[table]capposition=top

# Construction of Inter-Group Complementary Code Set and 2-D Z-Complementary
Array Code Set Based on Multivariable Functions

Abhishek Roy, and Sudhan Majhi Abhishek Roy is with the Department of
Mathematics, Indian Institute of Technology, Patna, Bihar, 801106 India
e-mail<EMAIL_ADDRESS>Majhi is with the department of Electrical
Engineering, Indian Institute of Science, Bangalore, Karnataka, 560012 India
e-mail<EMAIL_ADDRESS>
###### Abstract

The need of two dimensional (2-D) arrays with good 2-D correlation properties
have been found of great interest in research because of the rapid development
of the modern technologies in the field of wireless communications. 2-D arrays
have found various applications wireless communication technologies such as
ultra-wide band (UWB), 2-D synchronization, phased array antenna etc. In this
paper, we propose a direct construction of 2-D Z-complementary array code set
(ZCACS) by using multivariable function. For this purpose, we first proposed a
construction of inter-group complementary (IGC) code set using multivariable
function and use this construction to produce our desired 2-D ZCACS. The use
of multivariable function enables us to set the size of the 2-D ZCACS to be
$L_{1}\times L_{2}$, where $L_{1}=2^{m}$,
$L_{2}=2p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$, $m\geq 1$ and
$m_{\alpha}\geq 2$ for all $\alpha=1,2,\dots,k$, which is an improvement over
the traditional generalized Boolean function that produces sequences of length
of the form $2^{m}$.

###### Index Terms:

Golay complementary pair (GCP), inter-group complementary (IGC) code set,
multivariable function, Z-complementary array code set (ZCACS).

##  I Introduction

In the field of sequence design, the introduction of Golay complementary pair
(GCP) by Marcel J. E. Golay in 1951 marks a milestone [1]. It is a pair of
sequences whose aperiodic auto-correlation sum (AACS) is zero for all time
shifts except the zero shift. Due to its ideal auto-correlation properties, it
has been widely used in wireless communication engineering such as reducing
the peak-to-mean envelope power ratio (PMEPR) of orthogonal frequency division
multiplexing (OFDM) [2, 3, 4], channel estimation [5], radar [6] etc. In 1972,
Tseng and Liu in [7] extended the concept of GCP to complementary set (CS),
which refers to a set of sequences of same lengths having zero AACS for all
non-zero time shifts. Although there are many constructions of CSs in the
literature, the first direct construction was given by Paterson [8] by using
generalized Boolean function (GBF). Later, CS was extended to the concept of
mutually orthogonal Golay complementary set (MOGCS) which is a collection of
multiple CSs with the property that aperiodic cross-correlation sum (ACCS) of
the CSs is zero for all shifts. Complete complementary code (CCC) is the
special case of MOGCS when the set size achieves its upper bound and it was
introduced by Suehiro and Hatori in [9]. In the literature, there are many
constructions of CCCs [10, 11, 12, 13, 14, 15, 16].

There has been some drawbacks in CCCs regarding the set size, which is bounded
by the number of sequences in each constituting CS. To ovecome this drawbak,
later the concept of the Z-complementary code set (ZCCS) was introduced. A
ZCCS consists of codes which display zero ACCS wihin a range around the zero
shift, which is called the zero-correlation zone (ZCZ). Several direct and
indirect constructions of ZCCS can be found in the literature [17, 18, 19, 20]
using GBF method or other recursive tools. A special case of ZCCS is inter-
group complementary (IGC) code set where the codes are divided into some
groups and ACCS of codes from two different groups is zero for all time shifts
and within the same group it follows ZCCS properties. Several constructions of
IGC code sets can be found in the literture [21, 22, 23].

The discussed above are known as one dimensional (1-D) sequences. However,
these 1-D sequences can be generealized to two-dimensional (2-D) arrays. The
1-D GCP can be extended to 2-D and called 2-D Golay complementary array pair
(GCAP). Some constructions of 2-D GCAP are provided in [24, 25, 26]. Similarly
2-D MOGCS is a generalization of 1-D MOGCS. It was first introduced by Lüke in
1985 [27]. Later in 2003, Zhang, Lin and Hatori proposed an application of
2D-MOGCSs in ultra wide-band (UWB) multi-carrier code division multiple access
(MC-CDMA) system [28]. When the set size of a 2-D MOGCS is equal to the number
of constituent arrays, then it is called 2-D CCC [29]. Till date many
constructions of 2-D MOGCS have appered in the literature [30, 31, 32].

As the set size is limited in 2-D MOGCS, the 2-D generalization of 1-D ZCCS is
introduced by Xeng et al. in 2004, called 2-D Z-complementary array code set
(ZCACS) [33]. The applications of 2-D ZCACS include the high bandwidth
spectral efficient implementaion of UWB MC-CDMA in one dimension and reduction
of PMEPR in other dimension, multiple access system etc.

There has been some construction of 2-D ZCACS in the literature. Recently, Das
et al. [34] proposed a construction of 2-D ZCACS by using 2-D Z-paraunitary
(ZPU) matrices. But the construnctions in the literature including this does
not follow a direct appraoch. Motivated by this, we have developed a direct
construction method of 2-D ZCACS by using multivariable funcions. Recenty, the
concept of multivariable functions is introduced by Sarkar et al. in [16].

Motivated by this, in this paper, we propose a construction of IGC code set of
length of the form $p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$, where
$p_{\alpha}$’s are prime numbers and $m_{\alpha}\geq 2$ for all $\alpha$, by
using multivariable function. Then we propose construction of 2-D ZCACS by
using a set of multi-variable function, which produce the arrays corresponding
to the 2-D ZCACS. The size of the proposed 2-D ZCACS is of the form
$L_{1}\times L_{2}$, where $L_{1}=2^{m}$ and
$L_{2}=2p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$. The proposed
construction is a direct method.

The rest of the paper is organized as follows. In Section II, some definitions
and lemmas related to our proposed construction are discussed. The
construction of the IGC code set is described in section III. The section IV
contains the contrucion of 2-D ZCACS. Finally, conclusions are drawn in
section V.

##  II Preliminaries

###  II-A One dimensional sequences

######  Definition 1 (ACCF ([19]))

For two complex-valued sequences $\mathbf{a}=(a_{0},a_{1},\dots,a_{L_{1}})$
and $\mathbf{b}=(b_{0},b_{1},\dots,b_{L_{1}})$ of length $L$, the ACCF at the
time shift $\tau$ is defined by

$C(\mathbf{a},\mathbf{b})(\tau)=\begin{cases}\sum\limits_{i=0}^{L-\tau-1}a_{i}b_{i+\tau}^{*},&0\leq\tau<L;\\\
\sum\limits_{i=0}^{L+\tau-1}a_{i-\tau}b_{i}^{*},&-L<\tau\leq-1;\\\
0,&\mid\tau\mid\geq L,\end{cases}$ (1)

where $(\cdot)^{*}$ denotes the complex conjugate. If $\mathbf{a}=\mathbf{b}$,
then the function is called AACF and is denoted by $A(\mathbf{a})(\tau)$.

######  Definition 2 (ZCP [19])

A pair of length-$L$ sequences $(\mathbf{a},\mathbf{b})$ is called a ZCP if
they satisfy the following:

$A(\mathbf{a})(\tau)+A(\mathbf{b})(\tau)=0~{}~{}\text{for}~{}~{}0<\mid\tau\mid<Z,$
(2)

where $Z\leq L$. Here $Z$ is called the ZCZ width. If $Z=L$, then
$(\mathbf{a},\mathbf{b})$ is called a GCP.

######  Definition 3 (Code ([18]))

An ordered set of sequences
$\\{\mathbf{a}_{0},\mathbf{a}_{1},\dots,\mathbf{a}_{M-1}\\}$ of same length
$L$, is called a code. A code can be written in matrix form as follows:

$\mathcal{C}=\begin{bmatrix}\mathbf{a}_{0}\\\\[2.0pt] \mathbf{a}_{1}\\\
\vdots\\\ \mathbf{a}_{M-1}\end{bmatrix}_{M\times L}$ (3)

######  Definition 4 (ACCF of code ([18]))

Let
$\mathcal{C}_{1}=\\{\mathbf{a}_{0}^{1},\mathbf{a}_{1}^{1},\dots,\mathbf{a}_{M-1}^{1}\\}$,
$\mathcal{C}_{2}=\\{\mathbf{a}_{0}^{2},\mathbf{a}_{1}^{2},\dots,\mathbf{a}_{M-1}^{2}\\}$
be two $M\times L$ matrices or codes, then ACCF of $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ is defined by

$C(\mathcal{C}_{1},\mathcal{C}_{2})(\tau)=\displaystyle\sum_{\lambda=0}^{M-1}C(\mathbf{a}_{\lambda}^{1},\mathbf{a}_{\lambda}^{2})(\tau).$
(4)

######  Definition 5 (MOGCS([11]))

A collection
$\mathcal{C}=\\{\mathcal{C}_{0},\mathcal{C}_{1},\cdots,\mathcal{C}_{K-1}\\}$
of codes is called MOGCS if it satisfies

$C(\mathcal{C}_{\nu_{1}},\mathcal{C}_{\nu_{2}})(\tau)=\begin{cases}ML,&\tau=0,\quad\nu_{1}=\nu_{2};\\\
0,&0<|\tau|<L,\quad\nu_{1}=\nu_{2};\\\
0,&|\tau|<L,\quad\nu_{1}\neq\nu_{2}.\end{cases}$ (5)

When $K=M$, then $\mathcal{C}$ is called CCC.

######  Definition 6 (ZCCS ([18]))

A collection
$\mathcal{C}=\\{\mathcal{C}_{0},\mathcal{C}_{1},\cdots,\mathcal{C}_{K-1}\\}$
is called ZCCS if it satisfies

$C(\mathcal{C}_{\nu_{1}},\mathcal{C}_{\nu_{2}})(\tau)=\begin{cases}ML,&\tau=0,\quad\nu_{1}=\nu_{2};\\\
0,&0<|\tau|<Z,\quad\nu_{1}=\nu_{2};\\\
0,&|\tau|<Z,\quad\nu_{1}\neq\nu_{2},\end{cases}$ (6)

where $Z\leq L$. Here, $Z$ is called the ZCZ width. It is detnoted by
$(K,Z)$-$\text{ZCCS}_{M}^{L}$.

######  Lemma 1 ([18])

For any $(K,Z)$-$\text{ZCCS}_{M}^{L}$, the following inequality holds

$K\leq M\left\lfloor\frac{L}{Z}\right\rfloor,$ (7)

where $\left\lfloor\cdot\right\rfloor$ is the floor function. If equality
holds, then the ZCCS is said to be an optimal.

######  Definition 7 (IGC code set ([21]))

Let $I(K,M,L,Z)=\\{\mathcal{C}^{i}:i=0,1,\dots,K-1\\}$ be an optimal
$(K,Z)$-$\text{ZCCS}_{M}^{L}$. Then, it follows the bound $K=M\left\lfloor
L/Z\right\rfloor$. Let also be that the $K$ codes can be divided into $M$
number of code groups $I^{g}(g=0,1,\dots,M-1)$. Each group contains
$K/M=\left\lfloor L/Z\right\rfloor$ codes and the code set $I(K,M,L,Z)$
follows the following properties:

$C({\mathcal{C}^{i},\mathcal{C}^{j}})({\tau})=\begin{cases}ML,&i=j,\quad\tau=0,\\\
0,&i=j,\quad 0<|\tau|<Z,\\\ 0,&i\neq j,\quad\mathcal{C}^{i},\mathcal{C}^{j}\in
I^{g},~{}|\tau|<Z,\\\ 0,&\mathcal{C}^{i}\in I^{g_{1}},\quad\mathcal{C}^{j}\in
I^{g_{2}},~{}g_{1}\neq g_{2},|\tau|<L.\end{cases}\\\ $ (8)

Then, the code set $I(K,M,L,Z)$ is called an IGC code set.

###  II-B Two Dimensional Array

######  Definition 8 (2-D Array ([34]))

A 2-D array is a two dimensional matrix $\mathbf{X}$, an element denoted by
$X_{i,j}$, where $0\leq i<L_{1}$, $0\leq j<L_{2}$ and the size of the array is
$L_{1}\times L_{2}$.

######  Definition 9 (2-D ACCF ([34]))

The 2-D ACCF of two arrays $\mathbf{X}$ and $\mathbf{Y}$ both of size
$(L_{1},L_{2})$ at time shift $(\tau_{1},\tau_{2})$ is defined by

$C(\mathbf{X},\mathbf{Y})(\tau_{1},\tau_{2})=\\!\begin{cases}\sum\limits_{i=0}^{L_{1}-\tau_{1}-1}\sum\limits_{j=0}^{L_{2}-\tau_{2}-1}X_{i,j}Y^{*}_{i+\tau_{1},j+\tau_{2}},&\text{if}~{}0\leq\tau_{1}<L_{1},0\leq\tau_{2}<L_{2};\\\
\sum\limits_{i=0}^{L_{1}+\tau_{1}-1}\sum\limits_{j=0}^{L_{2}-\tau_{2}-1}X_{i-\tau_{1},j}Y^{*}_{i,j+\tau_{2}},&\text{if}~{}-L_{1}<\tau_{1}\leq-1,0\leq\tau_{2}<L_{2};\\\
\sum\limits_{i=0}^{L_{1}-\tau_{1}-1}\sum\limits_{j=0}^{L_{2}+\tau_{2}-1}X_{i,j-\tau_{2}}Y^{*}_{i+\tau_{1},j},&\text{if}~{}0\leq\tau_{1}\leq
L_{1},-L_{2}<\tau_{2}\leq-1;\\\
\sum\limits_{i=0}^{L_{1}+\tau_{1}-1}\sum\limits_{j=0}^{L_{2}+\tau_{2}-1}X_{i-\tau_{1},j-\tau_{2}}Y^{*}_{i,j},&\text{if}~{}-L_{1}\\!\\!<\\!\tau_{1}\\!\leq\\!\\!-1,-L_{2}\\!\\!<\\!\tau_{2}\\!\leq\\!\\!-1.\end{cases}$
(9)

Here $(\cdot)^{*}$ denotes the complex conjugate. When
$\mathbf{X}=\mathbf{Y}$, then it is called 2-D AACF and is denoted by
$A(\mathbf{X})(\tau_{1},\tau_{2})$.

######  Definition 10 (2-D ZCAC ([34, 35]))

A set $\mathcal{X}=\\{\mathbf{X}_{0},\mathbf{X}_{1},\dots,\mathbf{X}_{M-1}\\}$
of 2-D arrays with equal size $L_{1}\times L_{2}$ is called a 2-D
Z-complementary array code (ZCAC) if they satisfy:

$\sum_{i=0}^{M-1}A(\mathbf{X}_{i})(\tau_{1},\tau_{2})=\begin{cases}ML_{1}L_{2},&(\tau_{1},\tau_{2})=(0,0);\\\
0,&0\leq\mid\tau_{1}\mid<Z_{1},0\leq\mid\tau_{2}\mid<Z_{2},(\tau_{1},\tau_{2})\neq(0,0).\end{cases}$
(10)

where $Z_{1}\leq L_{1}$, $Z_{2}\leq L_{2}$. When $M=2$, then it is called a
2-D Z-complementary array pair (ZCAP).

######  Definition 11 (2-D ZCACS ([34]))

A collection of 2-D ZCAC
$\\{\mathcal{X}_{0},\mathcal{X}_{1},\dots,\mathcal{X}_{K-1}\\}$ where
$\mathcal{X}_{\lambda}=\\{\mathbf{X}_{0}^{\lambda},\mathbf{X}_{1}^{\lambda},\dots,\mathbf{X}_{M-1}^{\lambda}\\}$
and each $\mathbf{X}_{i}^{\lambda}$ has equal size of $L_{1}\times L_{2}$, is
called a 2-D ZCACS if they satisfy:

$\sum_{i=0}^{M-1}C(\mathbf{X}_{i}^{\lambda_{1}},\mathbf{X}_{i}^{\lambda_{2}})(\tau_{1},\tau_{2})=\begin{cases}ML_{1}L_{2},&\lambda_{1}=\lambda_{2},(\tau_{1},\tau_{2})=(0,0);\\\
0,&\lambda_{1}=\lambda_{2},0\leq\mid\tau_{1}\mid<Z_{1},0\leq\mid\tau_{2}\mid<Z_{2};(\tau_{1},\tau_{2})\neq(0,0);\\\
0,&\lambda_{1}\neq\lambda_{2},\mid\tau_{1}\mid<Z_{1},\mid\tau_{2}\mid<Z_{2},\\\
\end{cases}$ (11)

where $Z_{1}\leq L_{1}$, $Z_{2}\leq L_{2}$.

###  II-C Multivariable Function

A multivariable function can be defined as

$f:\mathbb{Z}_{p_{1}}^{m_{1}}\times\mathbb{Z}_{p_{2}}^{m_{2}}\times\dots\times\mathbb{Z}_{p_{k}}^{m_{k}}\rightarrow\mathbb{Z}_{\lambda}$

where $p_{1},p_{2},\dots,p_{k}$ are prime numbers and $\lambda$ is a positive
integer. For ease of understanding we define the following notations

  * •

$v_{p_{\alpha},1},v_{p_{\alpha},2},\dots,v_{p_{\alpha},m_{\alpha}}$ are the
$m_{\alpha}$ variables which take values from $\mathbb{Z}_{p_{\alpha}}$ for
$\alpha=1,2\dots,k$.

  * •

$u_{p_{\alpha},i_{\alpha}}$ is the vector representation of $i_{\alpha}$ with
base $p_{\alpha}$, i.e.,
$u_{p_{\alpha},i_{\alpha}}=(i_{\alpha,1},i_{\alpha,2},\dots,i_{\alpha,m_{\alpha}})$
where
$i_{\alpha}=\sum_{\gamma=1}^{m_{\alpha}}p_{\alpha}^{\gamma-1}i_{\alpha,\gamma}$.

  * •

The set $M^{r}$ of monomials of degree at most $r$ is given by

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

where $j_{\beta}^{\alpha}\in\\{0,1,\dots,p_{\alpha}-1\\}$ and $0\leq
r\leq\sum_{\alpha=1}^{k}m_{\alpha}$.

A multivariable function $f$ is a linear combination of the monomials from the
sets $M^{r}$ with coefficients from $\mathbb{Z}_{\lambda}$. The order of the
multi-varible function is the degree of the highest degree monomial which has
non-zero co-efficient.

A multivariable function
$f:\mathbb{Z}_{p_{1}}^{m_{1}}\times\mathbb{Z}_{p_{2}}^{m_{2}}\times\dots\times\mathbb{Z}_{p_{k}}^{m_{k}}\rightarrow\mathbb{Z}_{\lambda}$
can be associated with a $\mathbb{Z}_{\lambda}$-valued sequence of length
$p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$. For this, we define the
ordered set

$\begin{split}U=\bigg{(}(u_{p_{1},i_{1}},u_{p_{2},i_{2}},\dots,u_{p_{k},i_{k}}):i_{1}=0,1,\dots,p_{1}^{m_{1}}-1;&i_{2}=0,1,\dots,p_{2}^{m_{2}}-1;\dots;\\\
&\qquad i_{k}=0,1,\dots,p_{k}^{m_{k}}-1\bigg{)}.\end{split}$ (12)

The $\mathbb{Z}_{\lambda}$-valued sequence corresponding to the multivariable
function $f$ is given by

$\begin{split}\mathbf{f}=\bigg{(}f(u_{p_{1},i_{1}},u_{p_{2},i_{2}},\dots,u_{p_{k},i_{k}}):i_{1}=0,1,\dots,p_{1}^{m_{1}}-1;&i_{2}=0,1,\dots,p_{2}^{m_{2}}-1;\dots;\\\
&\qquad i_{k}=0,1,\dots,p_{k}^{m_{k}}-1\bigg{)}.\end{split}$ (13)

One can also associate a complex-valued sequence corresponding to a
multivariable function $f$, which is given by

$\begin{split}\psi(\mathbf{f})=\bigg{(}\omega_{\lambda}^{f(u_{p_{1},i_{1}},u_{p_{2},i_{2}},\dots,u_{p_{k},i_{k}})}:i_{1}=0,1,\dots,p_{1}^{m_{1}}-1;&i_{2}=0,1,\dots,p_{2}^{m_{2}}-1;\dots;\\\
&\qquad i_{k}=0,1,\dots,p_{k}^{m_{k}}-1\bigg{)},\end{split}$ (14)

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

###### Remark 1

When $k=1$ and $p_{1}=2$, then the multivariable function reduces to
$f:\mathbb{Z}_{2}^{m}\rightarrow\mathbb{Z}_{\lambda}$ which is the form of the
generalized Boolean function (GBF). Hence, multivariable function is the
generalization of the GBF.

We further introduce some notations to produce 2-D matrices by using
multivariable functions. Let
$F_{\mathbf{d}}:\mathbb{Z}_{p_{1}}^{m_{1}}\times\mathbb{Z}_{p_{1}}^{m_{2}}\times\dots\times\mathbb{Z}_{p_{\alpha}}^{m_{\alpha}}\rightarrow\mathbb{Z}_{\lambda}$
be a set of multivariable functions for
$\mathbf{d}=(d_{p^{\prime}_{1}},d_{p^{\prime}_{2}},\dots,d_{p^{\prime}_{w}})\in\prod_{\alpha=1}^{w}\mathbb{Z}_{p^{\prime}_{\alpha}}$,
where $p^{\prime}_{\alpha}$ are some prime numbers. Let
$K=\\{F_{\mathbf{d}}:\mathbf{d}\in\prod_{\alpha=1}^{w}\mathbb{Z}_{p^{\prime}_{\alpha}}\\}$
be an ordered set, which can be written as a $\mathbb{Z}_{\lambda}$-valued
matrix or array of size $L_{1}\times L_{2}$, where
$L_{1}=p^{\prime}_{1}p^{\prime}_{2}\dots p^{\prime}_{w}$ and
$L_{2}=p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$. The $i$-th row of $K$
is determined by the multivariable function $F_{\vec{i}}$, where $\vec{i}$ is
the vector representation of $i$ in
$\mathbb{Z}_{p^{\prime}_{1}}\times\mathbb{Z}_{p^{\prime}_{2}}\times\dots\times\mathbb{Z}_{p^{\prime}_{w}}$.
So, the $(i,j)$-th component of the matrix $K$ is defined by
$K_{i,j}=F_{\vec{i}}(\vec{j})$, where $\vec{j}$ is the vector representation
of $j$ in $\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}^{m_{\alpha}}$. We can
also associate a complex-valued matrix $\Psi(K)$ with $K$, where
$\left(\Psi(K)\right)_{i,j}=\omega_{\lambda}^{K_{i,j}}$ for all $i,j$.

##  III Construction of IGC Code Set

In this section, a construction of IGC code set for lengths of the form
$p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$, ($m_{\alpha}\geq 2$, for all
$\alpha$) is presented by using multivariable function. We take the
multivariable functions

$f_{\alpha}:\mathbb{Z}_{p_{\alpha}}^{m_{\alpha}-1}\rightarrow\mathbb{Z}_{\lambda},$

for $\alpha=1,2,\dots,k$. Each $f_{\alpha}$ is defined by

$\begin{split}f_{\alpha}(v_{p_{\alpha},1},v_{p_{\alpha},2},\dots,v_{p_{\alpha},m_{\alpha}-1})=&\frac{\lambda}{p_{\alpha}}\sum_{\beta=1}^{m_{\alpha}-2}v_{p_{\alpha},\pi_{\alpha}(\beta)}v_{p_{\alpha},\pi_{\alpha}(\beta+1)}+\sum_{\beta=1}^{m_{\alpha}-1}c_{\alpha,\beta}v_{p_{\alpha},\beta}+c_{\alpha},\end{split}$
(15)

where $c_{\alpha,\beta},c_{\alpha}\in\mathbb{Z}_{\lambda}$ for
$\beta=1,2,\dots,m_{\alpha}-1$; $m_{\alpha}\geq 2$ for all $\alpha$,
$\pi_{\alpha}$ is a permutation of the set $\\{1,2,\dots,m_{\alpha}-1\\}$ and
$\lambda$ is taken in such a way that $p_{\alpha}\mid\lambda$ for all
$\alpha=1,2,\dots,k$. Let the function
$f:\mathbb{Z}_{p_{1}}^{m_{1}-1}\times\mathbb{Z}_{p_{2}}^{m_{2}-1}\times\dots\times\mathbb{Z}_{p_{k}}^{m_{k}-1}\rightarrow\mathbb{Z}_{\lambda}$
be defined by

$f=f_{1}+f_{2}+\dots+f_{k}.$ (16)

Now, for
$\mathbf{\gamma}=(\gamma_{1},\gamma_{2},\dots,\gamma_{k}),\mathbf{s}=(s_{1},s_{2},\dots,s_{k}),\mathbf{t}=(t_{1},t_{2},\dots,t_{k})\in\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{2}}\times\dots\times\mathbb{Z}_{p_{k}}$,
we define the set of functions

$a_{\mathbf{s},\mathbf{t}}^{\mathbf{\gamma}}:\mathbb{Z}_{p_{1}}^{m_{1}-1}\times\dots\times\mathbb{Z}_{p_{k}}^{m_{k}-1}\times\mathbb{Z}_{p_{1}}\times\dots\times\mathbb{Z}_{p_{k}}\rightarrow\mathbb{Z}_{\lambda},$

given by

$\begin{split}a_{\mathbf{s},\mathbf{t}}^{\mathbf{\gamma}}=&f+\sum_{\alpha=1}^{k}\frac{\lambda}{p_{\alpha}}v_{p_{\alpha},\pi_{\alpha}(1)}\gamma_{\alpha}+\sum_{\alpha=1}^{k}\frac{\lambda}{p_{\alpha}}v_{p_{\alpha},\pi_{\alpha}(m_{\alpha}-1)}t_{\alpha}+\sum_{\alpha=1}^{k}\frac{\lambda}{p_{\alpha}}v^{\prime}_{p_{\alpha}}s_{\alpha}\\\
=&R_{\mathbf{t}}^{\mathbf{\gamma}}+T_{\mathbf{s}},\end{split}$ (17)

where
$R_{\mathbf{t}}^{\mathbf{\gamma}}:\mathbb{Z}_{p_{1}}^{m_{1}-1}\times\dots\times\mathbb{Z}_{p_{k}}^{m_{k}-1}\rightarrow\mathbb{Z}_{\lambda}$
is defined by

$R_{\mathbf{t}}^{\mathbf{\gamma}}=f+\sum_{\alpha=1}^{k}\frac{\lambda}{p_{\alpha}}v_{p_{\alpha},\pi_{\alpha}(1)}\gamma_{\alpha}+\sum_{\alpha=1}^{k}\frac{\lambda}{p_{\alpha}}v_{p_{\alpha},\pi_{\alpha}(m_{\alpha}-1)}t_{\alpha},$
(18)

and
$T_{\mathbf{s}}:\mathbb{Z}_{p_{1}}\times\dots\times\mathbb{Z}_{p_{k}}\rightarrow\mathbb{Z}_{\lambda}$
is defined by

$T_{\mathbf{s}}=\sum_{\alpha=1}^{k}\frac{\lambda}{p_{\alpha}}v^{\prime}_{p_{\alpha}}s_{\alpha}.$
(19)

Next, we define the code $\mathcal{C}_{\mathbf{s},\mathbf{t}}$ which is given
by

$\mathcal{C}_{\mathbf{s},\mathbf{t}}=[\psi(a_{\mathbf{s},\mathbf{t}}^{\mathbf{\gamma}})]_{\gamma\in\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{2}}\times\dots\times\mathbb{Z}_{p_{k}}}=\begin{bmatrix}\psi(a_{\mathbf{s},\mathbf{t}}^{(0)_{\gamma}})\\\\[2.0pt]
\psi(a_{\mathbf{s},\mathbf{t}}^{(1)_{\gamma}})\\\ \vdots\\\
\psi(a_{\mathbf{s},\mathbf{t}}^{(p_{1}p_{2}\dots
p_{k}-1)_{\gamma}})\end{bmatrix},$ (20)

where $(i)_{\gamma}$ is the vector representation of $i$ in
$\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{2}}\times\dots\times\mathbb{Z}_{p_{k}}$.

It can be noted that $\psi(a_{\mathbf{s},\mathbf{t}}^{\mathbf{\gamma}})$ can
be written as

$\psi(a_{\mathbf{s},\mathbf{t}}^{\mathbf{\gamma}})=\psi(T_{\mathbf{s}})\otimes\psi(R_{\mathbf{t}}^{\mathbf{\gamma}}),$
(21)

where $\otimes$ denotes the Kronecker product.

Also, for two pairs of sequences $(\mathbf{a}_{1},\mathbf{a}_{2})$ and
$(\mathbf{b}_{1},\mathbf{b}_{2})$ and $\tau\geq 0$, we have

$\begin{split}C(\mathbf{a}_{1}\otimes\mathbf{b}_{1},\mathbf{a}_{2}\otimes\mathbf{b}_{2})(\tau)=&C(\mathbf{a}_{1},\mathbf{a}_{2})\left(\left\lfloor\frac{\tau}{N}\right\rfloor\right)C(\mathbf{b}_{1},\mathbf{b}_{2})(\tau\mod
N)\\\
&+\Delta_{N}C(\mathbf{a}_{1},\mathbf{a}_{2})\left(\left\lfloor\frac{\tau}{N}\right\rfloor+1\right)C(\mathbf{b}_{1},\mathbf{b}_{2})(\tau\mod
N-N)\end{split}$ (22)

where, each $\mathbf{a}_{i}$ has length $M$ and each $\mathbf{b}_{i}$ has
length $N$ for $i=1,2$; and

$\Delta_{N}=\begin{cases}0,&\tau\mod N=0;\\\ 1,&\text{otherwise}.\end{cases}$
(23)

Now, we have our construction of IGC code set of length
$p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$.

###### Theorem 1

The code set
$I=\\{\mathcal{C}_{\mathbf{s},\mathbf{t}}:\mathbf{s},\mathbf{t}\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}\\}$
forms an $(K,M,L,Z)$-IGC code set with $K=(p_{1}p_{2}\dots p_{k})^{2}$,
$M=p_{1}p_{2}\dots p_{k}$, $L=p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$
and $Z=p_{1}^{m_{1}-1}p_{2}^{m_{2}-1}\dots p_{k}^{m_{k}-1}$. The
$\mathbf{t}$-th code group is expressed as
$I^{\mathbf{t}}=\\{\mathcal{C}_{\mathbf{s},\mathbf{t}}:\mathbf{s}\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}\\}$.

###### Proof:

Appendix A. ∎

##  IV Construction of 2-D ZCACS

In this section we give the construction of 2-D ZCACS of arbitrary length
using multivariable functions. But before the construction we give a lemma.

######  Lemma 2 ([8])

Let $\lambda$ be an even positive integer and
$f:\mathbb{Z}_{2}^{m}\rightarrow\mathbb{Z}_{\lambda}$ be a function given by

$f(x_{1},x_{2},\dots,x_{m})=\frac{\lambda}{2}\sum_{\beta=1}^{m-1}x_{\pi(\beta)}x_{\pi(\beta+1)}+\sum_{\beta=1}^{m}g_{\beta}x_{\beta},$
(24)

where $\pi$ is a permutation of the set $\\{1,2,\dots,m\\}$ and
$g_{\beta}\in\mathbb{Z}_{\lambda}$. Then the complex-valued sequences
$\mathbf{a}$ and $\mathbf{b}$, corresponding to the functions

$a(x_{1},x_{2},\dots,x_{m})=f(x_{1},x_{2},\dots,x_{m})+e,$

and

$b(x_{1},x_{2},\dots,x_{m})=f(x_{1},x_{2},\dots,x_{m})+\frac{\lambda}{2}x_{\pi(1)}+e^{\prime},$

respectively, forms a GCP of length $2^{m}$ for any
$e,e^{\prime}\in\mathbb{Z}_{\lambda}$.

Let, $2\mid\lambda$ and $a_{\mathbf{s},\mathbf{t}}^{\mathbf{\gamma}}$ be the
same function used in Theorem 1. For
$\mathbf{d}=(d_{1},d_{2},\dots,d_{m})\in\mathbb{Z}_{2}^{m}$,
$\mathbf{\gamma}=(\gamma_{1},\gamma_{2},\dots,\gamma_{k})\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}$
and $\mathbf{t}_{1}\neq\mathbf{t}_{2}$, we define a set of functions

$F_{\mathbf{d}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}:\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}^{m_{\alpha}-1}\times\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}\times\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{\lambda},$

given by

$\begin{split}F_{\mathbf{d}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}=(1-v^{\prime\prime})\\{a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}+a(\mathbf{d})\\}+v^{\prime\prime}\\{a_{\mathbf{s}_{2},\mathbf{t}_{2}}^{\mathbf{\gamma}}+b(\mathbf{d})\\},\end{split}$
(25)

where $v^{\prime\prime}$ is the variable correspoding to $\mathbb{Z}_{2}$ and
$a,b$ are the functions from Lemma 2.

Let the $\mathbb{Z}_{\lambda}$-valued 2-D arrays be given by
$K^{\mathbf{\gamma}}=\\{F_{\mathbf{d}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}:\mathbf{d}\in\mathbb{Z}_{2}^{m}\\}$
for $\mathbf{\gamma}\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}$ and we
define

$\mathbf{X}^{\mathbf{\gamma}}=\Psi(K^{\mathbf{\gamma}}).$ (26)

###### Theorem 2

The set
$\mathcal{X}=\\{\mathbf{X}^{\mathbf{\gamma}}:\mathbf{\gamma}\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}\\}$
of 2-D arrays is a 2-D ZCAC. Each array of size $L_{1}\times L_{2}$, where
$L_{1}=2^{m}$ and $L_{2}=2p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$ and
ZCZ size is $Z_{1}\times Z_{2}$, where $Z_{1}=2^{m}$ and
$Z_{2}=p_{1}^{m_{1}-1}p_{2}^{m_{2}-1}\dots p_{k}^{m_{k}-1}$.

###### Proof:

Appendix B. ∎

Now, we propose the construction of 2-D ZCACS by using multivariable function.
For that we define the set
$\Omega=\\{(\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}):\mathbf{t}_{1}\neq\mathbf{t}_{2}\\}$.
Let $\Lambda\subset\Omega$ such that for any two
$\zeta_{1}=(\mathbf{s}_{1}^{1},\mathbf{s}_{2}^{1},\mathbf{t}_{1}^{1},\mathbf{t}_{2}^{1}),\zeta_{2}=(\mathbf{s}_{1}^{2},\mathbf{s}_{2}^{2},\mathbf{t}_{1}^{2},\mathbf{t}_{2}^{2})\in\Lambda$,
$\mathbf{t}_{1}^{1},\mathbf{t}_{2}^{1},\mathbf{t}_{1}^{2}$ and
$\mathbf{t}_{2}^{2}$ are all distinct.

For any
$\zeta=(\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2})\in\Lambda$,
we let

$\mathcal{F}_{\mathbf{d}}^{\gamma,\zeta}:\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}^{m_{\alpha}-1}\times\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}\times\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{\lambda},$

be a set of functions defined by

$\begin{split}\mathcal{F}_{\mathbf{d}}^{\gamma,\zeta}=F_{\mathbf{d}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}.\end{split}$
(27)

Let the $\mathbb{Z}_{\lambda}$-valued 2-D arrays be given by
$\mathcal{K}^{\mathbf{\gamma},\zeta}=\\{F_{\mathbf{d}}^{\mathbf{\gamma},\zeta}:\mathbf{d}\in\mathbb{Z}_{2}^{m}\\}$
for $\mathbf{\gamma}\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}$ and we
define

$\mathbf{X}^{\mathbf{\gamma},\zeta}=\Psi(K^{\mathbf{\gamma},\zeta}).$ (28)

###### Theorem 3

The set $\\{\mathcal{X}^{\zeta}:\zeta\in\Lambda\\}$, where
$\mathcal{X}^{\zeta}=\\{\mathbf{X}^{\mathbf{\gamma},\zeta}:\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}\\}$
is a 2-D ZCACS of size $L_{1}\times L_{2}$, where $L_{1}=2^{m}$ and
$L_{2}=2p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$ and ZCZ size is
$Z_{1}\times Z_{2}$, where $Z_{1}=2^{m}$ and
$Z_{2}=p_{1}^{m_{1}-1}p_{2}^{m_{2}-1}\dots p_{k}^{m_{k}-1}$.

###### Proof:

Appendix C. ∎

##  V Conclusion

In this paper, we have investigated IGC code set and 2-D ZCACS. We have used
multivariable function to construct these sequences and arrays, so that the
length and array sizes could be flexible. We first proposed a class of
multivariables functions to generate the IGC code set of length of the form
$p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$ where $m_{\alpha}\geq 2$ for
all $\alpha$. Using Theorem 1, we have developed the construction of 2-D ZCAC
of array size $L_{1}\times L_{2}$ where $L_{1}=2^{m}$ and
$L_{2}=2p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$. And finally in Theorem
3, we have proposed our main construction of 2-D ZCACS of the same array sizes
using multi-varaible functions. Although our constructions cover a majority of
evern lengths for the array sizes of 2-D ZCACS, the construction for all
remaining lengths including the odd lengths are still an open problem.

##  Appendix A Proof of Theorem 1

We give two lemmas before the proof of Theorem 1.

######  Lemma 3 ([16])

For
$\mathbf{t}_{1},\mathbf{t}_{2}\in\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{2}}\times\dots\times\mathbb{Z}_{p_{k}}$,
we have

$\begin{split}\sum_{\mathbf{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}}C\left(\psi(R_{\mathbf{t}_{1}}^{\mathbf{\gamma}}),\psi(R_{\mathbf{t}_{2}}^{\mathbf{\gamma}})\right)(\tau)=\begin{cases}\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}},&\mathbf{t}_{1}=\mathbf{t}_{2},\tau=0;\\\
0,&\mathbf{t}_{1}=\mathbf{t}_{2},0<\mid\tau\mid<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1};\\\
0,&\mathbf{t}_{1}\neq\mathbf{t}_{2},\mid\tau\mid<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}.\end{cases}\end{split}$
(29)

###### Lemma 4

For
$\mathbf{s}_{1},\mathbf{s}_{2}\in\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{2}}\times\dots\times\mathbb{Z}_{p_{k}}$,
we have

$C\left(\psi(T_{\mathbf{s}_{1}}),\psi(T_{\mathbf{s}_{2}})\right)(0)=\begin{cases}\prod_{\alpha=1}^{k}p_{\alpha},&\mathbf{s}_{1}=\mathbf{s}_{2};\\\
0,&\mathbf{s}_{1}\neq\mathbf{s}_{2}.\end{cases}$ (30)

###### Proof:

To show that
$\\{\mathcal{C}_{\mathbf{s},\mathbf{t}}:\mathbf{s},\mathbf{t}\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}\\}$
is an IGC code set we need to prove that

$\begin{split}C(\mathcal{C}_{\mathbf{s}_{1},\mathbf{t}_{1}},\mathcal{C}_{\mathbf{s}_{2},\mathbf{t}_{2}})(\tau)=\begin{cases}\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}+1},&\mathbf{s}_{1}=\mathbf{s}_{2},\mathbf{t}_{1}=\mathbf{t}_{2},\tau=0;\\\
0,&\mathbf{s}_{1}=\mathbf{s}_{2},\mathbf{t}_{1}=\mathbf{t}_{2},0<\tau<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1};\\\
0,&\mathbf{s}_{1}\neq\mathbf{s}_{2},\mathbf{t}_{1}=\mathbf{t}_{2},\tau<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1};\\\
0,&\mathbf{t}_{1}\neq\mathbf{t}_{2},\tau<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}},\end{cases}\end{split}$
(31)

keeping in mind that
$C(\mathbf{a},\mathbf{b})(-\tau)=C^{*}(\mathbf{b},\mathbf{a})(\tau)$, for a
pair of sequences $(\mathbf{a},\mathbf{b})$ and $\tau\geq 0$.

Now,

$\begin{split}C(\mathcal{C}_{\mathbf{s}_{1},\mathbf{t}_{1}},\mathcal{C}_{\mathbf{s}_{2},\mathbf{t}_{2}})(\tau)=&\sum_{\mathbf{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}}C\left(\psi(a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}),\psi(a_{\mathbf{s}_{2},\mathbf{t}_{2}}^{\mathbf{\gamma}})\right)(\tau)\\\
=&\sum_{\mathbf{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}}C\left(\psi(T_{\mathbf{s}_{1}})\otimes\psi(R_{\mathbf{t}_{1}}^{\mathbf{\gamma}}),\psi(T_{\mathbf{s}_{2}})\otimes\psi(R_{\mathbf{t}_{2}}^{\mathbf{\gamma}})\right)(\tau)\\\
=&C\left(\psi(T_{\mathbf{s}_{1}}),\psi(T_{\mathbf{s}_{2}})\right)\left(\left\lfloor\frac{\tau}{N}\right\rfloor\right)\\\
&\times\sum_{\mathbf{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}}C\left(\psi(R_{\mathbf{t}_{1}}^{\mathbf{\gamma}}),\psi(R_{\mathbf{t}_{2}}^{\mathbf{\gamma}})\right)\left(\tau\mod
N\right)\\\
&+\Delta_{N}C\left(\psi(T_{\mathbf{s}_{1}}),\psi(T_{\mathbf{s}_{2}})\right)\left(\left\lfloor\frac{\tau}{N}\right\rfloor+1\right)\\\
&\times\sum_{\mathbf{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}}C\left(\psi(R_{\mathbf{t}_{1}}^{\mathbf{\gamma}}),\psi(R_{\mathbf{t}_{2}}^{\mathbf{\gamma}})\right)\left(\tau\mod
N-N\right)\end{split}$ (32)

where $N=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}$.

The rest of the proof is split into three cases.

  1. Case I:

Let $\tau=0$. Then $\left\lfloor\frac{\tau}{N}\right\rfloor=0$, $\tau\mod N=0$
and $\Delta_{N}=0$. Using Lemma 3 and Lemma 4, we have

$\begin{split}C(\mathcal{C}_{\mathbf{s}_{1},\mathbf{t}_{1}},\mathcal{C}_{\mathbf{s}_{2},\mathbf{t}_{2}})(0)=\begin{cases}\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}+1},&\mathbf{s}_{1}=\mathbf{s}_{2},\mathbf{t}_{1}=\mathbf{t}_{2};\\\
0,&\mathbf{s}_{1}\neq\mathbf{s}_{2},\mathbf{t}_{1}=\mathbf{t}_{2};\\\
0,&\mathbf{t}_{1}\neq\mathbf{t}_{2}.\end{cases}\end{split}$ (33)

  2. Case II:

Let $0<\tau<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}$. In this case
$\left\lfloor\frac{\tau}{N}\right\rfloor=0$, $0<\tau\mod
N<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}$,
$-\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}<\tau\mod N-N<0$ and
$\Delta_{N}=1$. Using Lemma 3, we have

$\begin{split}\sum_{\mathbf{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}}C\left(\psi(R_{\mathbf{t}_{1}}^{\mathbf{\gamma}}),\psi(R_{\mathbf{t}_{2}}^{\mathbf{\gamma}})\right)\left(\tau\mod
N\right)=0\end{split}$ (34)

and

$\begin{split}\sum_{\mathbf{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}}C\left(\psi(R_{\mathbf{t}_{1}}^{\mathbf{\gamma}}),\psi(R_{\mathbf{t}_{2}}^{\mathbf{\gamma}})\right)\left(\tau\mod
N-N\right)=0\end{split}$ (35)

for all values of $\mathbf{t}_{1}$ and $\mathbf{t}_{2}$. So, from (32), (34)
and (35), we get

$C(\mathcal{C}_{\mathbf{s}_{1},\mathbf{t}_{1}},\mathcal{C}_{\mathbf{s}_{2},\mathbf{t}_{2}})(\tau)=0$
(36)

for all values of $\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1}$ and
$\mathbf{t}_{2}$.

  3. Case III:

Let
$\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}\leq\tau<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}$
and $\mathbf{t}_{1}\neq\mathbf{t}_{2}$. In this case, we have two subcases.

    1. Sub-case 1:

When $\tau=\beta\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}$ for some
$\beta\in\\{1,2,\dots,\prod_{\alpha=1}^{k}p_{\alpha}-1\\}$, then
$\left\lfloor\frac{\tau}{N}\right\rfloor=\beta$, $\tau\mod N=0$ and
$\Delta_{N}=0$. Then, form Lemma 3, we have

$\sum_{\mathbf{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}}C\left(\psi(R_{\mathbf{t}_{1}}^{\mathbf{\gamma}}),\psi(R_{\mathbf{t}_{2}}^{\mathbf{\gamma}})\right)\left(\tau\mod
N\right)=0.$ (37)

So, from (32) and (37), we get

$C(\mathcal{C}_{\mathbf{s}_{1},\mathbf{t}_{1}},\mathcal{C}_{\mathbf{s}_{2},\mathbf{t}_{2}})(\tau)=0.$
(38)

    2. Sub-case 2:

Let
$\beta\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}<\tau<(\beta+1)\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}$
for some $\beta\in\\{1,2,\dots,\prod_{\alpha=1}^{k}p_{\alpha}-1\\}$. Then
$\left\lfloor\frac{\tau}{N}\right\rfloor=\beta$, $0<\tau\mod
N<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}$,
$-\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}<\tau\mod N-N<0$ and
$\Delta_{N}=1$. Then using Lemma 3, we have

$\begin{split}\sum_{\mathbf{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}}C\left(\psi(R_{\mathbf{t}_{1}}^{\mathbf{\gamma}}),\psi(R_{\mathbf{t}_{2}}^{\mathbf{\gamma}})\right)\left(\tau\mod
N\right)=0\end{split}$ (39)

and

$\begin{split}\sum_{\mathbf{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}}C\left(\psi(R_{\mathbf{t}_{1}}^{\mathbf{\gamma}}),\psi(R_{\mathbf{t}_{2}}^{\mathbf{\gamma}})\right)\left(\tau\mod
N-N\right)=0\end{split}$ (40)

So, from (32), (39) and (40), we get

$C(\mathcal{C}_{\mathbf{s}_{1},\mathbf{t}_{1}},\mathcal{C}_{\mathbf{s}_{2},\mathbf{t}_{2}})(\tau)=0.$
(41)

Hence, the thorem is proved. ∎

##  Appendix B Proof of Theorem 2

###### Proof:

We need to show that

$\begin{split}\sum_{\mathbf{\gamma}\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}A(\mathbf{X}^{\gamma})(\tau_{1},\tau_{2})=\begin{cases}2^{m+1}\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}+1},&(\tau_{1},\tau_{2})=(0,0);\\\
0,&0\leq\mid\tau_{1}\mid<2^{m},0\leq\mid\tau_{2}\mid<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1},\\\
&(\tau_{1},\tau_{2})\neq(0,0).\end{cases}\end{split}$ (42)

It can be observed that,
$A(\mathbf{X})(-\tau_{1},\tau_{2})=A(\mathbf{X})^{*}(\tau_{1},\tau_{2})$ and
$A(\mathbf{X})(\tau_{1},-\tau_{2})=A(\mathbf{X})^{*}(\tau_{1},\tau_{2})$. So,
it is sufficient to prove only for $\tau_{1},\tau_{2}\geq 0$.

Let $i^{\prime},j^{\prime}$ be defined by $i^{\prime}=i+\tau_{1}$ and
$j^{\prime}=j+\tau_{2}$. Also, the vector representations of $i,i^{\prime}$ in
$\mathbb{Z}_{2}^{m}$ and $j,j^{\prime}$ in
$\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}^{m_{\alpha}-1}\times\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}\times\mathbb{Z}_{2}$
are defined by $\vec{i},\vec{i^{\prime}}$ and $\vec{j},\vec{j^{\prime}}$,
respectively.

Now, for a fixed $\gamma$, we have

$\begin{split}A(\mathbf{X}^{\gamma})(\tau_{1},\tau_{2})=&\sum_{i=0}^{L_{1}-\tau_{1}-1}\sum_{j=0}^{L_{2}-\tau_{2}-1}\mathbf{X}^{\gamma}_{i,j}{\mathbf{X}^{\gamma}}^{*}_{i+\tau_{1},j+\tau_{2}}\\\
=&\sum_{i=0}^{L_{1}-\tau_{1}-1}\sum_{j=0}^{L_{2}-\tau_{2}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j^{\prime}})}\end{split}$
(43)

where $L_{1}=2^{m}$ and $L_{2}=2p_{1}^{m_{1}}p_{2}^{m_{2}}\dots
p_{k}^{m_{k}}$.

We prove the rest in two cases.

  1. Case I:

Let $(\tau_{1},\tau_{2})=(0,0)$. Then

$\begin{split}A(\mathbf{X}^{\gamma})(0,0)=&\sum_{i=0}^{L_{1}-1}\sum_{j=0}^{L_{2}-1}\mathbf{X}^{\gamma}_{i,j}{\mathbf{X}^{\gamma}}^{*}_{i,j}=\sum_{i=0}^{L_{1}-1}\sum_{j=0}^{L_{2}-1}1=2^{m+1}\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}\end{split}$
(44)

and hence

$\sum_{\mathbf{\gamma}\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}A(\mathbf{X}^{\gamma})(0,0)=2^{m+1}\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}+1}.$
(45)

  2. Case II:

Let $0\leq\tau_{1}<2^{m}$ and
$0<\tau_{2}<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}$. In this case, we
have

$A(\mathbf{X}^{\gamma})(\tau_{1},\tau_{2})=\sum_{i=0}^{L_{1}-\tau_{1}-1}[S_{1}+S_{2}+S_{3}],$
(46)

where

$S_{1}=\sum_{j=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j^{\prime}})},$
(47)
$S_{2}=\sum_{j=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j^{\prime}})},$
(48)

and

$S_{3}=\sum_{j=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}}^{2\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j^{\prime}})}.$
(49)

We let $\vec{j}=(\vec{j}_{res},j_{L})$ and
$\vec{j^{\prime}}=(\vec{j^{\prime}}_{res},j^{\prime}_{L})$, where
$\vec{j}_{res},\vec{j^{\prime}}_{res}$ are the vectors in
$\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}^{m_{\alpha}-1}\times\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}$
which are truncated from $\vec{j}$ and $\vec{j^{\prime}}$, respectively and
$j_{L},j^{\prime}_{L}$ are the last vectors corresponding to $\mathbb{Z}_{2}$.
We define $j_{res},j^{\prime}_{res}$ to be the decimal numbers correspoding to
the vectors $\vec{j}_{res},\vec{j^{\prime}}_{res}$, respectively.

We note that

$\begin{split}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j^{\prime}})}=&\omega_{\lambda}^{(1-j_{L})\\{a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}(\vec{j}_{res})+a(\vec{i})\\}}\omega_{\lambda}^{j_{L}\\{a_{\mathbf{s}_{2},\mathbf{t}_{2}}^{\mathbf{\gamma}}(\vec{j}_{res})+b(\vec{i})\\}}\\\
&~{}~{}\times\omega_{\lambda}^{-(1-j^{\prime}_{L})\\{a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})+a(\vec{i^{\prime}})\\}}\omega_{\lambda}^{-j^{\prime}_{L}\\{a_{\mathbf{s}_{2},\mathbf{t}_{2}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})+b(\vec{i^{\prime}})\\}}.\end{split}$
(50)

Now, when $0\leq j\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1$,
then $\tau_{2}\leq
j^{\prime}\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1$. So, $j_{L}=0$,
$j^{\prime}_{L}=0$ and $0\leq
j_{res}\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1$,
$\tau_{2}\leq
j^{\prime}_{res}\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1$. From (47)
and (50), we have

$\begin{split}S_{1}=&\omega_{\lambda}^{a(\vec{i})-a(\vec{i^{\prime}})}\sum_{j=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}(\vec{j}_{res})-a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})}\\\
=&\omega_{\lambda}^{a(\vec{i})-a(\vec{i^{\prime}})}\sum_{j_{res}=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}(\vec{j}_{res})-a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})}\\\
\end{split}$ (51)

But, from Theorem 1, we have

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}\sum_{j_{res}=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}(\vec{j}_{res})-a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})}=0.$
(52)

So, we have

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}S_{1}=0.$ (53)

Similarly, when $\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}\leq
j\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1$, then
$\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}\leq
j^{\prime}\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}+\tau_{2}-1$. So,
$j_{L}=0$, $j^{\prime}_{L}=1$ and
$\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}\leq
j_{res}\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1$, $0\leq
j^{\prime}_{res}\leq\tau_{2}-1$. From (48) and (50), we have

$\begin{split}S_{2}=&\omega_{\lambda}^{a(\vec{i})-b(\vec{i^{\prime}})}\sum_{j=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}(\vec{j}_{res})-a_{\mathbf{s}_{2},\mathbf{t}_{2}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})}\\\
=&\omega_{\lambda}^{a(\vec{i})-b(\vec{i^{\prime}})}\sum_{j_{res}=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}(\vec{j}_{res})-a_{\mathbf{s}_{2},\mathbf{t}_{2}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})}\\\
\end{split}$ (54)

From Theorem 1, we have

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}\sum_{j_{res}=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1},\mathbf{t}_{1}}^{\mathbf{\gamma}}(\vec{j}_{res})-a_{\mathbf{s}_{2},\mathbf{t}_{2}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})}=0.$
(55)

So, we have

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}S_{2}=0.$ (56)

Arguing in a similar manner, we get

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}S_{2}=0.$ (57)

Using (43), (46), (53), (56) and (57), we can say

$\sum_{\mathbf{\gamma}\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}A(\mathbf{X}^{\gamma})(\tau_{1},\tau_{2})=0.$
(58)

  3. Case III:

Let $0<\tau_{1}<2^{m}$ and $\tau_{2}=0$. In this case, $j=j^{\prime}$,
$j_{res}=j^{\prime}_{res}$ and $j_{L}=j^{\prime}_{L}$. So, from (50), we get

$\begin{split}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j^{\prime}})}=\omega_{\lambda}^{(1-j_{L})\left(a(\vec{i})-a(\vec{i^{\prime}})\right)}\omega_{\lambda}^{j_{L}\left(b(\vec{i})-b(\vec{i^{\prime}})\right)}\end{split}$
(59)

Now, we have

$A(\mathbf{X}^{\gamma})(\tau_{1},\tau_{2})=\sum_{i=0}^{L_{1}-\tau_{1}-1}[P_{1}+P_{2}],$
(60)

where

$P_{1}=\sum_{j=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j})},$
(61)

and

$P_{2}=\sum_{j=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}}^{2\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\mathbf{s}_{1},\mathbf{s}_{2},\mathbf{t}_{1},\mathbf{t}_{2}}(\vec{j})}.$
(62)

Now, when $0\leq j\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1$, then
$j_{L}=0$. So, from (59) and (61), we get

$\begin{split}P_{1}=&\sum_{j=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1}\omega_{\lambda}^{a(\vec{i})-a(\vec{i^{\prime}})}\\\
=&\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}\times\omega_{\lambda}^{a(\vec{i})-a(\vec{i^{\prime}})}\end{split}$
(63)

Similarly, we get

$\begin{split}P_{2}=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}\times\omega_{\lambda}^{b(\vec{i})-b(\vec{i^{\prime}})}\end{split}$
(64)

so that

$P_{1}+P_{2}=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}\left[\omega_{\lambda}^{a(\vec{i})-a(\vec{i^{\prime}})}+\omega_{\lambda}^{b(\vec{i})-b(\vec{i^{\prime}})}\right]$
(65)

But, from Lemma 2, we have

$\sum_{i=0}^{L_{1}-\tau_{1}-1}\left[\omega_{\lambda}^{a(\vec{i})-a(\vec{i^{\prime}})}+\omega_{\lambda}^{b(\vec{i})-b(\vec{i^{\prime}})}\right]=0.$
(66)

So, from (43), (60), (65) and (66), we have

$A(\mathbf{X}^{\gamma})(\tau_{1},\tau_{2})=0.$ (67)

Hence, the thorem is proved.

∎

##  Appendix C Proof of Theorem 3

###### Proof:

It follows from Theorem 2, that the set $\mathcal{X}^{\zeta}$ is a 2-D ZCAC
for any $\zeta\in\Lambda$. So, it is sufficient to prove that

$\sum_{\gamma\in\prod_{\alpha}^{k}\mathbb{Z}_{p_{\alpha}}}C\left(\mathbf{X}^{\mathbf{\gamma},\zeta_{1}},\mathbf{X}^{\mathbf{\gamma},\zeta_{2}}\right)(\tau_{1},\tau_{2})=0$
(68)

when $\zeta_{1}\neq\zeta_{2}$, $\mid\tau_{1}\mid<2^{m}$ and
$\mid\tau_{2}\mid<p_{1}^{m_{1}-1}p_{2}^{m_{2}-1}\dots p_{k}^{m_{k}-1}$.

we shall prove only for $\tau_{1}\geq 0$ as
$C(\mathbf{X}_{1},\mathbf{X}_{2})(-\tau_{1},\tau_{2})=C(\mathbf{X}_{2},\mathbf{X}_{1})^{*}(\tau_{1},-\tau_{2})$.

We split the proof in two cases.

  1. Case I:

Let $\tau_{1}\geq 0$ and
$0\leq\tau_{2}<\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}$. For a fixed
$\gamma$, we have

$\begin{split}C\left(\mathbf{X}^{\mathbf{\gamma},\zeta_{1}},\mathbf{X}^{\mathbf{\gamma},\zeta_{2}}\right)(\tau_{1},\tau_{2})=&\sum_{i=0}^{L_{1}-\tau_{1}-1}\sum_{j=0}^{L_{2}-\tau_{2}-1}\mathbf{X}^{\mathbf{\gamma},\zeta_{1}}_{i,j}{\mathbf{X}^{\mathbf{\gamma},\zeta_{2}}}^{*}_{i+\tau_{1},j+\tau_{2}}\\\
=&\sum_{i=0}^{L_{1}-\tau_{1}-1}\sum_{j=0}^{L_{2}-\tau_{2}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\zeta_{1}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\zeta_{2}}(\vec{j^{\prime}})}\end{split}$
(69)

where $L_{1}=2^{m}$ and $L_{2}=2p_{1}^{m_{1}}p_{2}^{m_{2}}\dots
p_{k}^{m_{k}}$.

Now, in a manner similar to the proof of Thorem 2, we have

$\begin{split}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\zeta_{1}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\zeta_{2}}(\vec{j^{\prime}})}=&\omega_{\lambda}^{(1-j_{L})\\{a_{\mathbf{s}_{1}^{1},\mathbf{t}_{1}^{1}}^{\mathbf{\gamma}}(\vec{j}_{res})+a(\vec{i})\\}}\omega_{\lambda}^{j_{L}\\{a_{\mathbf{s}_{2}^{1},\mathbf{t}_{2}^{1}}^{\mathbf{\gamma}}(\vec{j}_{res})+b(\vec{i})\\}}\\\
&\times\omega_{\lambda}^{-(1-j^{\prime}_{L})\\{a_{\mathbf{s}_{1}^{2},\mathbf{t}_{1}^{2}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})+a(\vec{i^{\prime}})\\}}\omega_{\lambda}^{-j^{\prime}_{L}\\{a_{\mathbf{s}_{2}^{2},\mathbf{t}_{2}^{2}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})+b(\vec{i^{\prime}})\\}}.\end{split}$
(70)

and

$C\left(\mathbf{X}^{\mathbf{\gamma},\zeta_{1}},\mathbf{X}^{\mathbf{\gamma},\zeta_{2}}\right)(\tau_{1},\tau_{2})=\sum_{i=0}^{L_{1}-\tau_{1}-1}[S_{1}+S_{2}+S_{3}],$
(71)

where

$S_{1}=\sum_{j=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\zeta_{1}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\zeta_{2}}(\vec{j^{\prime}})},$
(72)
$S_{2}=\sum_{j=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\zeta_{1}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\zeta_{2}}(\vec{j^{\prime}})},$
(73)

and

$S_{3}=\sum_{j=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}}^{2\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\zeta_{1}}(\vec{j})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\zeta_{2}}(\vec{j^{\prime}})}.$
(74)

When $0\leq j\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1$, then
$\tau_{2}\leq j^{\prime}\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1$.
So, $j_{L}=0$, $j^{\prime}_{L}=0$ and $0\leq
j_{res}\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1$,
$\tau_{2}\leq
j^{\prime}_{res}\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1$. From (70)
and (72), we have

$\begin{split}S_{1}=&\omega_{\lambda}^{a(\vec{i})-a(\vec{i^{\prime}})}\sum_{j=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1}^{1},\mathbf{t}_{1}^{1}}^{\mathbf{\gamma}}(\vec{j}_{res})-a_{\mathbf{s}_{1}^{2},\mathbf{t}_{1}^{2}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})}\\\
=&\omega_{\lambda}^{a(\vec{i})-a(\vec{i^{\prime}})}\sum_{j_{res}=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1}^{1},\mathbf{t}_{1}^{1}}^{\mathbf{\gamma}}(\vec{j}_{res})-a_{\mathbf{s}_{1}^{2},\mathbf{t}_{1}^{2}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})}\\\
\end{split}$ (75)

But, from Theorem 1, we have

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}\sum_{j_{res}=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-\tau_{2}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1}^{1},\mathbf{t}_{1}^{1}}^{\mathbf{\gamma}}(\vec{j}_{res})-a_{\mathbf{s}_{1}^{2},\mathbf{t}_{1}^{2}}^{\mathbf{\gamma}}(\vec{j^{\prime}}_{res})}=0.$
(76)

So, we have

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}S_{1}=0.$ (77)

Arguing in a similar manner we can get

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}S_{2}=0,$ (78)

and

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}S_{3}=0.$ (79)

So, from (71), (77), (78) and (79), we have

$\sum_{\gamma\in\prod_{\alpha}^{k}\mathbb{Z}_{p_{\alpha}}}C\left(\mathbf{X}^{\mathbf{\gamma},\zeta_{1}},\mathbf{X}^{\mathbf{\gamma},\zeta_{2}}\right)(\tau_{1},\tau_{2})=0.$
(80)

  2. Case II:

Let $\tau_{1}\geq 0$ and
$-\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}-1}<\tau_{2}<0$. Then for a fixed
$\gamma$, we have

$\begin{split}C\left(\mathbf{X}^{\mathbf{\gamma},\zeta_{1}},\mathbf{X}^{\mathbf{\gamma},\zeta_{2}}\right)(\tau_{1},\tau_{2})=\sum_{i=0}^{L_{1}-\tau_{1}-1}\sum_{j=0}^{L_{2}+\tau_{2}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\zeta_{1}}(\vec{j^{\prime\prime}})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\zeta_{2}}(\vec{j})}\end{split}$
(81)

where $L_{1}=2^{m}$, $L_{2}=2p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{k}^{m_{k}}$
and we have defined $\vec{j^{\prime\prime}}$ to be the vector representation
of $j^{\prime\prime}=j-\tau_{2}$.

We let
$\vec{j^{\prime\prime}}=(\vec{j^{\prime\prime}}_{res},j^{\prime\prime}_{L})$,
where $\vec{j^{\prime\prime}}_{res}$ is the vector in
$\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}^{m_{\alpha}-1}\times\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}$
which is truncated from $\vec{j^{\prime\prime}}$ and $j^{\prime\prime}_{L}$ is
the last vector corresponding to $\mathbb{Z}_{2}$.

Now, we have

$\begin{split}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\zeta_{1}}(\vec{j^{\prime\prime}})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\zeta_{2}}(\vec{j})}=&\omega_{\lambda}^{(1-j^{\prime\prime}_{L})\\{a_{\mathbf{s}_{1}^{1},\mathbf{t}_{1}^{1}}^{\mathbf{\gamma}}(\vec{j^{\prime\prime}}_{res})+a(\vec{i})\\}}\omega_{\lambda}^{j^{\prime\prime}_{L}\\{a_{\mathbf{s}_{2}^{1},\mathbf{t}_{2}^{1}}^{\mathbf{\gamma}}(\vec{j^{\prime\prime}}_{res})+b(\vec{i})\\}}\\\
&\times\omega_{\lambda}^{-(1-j_{L})\\{a_{\mathbf{s}_{1}^{2},\mathbf{t}_{1}^{2}}^{\mathbf{\gamma}}(\vec{j}_{res})+a(\vec{i^{\prime}})\\}}\omega_{\lambda}^{-j_{L}\\{a_{\mathbf{s}_{2}^{2},\mathbf{t}_{2}^{2}}^{\mathbf{\gamma}}(\vec{j}_{res})+b(\vec{i^{\prime}})\\}}.\end{split}$
(82)

Similar to Case I, we now have

$C\left(\mathbf{X}^{\mathbf{\gamma},\zeta_{1}},\mathbf{X}^{\mathbf{\gamma},\zeta_{2}}\right)(\tau_{1},\tau_{2})=\sum_{i=0}^{L_{1}-\tau_{1}-1}[P_{1}+P_{2}+P_{3}],$
(83)

where

$P_{1}=\sum_{j=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}+\tau_{2}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\zeta_{1}}(\vec{j^{\prime\prime}})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\zeta_{2}}(\vec{j})},$
(84)
$P_{2}=\sum_{j=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}+\tau_{2}}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\zeta_{1}}(\vec{j^{\prime\prime}})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\zeta_{2}}(\vec{j})},$
(85)

and

$P_{3}=\sum_{j=\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}}^{2\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}+\tau_{2}-1}\omega_{\lambda}^{F_{\vec{i}}^{\mathbf{\gamma},\zeta_{1}}(\vec{j^{\prime\prime}})}\omega_{\lambda}^{-F_{\vec{i^{\prime}}}^{\mathbf{\gamma},\zeta_{2}}(\vec{j})}.$
(86)

When $0\leq j\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}+\tau_{2}-1$, then
$-\tau_{2}\leq
j^{\prime\prime}\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1$. So,
$j_{L}=0$, $j^{\prime\prime}_{L}=0$ and $0\leq
j_{res}\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}+\tau_{2}-1$,
$-\tau_{2}\leq
j^{\prime\prime}_{res}\leq\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}-1$. From
(82) and (84), we have

$\begin{split}P_{1}=&\omega_{\lambda}^{a(\vec{i})-a(\vec{i^{\prime}})}\sum_{j=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}+\tau_{2}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1}^{1},\mathbf{t}_{1}^{1}}^{\mathbf{\gamma}}(\vec{j^{\prime\prime}}_{res})-a_{\mathbf{s}_{1}^{2},\mathbf{t}_{1}^{2}}^{\mathbf{\gamma}}(\vec{j}_{res})}\\\
=&\omega_{\lambda}^{a(\vec{i})-a(\vec{i^{\prime}})}\sum_{j_{res}=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}+\tau_{2}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1}^{1},\mathbf{t}_{1}^{1}}^{\mathbf{\gamma}}(\vec{j^{\prime\prime}}_{res})-a_{\mathbf{s}_{1}^{2},\mathbf{t}_{1}^{2}}^{\mathbf{\gamma}}(\vec{j}_{res})}\\\
\end{split}$ (87)

From Theorem 1, we have

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}\sum_{j_{res}=0}^{\prod_{\alpha=1}^{k}p_{\alpha}^{m_{\alpha}}+\tau_{2}-1}\omega_{\lambda}^{a_{\mathbf{s}_{1}^{1},\mathbf{t}_{1}^{1}}^{\mathbf{\gamma}}(\vec{j^{\prime\prime}}_{res})-a_{\mathbf{s}_{1}^{2},\mathbf{t}_{1}^{2}}^{\mathbf{\gamma}}(\vec{j}_{res})}=0.$
(88)

So, we have

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}P_{1}=0.$ (89)

Similarly, we can get

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}P_{2}=0,$ (90)

and

$\sum_{\gamma\in\prod_{\alpha=1}^{k}\mathbb{Z}_{p_{\alpha}}}P_{3}=0.$ (91)

Therefore, from (83), (89), (90) and (91), we have

$\sum_{\gamma\in\prod_{\alpha}^{k}\mathbb{Z}_{p_{\alpha}}}C\left(\mathbf{X}^{\mathbf{\gamma},\zeta_{1}},\mathbf{X}^{\mathbf{\gamma},\zeta_{2}}\right)(\tau_{1},\tau_{2})=0.$
(92)

This proves the theorem. ∎

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# Spectral Convolutional Conditional Neural Processes

Peiman Mohseni & Nick Duffield
Texas A&M University
{peiman.mohseni<EMAIL_ADDRESS>
###### Abstract

Conditional Neural Processes (CNPs) constitute a family of probabilistic
models that harness the flexibility of neural networks to parameterize
stochastic processes. Their capability to furnish well-calibrated predictions,
combined with simple maximum-likelihood training, has established them as
appealing solutions for addressing various learning problems, with a
particular emphasis on meta-learning. A prominent member of this family,
Convolutional Conditional Neural Processes (ConvCNPs), utilizes convolution to
explicitly introduce translation equivariance as an inductive bias. However,
ConvCNP’s reliance on local discrete kernels in its convolution layers can
pose challenges in capturing long-range dependencies and complex patterns
within the data, especially when dealing with limited and irregularly sampled
observations from a new task. Building on the successes of Fourier neural
operators (FNOs) for approximating the solution operators of parametric
partial differential equations (PDEs), we propose Spectral Convolutional
Conditional Neural Processes (SConvCNPs), a new addition to the NPs family
that allows for more efficient representation of functions in the frequency
domain.

##  1 Introduction

Stochastic processes are fundamental mathematical concepts extensively
employed across diverse disciplines to characterize the inherent randomness
observed in numerous natural phenomena. At its core, a stochastic process is
defined as a typically infinite collection of random variables, indexed by a
specific set. Particularly noteworthy is when these random variables pertain
to elements within a function space, as exemplified by entities like Gaussian
Processes (GPs). These processes find utility in representing real-valued
signals that are indexed by either time or space, leading to a growing
interest in the development of machine learning algorithms designed to handle
data generated by such processes (Mathieu et al., 2021; Dupont et al., 2021).

Unfortunately, the conventional construction of stochastic processes like GPs,
used in regression and classification tasks, can quickly become
computationally challenging as the size of the dataset or dimensionality
expands Garnelo et al. (2018a). The remarkable success of deep neural networks
in approximating functions, especially in the presence of large datasets, has
sparked a burgeoning line of research focused on using neural networks to
directly parameterize stochastic processes. Neural Processes (NPs, Garnelo et
al. (2018a; b)) are prominent outcomes of this perspective and have found
successful applications in the realm of meta-learning (Jha et al., 2022). The
use of neural networks is appealing since the majority of the computational
burden is concentrated during the training process Dutordoir et al. (2023).
The training regime of NPs which revolves around training over a large number
of small datasets, in conjunction with their formulation, naturally fosters an
exchange of information between different datasets while providing on-the-fly
uncertainty estimates (Gordon et al., 2019).

Since their initial introduction, there have been various follow-up works
aimed at addressing the limitations of vanilla CNPs, including issues such as
underfitting (Kim et al., 2019; Nguyen & Grover, 2022; Kim et al., 2022),
uncorrelated predictions (Louizos et al., 2019; Foong et al., 2020; Bruinsma
et al., 2021; Markou et al., 2022; Bruinsma et al., 2023), and the use of a
simple Gaussian predictive distribution (mohseni23a). In this article, we
focus specifically on Convolutional Conditional Neural Processes (ConvCNPs,
Gordon et al. (2019)), which represent a variant designed to handle symmetries
such as translation equivariance by integrating convolutional neural networks
(CNNs). Although ConvCNPs exhibit impressive performance, they may encounter
challenges when it comes to effectively integrating partial information
scattered irregularly across the domain to infer the underlying pattern. This
arises from the commonly used local discrete kernels, which often possess an
effective memory horizon considerably shorter than the length of the input
sequence (Romero et al., 2021). To circumvent this issue, we take a different
approach based on the developments of neural network based surrogates for
solving partial differential equations (PDEs, Raissi et al. (2019); Lu et al.
(2019); Kovachki et al. (2023)). In particular, we leverage the formulation of
Fourier Neural Operators (FNOs, Li et al. (2020a)) to perform global
convolution. The resulting model, which we refer to as Spectral Convolution
Conditional Neural Process (SConvCNP), is demonstrated to outperform baselines
on challenging regression problems111Code is available at
https://github.com/peiman-m/SpecConvCNP.

##  2 Preliminaries

###  2.1 Fourier Neural Operators

Neural operators (Li et al., 2020b; a; Gupta et al., 2021; Tran et al., 2021;
Kovachki et al., 2023; Liu et al., 2023; Helwig et al., 2023) are a class of
models that leverage neural networks to construct a parametric approximation
of mappings between Banach spaces. Much like the conventional design of a
standard neural network layer, which relies on an affine transformation of the
input followed by a pointwise nonlinearity, neural operators apply the linear
integral operator $\mathcal{K}(v)(t)=\int\kappa(x,t)v(x)dx$ with kernel
$\kappa$ on the input function $v\in C_{b}({\mathbb{X}},{\mathbb{Y}})$, which
is subsequently subjected to a nonlinear activation function (Kovachki et al.,
2023). When $\kappa$ is stationary, $\mathcal{K}(v)$ simplifies to $\kappa\ast
v$, the convolution of $\kappa$ and $v$. By the convolution theorem,
$\mathcal{F}(\kappa\ast v)$, i.e. the Fourier transform of $\kappa\ast v$,
coincides with the pointwise product of $\mathcal{F}(\kappa)$ and
$\mathcal{F}(v)$. A particularly interesting scenario arises when $\kappa$ is
assumed to be periodic, leading to a Fourier transform that adopts the shape
of a Dirac comb with discrete support. As suggested by Li et al. (2020a), both
$\mathcal{F}(\kappa)$ and $\mathcal{F}(v)$ are truncated at $k$ Fourier modes,
thus allowing for direct parameterization of the kernel in the frequency
domain with a finite number of complex-valued weights $W_{\kappa}$ leading to
the following formulation of the integral operator

$\mathcal{K}(v)=\mathcal{F}^{-1}\left(W_{k}\,.\,\mathcal{F}\left(v\right)\right).$
(1)

Augmented by the element-wise affine transformation $W_{\alpha}v+b_{\alpha}$
of $v$ in the physical space, each operator layer is expressed as

$\mathcal{L}(v)=\sigma(W_{\alpha}v+b_{\alpha}+\mathcal{K}(v)),$ (2)

where $\sigma$ is a pointwise non-linear function. In practice, when we have
access to a finite set of samples of $v$, $\mathcal{F}(v)$ (and similarly its
transform) is computed using the Fast Fourier Transform (FFT). Through the
composition of operator layers as depicted in Equation (3), FNOs can
effectively approximate highly intricate operators.

$\mathcal{G}(v)=(\mathcal{Q}\circ\mathcal{L}^{(L)}\circ\cdots\circ\mathcal{L}^{(1)}\circ\mathcal{P})(v).$
(3)

In this equation, $L$ represents the number of layers, and $\mathcal{P}$ and
$\mathcal{Q}$ correspond to element-wise mappings to and from a latent space.

##  3 Method

###  3.1 Convolutional Conditional Neural Processes

Conditional Neural Processes (CNPs, Garnelo et al. (2018a; b)) are meta-
learning methods that use neural networks to parameterize the predictive
distribution $p(y\,|\,x,\mathcal{D})$ where $(x,y)$ and
$\mathcal{D}=\\{(x_{i},y_{i})\in{\mathbb{X}}\times{\mathbb{Y}}\\}_{i=1}^{N}$
correspond to a target input-output pair and a set of observations sampled
from a task $\mathcal{T}$, respectively. In CNPs, first, the set of
observations $\mathcal{D}$, also known as the context set, are embedded into a
latent space ${\mathbb{H}}$ through an encoder
$\varphi_{e}\colon\cup_{k=1}^{\infty}({\mathbb{X}}\times{\mathbb{Y}})^{k}\rightarrow{\mathbb{H}}$.
Next, a decoder
$\varphi_{d}\colon{\mathbb{X}}\times{\mathbb{H}}\rightarrow\Theta$ maps target
input $x$ and the context embedding to the parameters $\theta\in\Theta$ of the
conditional distribution $p_{\theta}(y\,|\,x,\mathcal{D})$. Unlike the
majority of CNP variants where the latent space ${\mathbb{H}}$ is chosen to be
a finite-dimensional Euclidean space, ConvCNP maps the context points to an
infinite-dimensional space of functions operating on ${\mathbb{X}}$.
Consequently, the translation of inputs can be reflected in the context
embedding meaningfully. More precisely, ConvCNP employs a kernel density
estimator along with the Nadaraya-Watson estimator to form the functional
representations
$E(\mathcal{D})\colon\cup_{k=1}^{\infty}({\mathbb{X}}\times{\mathbb{Y}})^{k}\rightarrow{\mathbb{H}}$
of the context set as follows

$E(\mathcal{D})(t)=\left[\sum_{i=1}^{N}\psi_{e}(t,x_{i})\,,\,\sum_{i=1}^{N}\frac{\psi_{e}(t,x_{i})\,y_{i}}{\psi_{e}(t,x_{i})}\right],$
(4)

where $\psi_{e}$ is a kernel function usually taken as the Gaussian kernel
with a learnable length scale. This functional representation is then
transformed through a translation-equivariant operator
$\Phi\colon{\mathbb{H}}\rightarrow C_{b}({\mathbb{X}},{\mathbb{Y}})$ to form
the final context representation as

$\varphi_{e}(\mathcal{D})=\Phi\left(E\left(\mathcal{D}\right)\right)$ (5)

In practice, the operator $\Phi$ is implemented using a convolutional neural
network (CNN) since any deep translation-equivariant model can be represented
equivalently using CNNs (Kondor & Trivedi, 2018). Before applying the CNN, the
range, which encompasses both the context and target inputs, is discretized
uniformly at $(t_{j})_{j=1}^{M}\subset{\mathbb{X}}$ with a predetermined
density. This step, however, can be bypassed if the data is already organized
on a grid that can serve as our discretization. In the end, the decoder
$\varphi_{d}$ uses the discretized context representation
$(\varphi_{e}(\mathcal{D})(t_{j}))_{j=1}^{M}$ to regress the parameters of
$p_{\theta}(y\,|\,x,\mathcal{D})$ as follows

$\begin{split}\theta&=\varphi_{d}\left(x\,,\,\left(\varphi_{e}\left(\mathcal{D}\right)\left(t_{j}\right)\right)_{j=1}^{M}\right)\\\
&=\sum_{j=1}^{M}\varphi_{e}(\mathcal{D})(t_{j})\,\psi_{d}(x,\,t_{j}),\end{split}$
(6)

where $\psi_{d}$ is a kernel function. Additional constraints, such as
ensuring the positivity of the scale parameter of the distribution, can be
enforced by employing a suitable activation function to the output of the
decoder. Much like the preceding step, this process can be streamlined for on-
the-grid data by integrating it into the CNN (Gordon et al., 2019).

ConvCNP
---
SConvCNP

Figure 1:  Illustrative functions acquired by ConvCNP (upper row) and SConvCNP
(lower row) during training on a periodic kernel (first column), sawtooth
function (second column), and square wave function (third column). The plots
display means and two standard deviations.

###  3.2 Spectral Convolutional Conditional Neural Processes

By a substantial margin, most convolutional kernels used in CNNs are designed
as finite sequences of independent weights. The CNNs’ effective memory
horizon, which is determined by the kernel size, is generally much smaller
than the input length and must be specified beforehand (Romero et al., 2021).
As a result, the capability to capture dependencies that extend beyond the
kernel’s effective range is hindered. This limitation becomes even more
pronounced when dealing with irregularly sampled or partially observed data, a
situation frequently encountered in contexts where NPs are utilized. It’s
important to note that while ConvCNP assesses the functional representation as
described in Equation (4) over a uniform grid to facilitate the use of
discrete convolution kernels, the predictions derived from the kernel
regression are primarily reliable in areas with a higher density of data
points. Consequently, relying solely on local convolution might be
insufficient for capturing the underlying patterns within the data by
stitching together information from various regions. This problem, which is
ubiquitous in meta-learning scenarios where limited observations from a new
task are available, can be alleviated by performing global convolution.
Building upon this, we suggest substituting the CNNs employed to implement the
operator $\Phi$ in Equation (5) with FNOs.

##  4 Experiments

We perform experiments on synthetic one-dimensional regression tasks to
evaluate the efficacy of our proposed framework. The assessment involves
comparing SConvCNP against three baseline models: Vanilla Conditional Neural
Process (CNP), Attentive Conditional Neural Process (AttCNP, Kim et al.
(2019)), and Convolutional Conditional Neural Process (ConvCNP). For the
baseline models, we utilize the open-source implementation provided by
Bruinsma et al. (2023), which is available at
222https://github.com/wesselb/neuralprocesses. Default parameters, extensively
tested and optimized across diverse tasks, are applied. Our implementation of
SConvCNP closely adheres to the ConvCNP framework outlined in Bruinsma et al.
(2023), with the primary distinction lying in the parameterization of the
convolutional network, further detailed in Appendix A.2.

The summarized predictive log-likelihood of these methods on testing data is
presented in Table 1. It is evident that SConvCNP either matches or exceeds
the performance of the baseline models. Notably, the performance improvement
is more pronounced in scenarios where the underlying function is periodic.
This suggests that the use of global convolution may offer a more robust
representation of underlying patterns by considering information collectively,
as also demonstrated in the examples provided in Figure 1 where SConvCNP
produces superior fits. Additional details on experimental setups can be found
in Appendix A.1.

Table 1: Comparison of predictive log-likelihood obtained by different methods
over synthetically generated tasks ($5$ Seeds).

| RBF | Matérn 5/2 | Periodic | Sawtooth | Square Wave
---|---|---|---|---|---
CNP | ${-0.010_{\pm 0.015}}$ | ${-0.286_{\pm 0.010}}$ | ${-0.935_{\pm 0.000}}$ | ${-0.549_{\pm 0.000}}$ | ${-1.112_{\pm 0.000}}$
AttCNP | ${0.171_{\pm 0.124}}$ | ${-0.081_{\pm 0.081}}$ | ${-0.947_{\pm 0.000}}$ | ${-0.550_{\pm 0.000}}$ | ${-0.624_{\pm 0.065}}$
ConvCNP | $1.191_{\pm 0.069}$ | $\bm{0.675}_{\pm 0.004}$ | $0.883_{\pm 0.022}$ | $1.725_{\pm 0.012}$ | $0.928_{\pm 0.069}$
SConvCNP | $\bm{1.197}_{\pm 0.017}$ | $0.635_{\pm 0.017}$ | $\bm{1.349}_{\pm 0.018}$ | $\bm{1.893}_{\pm 0.007}$ | $\bm{1.446}_{\pm 0.093}$

##  5 Conclusion

In this work, we introduced Spectral Convolutional Conditional Neural
Processes (SConvCNPs), a new addition to the CNP family that harnesses
advancements in operator learning to enhance the expressive capabilities of
Convolutional Conditional Neural Processes (ConvCNPs) when modeling stationary
stochastic processes. Our experiments, conducted on synthetic datasets,
demonstrated that SConvCNPs enhance the predictive performance of ConvCNPs in
regression tasks, as evidenced by improvements in log-likelihood. Furthermore,
they adeptly capture global symmetries, including the prevalent periodic
patterns in the data.

#### Acknowledgments

We would like to thank Arman Hasanzadeh and Jonathan W. Siegel for their
valuable feedback and constructive suggestions. Additionally, we appreciate
the support from Texas A&M High Performance Research Computing, which provided
the computational resources necessary to conduct the experiments in this
study.

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##  Appendix A Experimental Details

###  A.1 Data Generation

Table 2: Synthetic processes used in 1D regression experiments. Here
$\left\lfloor\cdot\right\rfloor$ indicates the floor function, and $\llbracket
P\rrbracket$ takes on a value of 1 if the statement $P$ is true, and 0
otherwise.

Process | $\hskip 110.0ptg(\cdot)$
---|---
GP (RBF) | $\qquad g\sim\mathcal{GP}(0,C)\,,\,C(x,x^{\prime})=\sigma^{2}\exp{(-\frac{{\lVert x-x^{\prime}\rVert}_{2}^{2}}{2\mathcal{\ell}})}+\delta$
GP (Matérn 5/2) | $\qquad g\sim\mathcal{GP}(0,C)\,,\,C(x,x^{\prime})=\sigma^{2}(1+\frac{\sqrt{5}d}{\ell}+\frac{5d^{2}}{3\ell^{2}})+\delta,\,d={\lVert x-x^{\prime}\rVert}_{2}$
GP (Periodic) | $\qquad g\sim\mathcal{GP}(0,C)\,,\,C(x,x^{\prime})=\sigma^{2}\exp{(-\frac{2\sin^{2}\left(\pi{\lVert x-x^{\prime}\rVert}_{1}/p\right)}{\mathcal{\ell}^{2}})}+\delta$
Sawtooth (Gordon et al., 2019) | $\qquad g(x)=\frac{\alpha}{2}-\frac{\alpha}{\pi}\sum_{k=1}^{K}(-1)^{k}\frac{\sin{(2\pi k\omega(x+\delta)})}{k}$
Square Wave (Bonito et al., 2023) | $\qquad g(x)=\llbracket\left[\left\lfloor\omega x-\phi\right\rfloor\right]\,\mathrm{mod}\,\,\,2=0\rrbracket$

We consider data arising from the three stochastic processes described in
table 2 with the following choice of parameters:

  * •

RBF: $\ell=0.25$, $\sigma=0.75$, $\delta=0.02$

  * •

Matérn 5/2: $\ell=0.25$, $\sigma=0.75$, $\delta=0.02$

  * •

Periodic: $\ell=0.6$, $\sigma=0.75$, $p=1$, $\delta=0.02$

  * •

Sawtooth: $\alpha\sim\mathcal{U}(1,2)$, $\omega\sim(3,5)$,
$\delta\sim\mathcal{U}(-5,5)$, $K\sim\mathcal{U}(10,20)$

  * •

Square Wave: $\alpha\sim\mathcal{U}(1,2)$, $\omega\sim(2,5)$,
$\phi\sim\mathcal{U}(\frac{1}{\omega},1)$

For every functional form, the models undergo training and testing across
$10^{5}$ and $5\times 10^{4}$ tasks. Each task involves samples from 16
functions drawn from a stochastic process. For each function, we restrict the
inputs domain to $(-2,\,2)$. In both the training and testing phases, the
count of context points for each task is randomly chosen as any integer
between 3 and 50. This selection is mirrored for the target points.

###  A.2 Model Architectures

Consider a Fourier layer denoted by
$\mathrm{F}(d_{\mathrm{in}},\,d_{\mathrm{out}},\,l_{\mathrm{out}},\,m_{\mathrm{f}})$
where $d_{\mathrm{in}}$, $d_{\mathrm{out}}$, $l_{\mathrm{out}}$ and
$m_{\mathrm{f}}$ represent the input channels, output channels, output length,
and the number of kept Fourier modes, respectively. Additionally, let
$\mathrm{MLP}(d_{\mathrm{in}},\,d_{\mathrm{h_{1}}},\,\cdots,\,d_{\mathrm{h_{k}}},\,d_{\mathrm{out}})$
represent a fully connected network with Gaussian error linear unit (GeLU)
activation (Hendrycks & Gimpel, 2016). This network has $d_{\mathrm{in}}$
input channels, $d_{\mathrm{out}}$ output channels, and comprises $k$ hidden
layers with sizes $d_{\mathrm{h_{1}}},\,\cdots,\,d_{\mathrm{h_{k}}}$. In
contrast to the UNet architecture (Ronneberger et al., 2015) utilized in
ConvCNP, we incorporate the following layers:

  * •

$\mathrm{L_{0}}=\mathrm{MLP}(d_{\mathrm{in}}=16,\,d_{\mathrm{h_{1}}}=32,\,d_{\mathrm{out}}=64)$

  * •

$\mathrm{L_{1}}=\mathrm{F}(d_{\mathrm{in}}=64,\,d_{\mathrm{out}}=64,\,l_{\mathrm{out}}=\left\lfloor
l_{0}/2\right\rfloor,\,m_{\mathrm{f}}=32)$

  * •

$\mathrm{L_{2}}=\mathrm{F}(d_{\mathrm{in}}=64,\,d_{\mathrm{out}}=128,\,l_{\mathrm{out}}=\left\lfloor
l_{0}/4\right\rfloor,\,m_{\mathrm{f}}=32)$

  * •

$\mathrm{L_{3}}=\mathrm{F}(d_{\mathrm{in}}=128,\,d_{\mathrm{out}}=256,\,l_{\mathrm{out}}=\left\lfloor
l_{0}/4\right\rfloor,\,m_{\mathrm{f}}=32)$

  * •

$\mathrm{L_{4}}=\mathrm{F}(d_{\mathrm{in}}=256,\,d_{\mathrm{out}}=128,\,l_{\mathrm{out}}=\left\lfloor
l_{0}/2\right\rfloor,\,m_{\mathrm{f}}=32)$

  * •

$\mathrm{L_{5}}=\mathrm{F}(d_{\mathrm{in}}=192,\,d_{\mathrm{out}}=64,\,l_{\mathrm{out}}=l_{0},\,m_{\mathrm{f}}=32)$

  * •

$\mathrm{L_{6}}=\mathrm{MLP}(d_{\mathrm{in}}=128,\,d_{\mathrm{h_{1}}}=32,\,d_{\mathrm{out}}=16)$

where $l_{0}$ is the length of $\mathrm{L_{0}}$’s input. The $\mathrm{MLP}$
layers serve the purpose of elevating the input channels to a higher dimension
and projecting them back at the end of the process. During our experiments, we
observed that introducing skip layers between coupled Fourier layers enhances
performance, akin to the approach in UNOs (Rahman et al., 2022). The following
outlines the concatenation of layers, denoted as
$\mathrm{L_{i}}\leftarrow[\mathrm{L_{j}},\mathrm{L_{k}}]$, signifying that the
input to layer $i$ is the concatenation of activations from layers $j$ and
$k$:

  * •

$\mathrm{L_{5}}\leftarrow[\mathrm{L_{4}},\mathrm{L_{1}}]$

  * •

$\mathrm{L_{6}}\leftarrow[\mathrm{L_{5}},\mathrm{L_{0}}]$

The training of all models was conducted with a batch size set to 16,
utilizing the Adam optimizer with a learning rate of 0.0005.