$
(49)

where $F=f_{1}+f_{2}+\ldots+f_{n}$. As, the statement is true for $k=2,n$,
from (48) and (49), we have

$\begin{split}\psi(f)=\psi(F+f_{n+1})=\psi(F)\otimes\psi(f_{n+1})&=\psi(f_{1}+f_{2}+\cdots+f_{n})\otimes\psi(f_{n+1})\\\
&=\psi(f_{1})\otimes\psi(f_{2})\otimes\ldots\otimes\psi(f_{n})\otimes\psi(f_{n+1}).\end{split}$
(50)

Therefore, the statement is true for $k=n+1$ also and hence it is true for all
values of $k$. ∎

The Lemma 4 will be used in the proof of the next proposed theorem of CCC
construction. Now, we present another lemma which will be used in our next
theorem.

###### Lemma 5

Let us assume that $p_{\alpha}|q$, $\alpha=1,2,\ldots,k$. Then the ordered set
$C_{\mathbf{t}_{k}}^{\mathbf{p}_{k}}$ can be expressed as

$C_{\mathbf{t}_{k}}^{\mathbf{p}_{k}}=C_{t_{1}}^{p_{1}}\otimes
C_{t_{2}}^{p_{2}}\otimes\ldots\otimes C_{t_{k}}^{p_{k}},$ (51)

where $C_{t_{i}}^{p_{i}}$ can be obtained from (15) by substituting $t=t_{i}$
and $p=p_{i}$, $i=1,2,\ldots,k$.

###### Proof:

We prove the statement given in (51) using mathematical induction. For $k=1$,
(51) is satisfied trivially. For $k=2$, from (43), we have

$\begin{split}C_{\mathbf{t}_{2}}^{\mathbf{p}_{2}}=\left[\psi(a_{\mathbf{t}_{2}}^{\bm{\alpha}_{2},\mathbf{p}_{2}})\right]_{\bm{\alpha}_{2}\in
I_{2}}.\end{split}$ (52)

Using Lemma 4 in (41), we have

$\psi(a_{\mathbf{t}_{2}}^{\bm{\alpha}_{2},\mathbf{p}_{2}})=\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\otimes\psi(a_{t_{2}}^{\alpha_{2},p_{2}}).$
(53)

Therefore

$\begin{split}C_{\mathbf{t}_{2}}^{\mathbf{p}_{2}}=\left[\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\otimes\psi(a_{t_{2}}^{\alpha_{2},p_{2}})\right]_{\bm{\alpha}\in
I_{2}}&=\begin{bmatrix}\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\otimes\psi(a_{t_{2}}^{0,p_{2}})\\\
\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\otimes\psi(a_{t_{2}}^{1,p_{2}})\\\
\vdots\\\
\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\otimes\psi(a_{t_{2}}^{p_{2}-1,p_{2}})\end{bmatrix}_{\alpha_{1}\in
I_{1}}\\\ &=\left[\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\right]_{\alpha_{1}\in
I_{1}}\otimes\begin{bmatrix}\psi(a_{t_{2}}^{0,p_{2}})\\\
\psi(a_{t_{2}}^{1,p_{2}})\\\ \vdots\\\
\psi(a_{t_{2}}^{p_{2}-1,p_{2}})\end{bmatrix}\\\
&=\begin{bmatrix}\psi(a_{t_{1}}^{0,p_{1}})\\\ \psi(a_{t_{1}}^{1,p_{1}})\\\
\vdots\\\
\psi(a_{t_{1}}^{p_{1}-1,p_{1}})\end{bmatrix}\otimes\begin{bmatrix}\psi(a_{t_{2}}^{0,p_{2}})\\\
\psi(a_{t_{2}}^{1,p_{2}})\\\ \vdots\\\
\psi(a_{t_{2}}^{p_{2}-1,p_{2}})\end{bmatrix}\\\ &=C_{t_{1}}^{p_{1}}\otimes
C_{t_{2}}^{p_{2}}.\end{split}$ (54)

Hence, the statement is true for $k=2$. We assume that the statement is true
for $k=n$ and hence

$C_{\mathbf{t}_{n}}^{\mathbf{p}_{n}}=\left[\psi(a_{\mathbf{t}_{n}}^{\bm{\alpha}_{n},\mathbf{p}_{n}})\right]_{\bm{\alpha}_{n}\in
I_{n}}=C_{t_{1}}^{p_{1}}\otimes C_{t_{2}}^{p_{2}}\otimes\ldots\otimes
C_{t_{n}}^{p_{n}}.$ (55)

For $k=n+1$, using Lemma 4 in (43), we have

$\begin{split}C_{\mathbf{t}_{n+1}}^{\mathbf{p}_{n+1}}&=\left[\psi(a_{\mathbf{t}_{n+1}}^{\bm{\alpha}_{n+1},\mathbf{p}_{n+1}})\right]_{\bm{\alpha}_{n+1}\in
I_{n+1}}\\\
&=\left[\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\otimes\psi(a_{t_{2}}^{\alpha_{2},p_{2}})\otimes\ldots\otimes\psi(a_{t_{n+1}}^{\alpha_{n+1},p_{n+1}})\right]_{\bm{\alpha}\in
I_{n+1}}\\\
&=\begin{bmatrix}\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\otimes\psi(a_{t_{2}}^{\alpha_{2},p_{2}})\otimes\ldots\otimes\psi(a_{t_{n}}^{\alpha_{n},p_{n}})\otimes\psi(a_{t_{n+1}}^{0,p_{n+1}})\\\
\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\otimes\psi(a_{t_{2}}^{\alpha_{2},p_{2}})\otimes\ldots\otimes\psi(a_{t_{n}}^{\alpha_{n},p_{n}})\otimes\psi(a_{t_{n+2}}^{1,p_{n+1}})\\\
\vdots\\\
\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\otimes\psi(a_{t_{2}}^{\alpha_{2},p_{2}})\otimes\ldots\otimes\psi(a_{t_{n}}^{\alpha_{n},p_{n}})\otimes\psi(a_{t_{n+1}}^{p_{n+1}-1,p_{n+1}})\end{bmatrix}_{(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\in
I_{n}}\end{split}$ (56)
$\begin{split}&=\left[\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\otimes\psi(a_{t_{2}}^{\alpha_{2},p_{2}})\otimes\ldots\otimes\psi(a_{t_{n}}^{\alpha_{n},p_{n}})\right]_{\bm{\alpha}_{n}\in
I_{n}}\otimes\begin{bmatrix}\psi(a_{t_{n+1}}^{0,p_{n+1}})\\\
\psi(a_{t_{n+1}}^{1,p_{n+1}})\\\ \vdots\\\
\psi(a_{t_{n+1}}^{p_{n+1}-1,p_{n+1}})\end{bmatrix}\\\
&=\left[\psi(a_{t_{1}}^{\alpha_{1},p_{1}})\otimes\psi(a_{t_{2}}^{\alpha_{2},p_{2}})\otimes\ldots\otimes\psi(a_{t_{n}}^{\alpha_{n},p_{n}})\right]_{\bm{\alpha}_{n}\in
I_{n}}\otimes C_{t_{n+1}}^{p_{n+1}}\\\
&=\left[\psi(a_{\mathbf{t}_{n}}^{\bm{\alpha}_{n},\mathbf{p}_{n}})\right]_{\bm{\alpha}_{n}\in
I_{n}}\otimes C_{t_{n+1}}^{p_{n+1}}\\\ \end{split}$

∎

For $k=n+1$, from (55) and (56), we have

$C_{\mathbf{t}_{n+1}}^{\mathbf{p}_{n+1}}=C_{t_{1}}^{p_{1}}\otimes
C_{t_{2}}^{p_{2}}\otimes\ldots\otimes C_{t_{n}}^{p_{n}}\otimes
C_{t_{n+1}}^{p_{n+1}}.$ (57)

Therefore, the statement is true for $k=n+1$ also and hence it is true for all
values of $k$.

###### Theorem 2

Let us assume that $p_{\alpha}|q$, $\alpha=1,2,\ldots,k$. Then the collection
of ordered set or codes

$S=\left\\{C_{\mathbf{t}_{k}}^{\mathbf{p}_{k}}:t_{1}=0,1,\ldots,p_{1}-1,t_{2}=0,1,\ldots,p_{2}-1,\ldots,t_{k}=0,1,\ldots,p_{k}-1\right\\},$

forms
$(\prod_{i=1}^{k}p_{i},\prod_{i=1}^{k}p_{i},\prod_{i=1}^{k}p_{i}^{m_{i}})$-CCC.

###### Proof:

We prove the statement using mathematical induction. Using Lemma 5, the set
$S$ can be expressed as

$\begin{split}S=\left\\{C_{t_{1}}^{p_{1}}\otimes
C_{t_{2}}^{p_{2}}\otimes\ldots\otimes
C_{t_{k}}^{p_{k}}:t_{1}=0,1,\ldots,p_{1}-1,t_{2}=0,1,\ldots,p_{2}-1,\ldots,\right.\\\
\left.t_{k}=0,1,\ldots,p_{k}-1\right\\}.\end{split}$ (58)

From Theorem 1, $\\{C_{t_{i}}^{p_{i}}:t_{i}=0,1,\ldots,p_{i}-1\\}$ forms
$(p_{i},p_{i},p_{i}^{m_{i}})$-CCC, $i=1,2,\ldots,k$. For $k=1$, the statement
is satisfied trivially. For $k=2$, the set $S$ reduces to

$S=\left\\{C_{t_{1}}^{p_{1}}\otimes
C_{t_{2}}^{p_{2}}:t_{1}=0,1,\ldots,p_{1}-1,t_{2}=0,1,\ldots,p_{2}-1\right\\}.$

Since, $\\{C_{t_{i}}^{p_{i}}:t_{i}=0,1,\ldots,p_{i}-1\\}$ forms
$(p_{i},p_{i},p_{i}^{m_{i}})$-CCC, $i=1,2$, using Lemma 3, $S$ forms
$(p_{1}p_{2},p_{1}p_{2},p_{1}^{m_{1}}p_{2}^{m_{2}})$-CCC. Therefore, the
statement is true for $k=2$. Let us assume that the statement is true for
$k=n$. Hence,

$\left\\{C_{t_{1}}^{p_{1}}\otimes C_{t_{2}}^{p_{2}}\otimes\ldots\otimes
C_{t_{n}}^{p_{n}}:t_{1}=0,1,\ldots,p_{1}-1,t_{2}=0,1,\ldots,p_{2}-1,\ldots,t_{n}=0,1,\ldots,p_{n}-1\right\\},$

forms
$(\prod_{i=1}^{n}p_{i},\prod_{i=1}^{n}p_{i},\prod_{i=1}^{n}p_{i}^{m_{i}})$-CCC.
For $k=n+1$, the set $S$ reduces to

$\begin{split}S&=\left\\{C_{t_{1}}^{p_{1}}\otimes
C_{t_{2}}^{p_{2}}\otimes\ldots\otimes
C_{t_{n+1}}^{p_{n+1}}:t_{1}=0,1,\ldots,p_{1}-1,t_{2}=0,1,\ldots,p_{2}-1,\ldots,\right.\\\
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\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \left.t_{n+1}=0,1,\ldots,p_{n+1}-1\right\\}.\end{split}$
(59)

As per our assumption,

$\left\\{C_{t_{1}}^{p_{1}}\otimes C_{t_{2}}^{p_{2}}\otimes\ldots\otimes
C_{t_{n}}^{p_{n}}:t_{1}=0,1,\ldots,p_{1}-1,t_{2}=0,1,\ldots,p_{2}-1,\ldots,t_{n}=0,1,\ldots,p_{n}-1\right\\},$

forms
$(\prod_{i=1}^{n}p_{i},\prod_{i=1}^{n}p_{i},\prod_{i=1}^{n}p_{i}^{m_{i}})$-CCC
and since $\\{C_{t_{n+1}}^{p_{n+1}}:t_{n}=0,1,\ldots,p_{n+1}-1\\}$ forms
$(p_{n+1},p_{n+1},p_{n+1}^{m_{n+1}})$-CCC, using Lemma 3, the set $S$ forms
$(\prod_{i=1}^{n+1}p_{i},\prod_{i=1}^{n+1}p_{i},\prod_{i=1}^{n+1}p_{i}^{m_{i}})$-CCC.
Therefore, the statement is true for $k=n+1$ and hence it is true for all
values of $k$. ∎

Lemma 4 and Lemma 5 show that the set of CSs, $S$ in Theorem 2 is obtained
from multivariable function $f$ which is defined in (40). Also, the CSs in $S$
can be expressed as Kronecker product of other CSs which can be obtained from
Theorem 1. Therefore, Theorem 2 provides a link between the proposed
construction and Kronecker product based constructions as introduced in [33,
Th. 1].

Below, we present an example to illustrate Theorem 2.

###### Example 3

Let us choose $k=2$, $m_{1}=2$, $m_{2}=2$, $p_{1}=2$, $p_{2}=3$, $q=6$,
$(\pi_{1}(1),\pi_{1}(2))=(1,2)$ and $(\pi_{2}(1),\pi_{2}(2))=(1,2)$.
Therefore, $\lambda=l.c.m(p_{1}p_{2},q)=6$ and from (39), we have

$f=f_{1}+f_{2},$ (60)

where

$\begin{split}f_{1}(v_{2,1},v_{2,2})&=3v_{2,1}v_{2,2}+v_{2,1}+1\\\
f_{2}(v_{3,1},v_{3,2})&=2v_{3,1}v_{3,2}+2v_{3,1}+2.\end{split}$ (61)

From (41), we have

$\begin{split}a_{\mathbf{t}_{2}}^{\bm{\alpha}_{2},\mathbf{p}_{2}}=f+3\alpha_{1}v_{2,1}+2\alpha_{2}v_{3,1}+3t_{1}v_{2,2}+2t_{2}v_{3,2},\end{split}$
(62)

where $\bm{\alpha}_{2}=(\alpha_{1},\alpha_{2})$,
$\mathbf{p}_{2}=(p_{1},p_{2})$, $\mathbf{t}_{2}=(t_{1},t_{2})$,
$\alpha_{1},t_{1}\in\\{0,1\\}$ and $\alpha_{2},t_{2}\in\\{0,1,2\\}$.

From (43), we have

$\begin{split}C_{(0,0)}^{(2,3)}&=\begin{bmatrix}\psi(f+3\cdot 0\cdot
v_{2,1}+2\cdot 0\cdot v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 0\cdot v_{3,2})\\\
\psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 0\cdot
v_{2,2}+2\cdot 0\cdot v_{3,2})\\\ \psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 2\cdot
v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 0\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot
v_{2,1}+2\cdot 0\cdot v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 0\cdot v_{3,2})\\\
\psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 0\cdot
v_{2,2}+2\cdot 0\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 2\cdot
v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 0\cdot v_{3,2})\end{bmatrix}\\\
&=\begin{bmatrix}351315333402420444351315333135153111\\\
315333351420444402315333351153111135\\\
333351315444402420333351315111135153\\\
351315333135153111351315333402420444\\\
315333351153111135315333351420444402\\\
333351315111135153333351315444402420\end{bmatrix}\\\
C_{(0,1)}^{(2,3)}&=\begin{bmatrix}\psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 0\cdot
v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 1\cdot v_{3,2})\\\ \psi(f+3\cdot 0\cdot
v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 1\cdot v_{3,2})\\\
\psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 2\cdot v_{3,1}+3\cdot 0\cdot
v_{2,2}+2\cdot 1\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 0\cdot
v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 1\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot
v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 1\cdot v_{3,2})\\\
\psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 2\cdot v_{3,1}+3\cdot 0\cdot
v_{2,2}+2\cdot 1\cdot v_{3,2})\end{bmatrix}\\\
&=\begin{bmatrix}351531111402042222351531111135315555\\\
315555135420000240315555135153333513\\\
333513153444024204333513153111351531\\\
351531111135315555351531111402042222\\\
315555135153333513315555135420000240\\\
333513153111351531333513153444024204\end{bmatrix}\\\
C_{(0,2)}^{(2,3)}&=\begin{bmatrix}\psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 0\cdot
v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 2\cdot v_{3,2})\\\ \psi(f+3\cdot 0\cdot
v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 2\cdot v_{3,2})\\\
\psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 2\cdot v_{3,1}+3\cdot 0\cdot
v_{2,2}+2\cdot 2\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 0\cdot
v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 2\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot
v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 0\cdot v_{2,2}+2\cdot 2\cdot v_{3,2})\\\
\psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 2\cdot v_{3,1}+3\cdot 0\cdot
v_{2,2}+2\cdot 2\cdot v_{3,2})\end{bmatrix}\end{split}$ (63)
$\begin{split}&=\begin{bmatrix}351153555402204000351153555135531333\\\
315111513420222024315111513153555351\\\
333135531444240042333135531111513315\\\
351153555135531333351153555402204000\\\
315111513153555351315111513420222024\\\
333135531111513315333135531444240042\end{bmatrix}\\\
C_{(1,0)}^{(2,3)}&=\begin{bmatrix}\psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 0\cdot
v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 0\cdot v_{3,2})\\\ \psi(f+3\cdot 0\cdot
v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 0\cdot v_{3,2})\\\
\psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 2\cdot v_{3,1}+3\cdot 1\cdot
v_{2,2}+2\cdot 0\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 0\cdot
v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 0\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot
v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 0\cdot v_{3,2})\\\
\psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 2\cdot v_{3,1}+3\cdot 1\cdot
v_{2,2}+2\cdot 0\cdot v_{3,2})\end{bmatrix}\\\
&=\begin{bmatrix}351315333402420444024042000402420444\\\
315333351420444402042000024420444402\\\
333351315444402420000024042444402420\\\
351315333135153111024042000135153111\\\
315333351153111135042000024153111135\\\
333351315111135153000024042111135153\end{bmatrix}\\\
C_{(1,1)}^{(2,3)}&=\begin{bmatrix}\psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 0\cdot
v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 1\cdot v_{3,2})\\\ \psi(f+3\cdot 0\cdot
v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 1\cdot v_{3,2})\\\
\psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 2\cdot v_{3,1}+3\cdot 1\cdot
v_{2,2}+2\cdot 1\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 0\cdot
v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 1\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot
v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 1\cdot v_{3,2})\\\
\psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 2\cdot v_{3,1}+3\cdot 1\cdot
v_{2,2}+2\cdot 1\cdot v_{3,2})\end{bmatrix}\\\
&=\begin{bmatrix}351531111402042222024204444402042222\\\
315555135420000240042222402420000240\\\
333513153444024204000240420444024204\\\
351531111135315555024204444135315555\\\
315555135153333513042222402153333513\\\
333513153111351531000240420111351531\end{bmatrix}\end{split}$
$\begin{split}C_{(1,2)}^{(2,3)}&=\begin{bmatrix}\psi(f+3\cdot 0\cdot
v_{2,1}+2\cdot 0\cdot v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 2\cdot v_{3,2})\\\
\psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 1\cdot
v_{2,2}+2\cdot 2\cdot v_{3,2})\\\ \psi(f+3\cdot 0\cdot v_{2,1}+2\cdot 2\cdot
v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 2\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot
v_{2,1}+2\cdot 0\cdot v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 2\cdot v_{3,2})\\\
\psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 1\cdot v_{3,1}+3\cdot 1\cdot
v_{2,2}+2\cdot 2\cdot v_{3,2})\\\ \psi(f+3\cdot 1\cdot v_{2,1}+2\cdot 2\cdot
v_{3,1}+3\cdot 1\cdot v_{2,2}+2\cdot 2\cdot v_{3,2})\end{bmatrix}\\\
&=\begin{bmatrix}351153555402204000024420222402204000\\\
315111513420222024042444240420222024\\\
333135531444240042000402204444240042\\\
351153555135531333024420222135531333\\\
315111513153555351042444240153555351\end{bmatrix}\end{split}$ (64)

Figure 2: Correlation plots of $(6,6,36)$-CCC.

Fig. 2(a) represents aperiodic auto-correlation sum of any code from the set
of codes $\\{C_{0,0}^{2,3},C_{0,1}^{2,3},$
$C_{0,2}^{2,3},C_{1,0}^{2,3},C_{1,1}^{2,3},C_{1,2}^{2,3}\\}$ and Fig. 2(b)
represents aperiodic cross-correlation sum between any two distinct codes from
the same set of codes. Therefore,
$\\{C_{0,0}^{2,3},C_{0,1}^{2,3},C_{0,2}^{2,3},C_{1,0}^{2,3},C_{1,1}^{2,3},C_{1,2}^{2,3}\\}$
forms $(6,6,36)$-CCC.

Table I: Comparison with [4, 8, 24, 25]

Sequence Class | Based on |  Length ($L$) | Constraints
---|---|---|---
($K,K,L$)-CCC [4] | Second-order GBFs | $2^{m}$ |  $K=2^{k+1}$ $m\in\mathbb{Z}^{+}\setminus\\{0,1\\}$, $k\in\mathbb{Z}^{+}$
($K,K,L$)-CCC [8] |  Higher-order ($\geq 2$) GBFs | $2^{m}$ |  $K=2^{k+1}$, $m\in\mathbb{Z}^{+}\setminus\\{0,1\\}$, $k\in\mathbb{Z}^{+}$
($K,K,L$)-CCC [24] |  Higher-order ($\geq 2$) GBFs | $2^{m}$ |  $K=2^{k+1}$, $m\in\mathbb{Z}^{+}\setminus\\{0,1\\}$, $k\in\mathbb{Z}^{+}$
($K,M,L$)-MOGCS [25] | Second-order GBFs | $2^{m-1}+2^{t}$ |  $K\leq M/2$, $M=2^{k+1}$ $m\in\mathbb{Z}^{+}\setminus\\{0,1\\}$, $k,t\in\mathbb{Z}^{+}$, $0\leq t\leq m-1$
($K,K,L$)-CCC Theorem 1 | Second-order multivariable functions | $p^{m}$ |  $m\in\mathbb{Z}^{+}\setminus\\{0\\}$, $K=p$, $p$ is a prime number
($K,K,L$)-CCC Theorem 2 | Second-order multivariable functions | $p_{1}^{m_{1}}p_{2}^{m_{2}}\cdots p_{k}^{m_{k}}$ |  $K=p_{1}p_{2}\cdots p_{k}$, $m_{i}\in\mathbb{Z}^{+}\setminus\\{0\\}$, $p_{i}$ is a prime number, $i=1,2,\ldots,k$

###### Remark 1

From Theorem 2, we obtain
$(\prod_{i=1}^{k}p_{i},\prod_{i=1}^{k}p_{i},\prod_{i=1}^{k}p_{i}^{m_{i}})$-CCC
where the length of the CCC is given by

$L=\prod_{i=1}^{k}p_{i}^{m_{i}}=p_{1}^{m_{1}}p_{2}^{m_{2}}\cdots
p_{k}^{m_{k}}.$

Let us consider $p_{1},p_{2},\ldots,p_{k}$ as distinct prime numbers. Based on
the different choices of $p_{1},p_{2},\ldots,p_{k}$,
$m_{1},m_{2},\ldots,m_{k}$ and $k\in\mathbb{Z}^{+}\setminus\\{0\\}$, and from
the statement of unique factorization theorem given in Lemma 1, it can be
observed that the parameter $L$ attains any positive integer values greater
than 1.

Remark 1 shows that the proposed construction in Theorem 2 generates CCCs of
all possible lengths greater than $1$.

###### Remark 2

The GBFs based constructions reported in [2] and [21] generates
$(2^{k},2^{m})$-CS, where $k\leq m$. The constructions reported in [4], [8],
and [24] produce $(2^{k},2^{k},2^{m})$-CCC. For $p_{1}=p_{2}=\cdots=p_{k}=2$
and $m_{1}+m_{2}+\cdots+m_{k}=m$, Theorem 2 generate
$(2^{k},2^{k},2^{m})$-CCC. Therefore, the proposed construction in Theorem 2
is also able to generate CSs and CCCs with the same parametes, $K=2^{k}$,
$M=2^{k}$, and $L=2^{m}$, as reported in [4], [8], and [24] for the
aforementioned fixed values of $p_{1},p_{2},\ldots,p_{k}$ and
$m_{1},m_{2},\ldots,m_{k}$.

##  IV Comparison with the Existing Work [31]

In [31], Wang _et al_ proposed PU matrix based construction of CCCs and then a
systematic approach has been introduced to extract multivariable functions as
the generators for CCCs of non-power-of-two lengths. The construction method
in [31] needs to perform the following steps unlike our proposed construction
to obtain the multivariable functions to generate CCCs:

  * •

It needs to follow a general framework for constructing desired PU matrices.

  * •

A series of operations need to be performed to extract multivariable functions
from the constructed PU matrices.

The multivariable functions extraction process also depends on the existence
suitable PU matrices which are known as seed PU matrices and it heavily rely
on a series of operations to extract multivariable functions from the seed PU
matrices. Unlike, the construction reported in [31], the proposed
multivariable functions generate CCCs without performing a series of
operations through multiple steps. Therefore, the proposed construction is
more suitable for rapid hardware generations of CCCs as compared to the work
reported in [31].

In addition to the above comparison with [31], we also compare the propsed
constructions with previous GBFs based constructions of CCCs [4, 8, 24] and
MOGCS [25] in TABLE I.

##  V Conclusions

In this paper, we have presented constructions of CCCs using multivariable
functions. First we have presented a construction of CCC of lengths in the
form $p^{m}$ in Theorem 1 which includes [6, Th. 3] as special case. Then we
have presented a class of multivariable functions in Theorem 2 with the help
of the proposed multivariable functions in Theorem 1. The proposed
multivariable functions in Theorem 2 generate CCCs of lengths in the form
$\prod_{i=1}^{k}p_{i}^{m_{i}}$. which covers possible lengths greater than
$1$, without having a series of sequence operation on some special sequences.
The proposed construction is also able to provide a flexibility on the number
of constituent sequences. Due flexibility in the lengths and the number of
constituent sequences, the proposed construction covers all possible
parameters which are introduced by the existing GBFs based constructions of
CCCs. We also establish a link, between the proposed multivariable functions
based CCCs and the CCCs which can be obtained by applying Kronecker product
operation on existing $(p,p,p^{m})$-CCCs, in Lemma 5.

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