A Direct Construction of 2D-CCC]A Direct Construction of 2D-CCC with Arbitrary Array Size and Flexible Set Size Using Multivariable Function

1]Gobinda<EMAIL_ADDRESS>
[2]Sudhan<EMAIL_ADDRESS>
[1]Mathematics, IIT Patna, Bihta, Patna, 801103, Bihar, India

*[2]Electrical Communication Engineering, IISc Bangalore, CV Raman Rd, Bengaluru, 560012, Karnataka, India

In this paper, we propose
direct construction of two-dimensional complete complementary codes (2D-CCCs) with arbitrary array size and flexible set size using multivariable functions (MVF). We investigate row and column sequences peak to mean envelope ratio (PMEPR) of the proposed construction 2D-CCCs. We have demonstrated that in the case of 2D-CCCs with array and set sizes as power-of-two, the maximum PMEPR for both rows and columns is upper-bounded by a value of two. The proposed construction generalizes many of the existing state-of-the-art sequence designs such as Golay complementary pair (GCP), Golay complementary set (GCS), one-dimensional (1D)-CCC, 2D Golay complementary array set (2D-GCAS), and 2D-CCC.

§ INTRODUCTION

A pair of sequences is called  a  Golay
complementary pair (GCP) if the aperiodic
auto-correlation sum (AACS) is zero for any non-zero
time-shift [1]. Because of its ideal auto-correlation property, it is used in wireless communication engineering for various purposes [2, 3, 4, 5, 6]. For example, GCP has been used to reduce the peak-to-mean envelope power ratio (PMEPR) in orthogonal frequency division multiplexing (OFDM)
[2, 3, 4], to estimate channel [5], and also use for radar communication [6]. In [7], Tseng and Liu generalized the concept of GCPs by introducing Golay complementary sets (GCSs) and mutually
orthogonal Golay complementary sets (MOGCS). GCS refers to a set of sequences of the same lengths having zero AACS for all non-zero time shifts. If the set size of GCS is $2$, it is called GCP.
The PMEPR of  GCS-based orthogonal frequency-division multiplexing (OFDM) is limited by the set size of GCS [3, 8, 9, 10].
On the other hand, MOGCS is a collection of GCSs in which each GCS is orthogonal to each other in terms of their zero cross-correlation sums for all time shifts.

When the set size of MOGCS reaches its upper bound, it is called
complete complementary code (CCC), first introduced by Suehiro and Hatori in  [11].

There are many constructions of CCCs [12, 13, 14, 15, 16, 17] that have a wide range of applications in wireless communications [18, 19], information hiding [20, 21], etc.

The sequences discussed above are one-dimensional (1D) sequences. Recently, Liu et al.  constructed two-dimensional (2D)-Golay complementary array
sets (GCAS) and demonstrated their applications in massive multiple input multiple output (MIMO) systems equipped with a uniform rectangular array (URA)  [22]. There are many constructions of GCAS available in [22, 23, 24, 25].
The authors in [26] implemented two-dimensional mutually orthogonal complementary array set (2D-MOCASs) in an ultra-wideband (UWB) multi-carrier code-division multiple access (MC-CDMA) system to deliver a low  PMEPR value in addition to good bandwidth efficient transmission. A 2D-MOCAS becomes a 2D-CCC when the number of component arrays is equal to the set size [24]. One of the most important uses of 2D-CCC and 2D-GCAS are in omnidirectional precoding (OP) based massive (MIMO) systems [27, 22, 28, 29, 30].
Other than this, 2D-CCC has other applications such as
multiple access interference (MAI)-free transmission [31],
image change and motion detection   [32],
and application in 2D-MC-CDMA system[33].
even though 2D-CCC has several essential applications, their direct constructions, which do not depend on initial sequences and matrices, are still not available in the literature. Most of the 2D-CCCs are constructed using different 1D or 2D sequences with certain operations
[34, 35, 36]. In [34], 2D-CCC with array size $2^{m}M \times 2^{m}M$ and set size $M^{2}$ is constructed for $m\geq 1$ and $M\geq 2$. In [33], 2D-CCC with array size $M^{2}\times K^{2}$ and set size $MK$ is constructed for $M,K\geq 2$. In [36], 2D-CCC with array size $M\times M$ and set size $M$ is constructed for $M\geq 2$. All the above constructions depend heavily on some choices of initial sequences and matrices.

Pai and Chen in  [28] used GBF to construct 2D-CCC of set size $2^{k}$ and  array size $2^{n}\times2^{m}$ where $m,n,k\geq 1$ and $m,n\geq k$. To the best of our knowledge, there is no direct construction of 2D-CCC with all possible array sizes and flexible set sizes available in the existing literature.

The limitations in array and set sizes of 2D-CCC found in direct constructions within existing literature motivate us to explore the use of multivariable functions (MVF) to achieve greater flexibility in both array and set dimensions. Motivated by the scarcity of 2D-CCC with flexible parameters, in this study, we
construct  2D-CCC of all possible array sizes of the form $m\times n$ where $m=\prod_{i=1}^{a}p_{i}^{m_{i}}$ $n=\prod_{j=1}^{b}q_{j}^{n_{j}}$, and flexible set size of the form $\prod_{i=1}^{a}p^{k_{i}}_{i}\prod_{j=1}^{b}q^{l_{j}}_{j}$ where $p_{i},q_{j}$ are prime numbers, and $m_{i},n_{j},k_{i},l_{j}$are arbitrary positive numbers by MVF. Since 2D-CCC is a collection of 2D-GCAS, the proposed construction also produces 2D-GCAS with arbitrary array size and flexible set size which has not been reported before either by indirect or direct construction.
The row and column
sequence PMEPR of the 2D-CCC array has been derived which is also not been reported before. The proposed construction has the following generalization abilities:

* The proposed construction provides 2D-GCAS with  array size $p^{m}\times q^{n}$ and set size $p^{k_{1}}q^{k_{2}}$ where $m,n,k_{1}$ and $k_{2}$ are positive integers which generalizes 2D-GCAS given in [22],

* The proposed construction yields 2D-CCC with array size $M^{2}\times K^{2}$ and set size $MK$ where $M,K\geq 2$, therefore, the proposed work generalizes the 2D-CCC given in  [33].

* The proposed construction  produces 1D-CCC with sequence length $2^{m}$ and set size  $2^{k}$, where $k,m \geq 1$. As a result, the 1D-CCC constructions in [37, 12] have appeared as a special case of the proposed work.

* The proposed construction produces GCAS with array size $2^{m}\times 2^{n}$ and set size $2^{k}$ where $m,n,k\geq 1$. Therefore, the GCAS proposed in [23, 38] can be constructed by the proposed method.

* The construction of 1D-CCC in  [15] with various lengths is covered by the proposed construction,

* The proposed method presents $2^{k}$ GCAS with array size $2^{kr}\times2^{ks}$ and set size $2^{k}$ where $k,r,s$ are positive integer.  Therefore, the GCAS proposed in wang2021new <cit.> appeared as a special case of our proposed construction.

The rest of the paper is structured as follows.
Section <ref> provides useful definitions. Section <ref> describes 2D-CCC construction. Section <ref> examines the PMEPR of row and column sequences in 2D-CCC arrays and provides generalizations of the proposed 2D-CCC. Section <ref> compares the proposed 2D-CCC to the current state-of-the-art. Section <ref> concludes.

§ PRELIMINARIES

§.§ One-Dimensional Sequences

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 aperiodic cross-correlation sum (ACCS) of sequence at the time shift $u$ is defined by

\begin{equation}
\boldsymbol{C}(\mathbf{a}, \mathbf{b}) (u) =
\begin{cases}
\sum\limits_{i=0}^{l-u-1} a_i b_{i+u}^*,  & 0\leq u < l ;\\
\sum\limits_{i=0}^{l+u-1} a_{i-u} b_{i}^*, & -l < u \leq -1 ;
\end{cases}
\end{equation}

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

Two sequences $\mathbf{a}, \mathbf{b}$ each of length $l$ is called Golay complementary pair (GCP) if

\begin{equation}
\boldsymbol{C}(\mathbf{a})(u)+ \boldsymbol{C}(\mathbf{b}) (u) =
\begin{cases}
2l,  & u=0 ;\\
0, & \text{otherwise}.
\end{cases}
\end{equation}

An ordered set of sequences $\mathcal{A}=\{\mathbf{a}_0, \mathbf{a}_1, \dots, \mathbf{a}_{r-1}\}$ of the same length $l$, is called a code. The code $\mathcal{A}$ can be written in a matrix form of order $r\times l$ as well, where $i$th row is given by the sequence $\mathbf{a}_{i}$ where $0\leq i \leq r-1$.

Let $\mathcal{A}_{1}=\{\mathbf{a}_0^1, \mathbf{a}_1^1, \dots, \mathbf{a}_{r-1}^{1}\}$, $ \mathcal{A}_{2}=\{\mathbf{a}_0^2, \mathbf{a}_1^2, \dots, \mathbf{a}_{r-1}^2\}$ be two codes, where $\mathbf{a}_{i}^{1}$ and $\mathbf{a}_{i}^{2}$ are sequences of length $l$. The ACCS of  $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ is defined by

\begin{equation}
\boldsymbol{C}(\mathcal{A}_{1},\mathcal{A}_{2})(u)=
\displaystyle \sum _{i=0}^{r-1} \boldsymbol{C}(\mathbf {a}_i ^{1},\mathbf {a}_i ^{2})(u).
\end{equation}

A collection $\mathcal{X}= \{\mathcal{A}_0, \mathcal{A}_1,\ldots, \mathcal{A}_{k-1}\}$  of codes each of size $r\times l$ is called mutually orthogonal Golay complementary sets (MOGCS) if it satisfies

\begin{equation}
\boldsymbol{C}(\mathcal{A}_{\nu_{1}},\mathcal{A}_{\nu_{2}})(u) =
\begin{cases}
rl, & u =0,\quad \nu_{1}=\nu_{2};\\
0, & 0 < |u | < l, \quad \nu_{1}=\nu_{2};\\
0, & |u | < l,\quad \nu_{1}\neq \nu_{2}.
\end{cases}
\end{equation}

When $k=r$, then $\mathcal{X}$ is called CCC.

§.§ Two-Dimensional Array

Let $\mathbf{A}=\left(a_{i,j}\right)$ and $\mathbf{B}=\left(b_{i,j}\right)$ be a complex valued array of size $l_{1} \times l_{2}$ where $0 \leq i<l_{1},0 \leq j<l_{2}$.
The 2D-ACCS of arrays $\mathbf{A}$ and $\mathbf{B}$ at shift $\left(u_{1}, u_{2}\right)$ is defined as

\begin{equation*}
\begin{split}
&\boldsymbol{C}\left(\mathbf{A},\mathbf{B}\right)\left(u_{1}, u_{2}\right)=\\
\sum_{i=0}^{l_{1}-1-u_{1}}\sum_{j=0}^{l_{2}-1-u_{2}} a_{i, j}b^{*}_{i+u_{1}, j+u_{2}},~~~\text{if}~~0 \leq u_{1}<l_{1},0 \leq u_{2}<l_{2};\\
\sum_{i=0}^{l_{1}-1-u_{1}}\sum_{j=0}^{l_{2}-1+u_{2}} a_{i, j-u_{2}}b^{*}_{i+u_{1}, j},~~~\text{if}~~0 \leq u_{1}<l_{1}, -l_{2} < u_{2}<0;\\
\sum_{i=0}^{l_{1}-1+u_{1}}\sum_{j=0}^{l_{2}-1-u_{2}} a_{i-u_{1}, j}b^{*}_{i, j+u_{2}},~~~\text{if}~-l_{1} < u_{1}<0,0 \leq u_{2}<l_{2};\\
\sum_{i=0}^{l_{1}-1+u_{1}}\sum_{j=0}^{l_{2}-1+u_{2}} a_{i-u_{1},j-u_{2}}b^{*}_{i,j},~~~\text{if}~-l_{1} < u_{1}<0,-l_{2} < u_{2}<0;
\end{cases}
\end{split}
\end{equation*}

If $\mathbf{A}=\mathbf{B}, \boldsymbol{C}\left(\mathbf{A}, \mathbf{B}\right)\left(u_{1},u_{2}\right)$ is called the 2D-AACS of $\mathbf{A}$ and referred to as $\boldsymbol{C}\left(\mathbf{A}\right)\left(u_{1},u_{2}\right)$.

Let $s\geq2$ be an integer, and define the set $\mathcal{N}=\{\mathbf{A}_{0},\mathbf{A}_{1},\ldots,\mathbf{A}_{s-1}\}$, where $\mathbf{A}_{i}$'s are arrays of  size $l_{1}\times l_{2}$ where $1\leq i\leq s-1$. We refer to the set $\mathcal{N}$ as a two-dimensional Golay complementary array set (2D-GCAS) if

\begin{equation}
\sum_{i=0}^{s-1}\boldsymbol{C}(\mathbf{A}_{i})(u_{1},u_{2})=
\begin{cases}
sl_{1}l_{2},  & (u_{1},u_{2})=(0,0) ;\\
0, & \text{otherwise}.
\end{cases}
\end{equation}

Consider the set $\mathcal{M}=\left\{\mathbf{A}^{k} \mid k=\right.$ $0,1, \ldots, s-1\}$, where each set $\mathbf{A}^{k}=\left\{\mathbf{A}_{0}^{k}, \mathbf{A}_{1}^{k}, \ldots, \mathbf{A}_{s-1}^{k}\right\}$ is composed of $s$ arrays of size $l_{1} \times l_{2}$, i.e., each $\mathbf{A}_{i}^{k}$ is an array of size $l_{1}\times l_{2}$ . The set $\mathcal{M}$ is said to be $(s,s,l_{1},l_{2})$ 2D-CCC if the following holds

\begin{equation}
\begin{split}
&=\sum_{i=0}^{s-1} \boldsymbol{C}\left(\mathbf{A}_{i}^{k}, \mathbf{A}_{i}^{k^{\prime}}\right)\left(u_{1}, u_{2}\right)\\
&= \begin{cases}sl_{1}l_{2}, \quad\left(u_{1}, u_{2}\right)=(0,0), k=k^{\prime} ;\\
0, \quad\left(u_{1}, u_{2}\right)\neq(0,0), k=k^{\prime} ;\\
0,     ~~~~k\neq k^{\prime}.
\end{cases}
\end{split}
\end{equation}

§.§ Multivariable Function [44]

Consider two positive integers $a$ and $b$ and consider $i$ and $j$ such that $1\leq i\leq a$ and $1\leq j\leq b$. For every value of $i$ and $j$ within these ranges, associate a positive integer, denoted $m_i$ and $n_j$ respectively. Additionally, associate a prime number, denoted $p_{i}$ and $q_{j}$ respectively. Now, consider the sets $\mathbb{A}_{p_{i}}=\{0,1,\ldots,p_{i}-1\}$ and $\mathbb{A}_{q_{j}}=\{0,1,\ldots,q_j-1\}$, which are subsets of the ring of integers $\mathbb{Z}$ under usual addition and multiplication.
A multi-variable function (MVF) can be defined with a domain constructed from cartesian products of the power sets of $\mathbb{A}_{p_i}$ and $\mathbb{A}_{q_j}$, and a range in the ring of integers modulo a positive integer $\lambda$, i.e., $\mathbb{Z}_\lambda=\{0,1,\ldots,\lambda-1\}$ as

$$f: \left(\mathbb{A}_{p_1}^{m_1}\!\!\times \!\mathbb{A}_{p_2}^{m_2}\!\times \dots \times \mathbb{A}_{p_a}^{m_a}\right)\!\times\!\left(\mathbb{A}_{q_1}^{n_1}\!\!\times \mathbb{A}_{q_2}^{n_2} \times \dots \times \mathbb{A}_{q_b}^{n_b}\right)\!\!\rightarrow\!\! \mathbb{Z}_\lambda,$$

Next, we consider two integers $c$ and $d$ in the range of  $0\leq c \leq p_{1}^{m_{1}}p_{2}^{m_{2}}\ldots p_{a}^{m_{a}}-1$ and  $0\leq d\leq q_{1}^{n_{1}}q_{2}^{n_{2}}\ldots q_{b}^{n_{b}}-1$.  The integers $c$ and $d$ can be decomposed as per the following equations:

\begin{equation}
\label{c,d}
\begin{split}
&c=c_{1}+c_{2}p_{1}^{m_{1}}+\ldots+c_{a}p_{1}^{m_{1}}p_{2}^{m_{2}}\ldots p_{a-1}^{m_{a-1}},\\
&d=d_{1}+d_{2}q_{1}^{n_{1}}+\ldots+d_{b}q_{1}^{n_{1}}q_{2}^{n_{2}}\ldots q_{b-1}^{n_{b-1}},
\end{split}
\end{equation}

where $0\leq c_{i}\leq p_{i}^{m_{i}}-1$ and $0\leq d_{j}\leq q_{j}^{n_{j}}-1$.
Let $\mathbf{C}_{i}=(c_{i,1},c_{i,2},\ldots,c_{i,m_{i}})\in \mathbb{A}_{p_{i}}^{m_{i}}$
be the vector representation of $c_{i}$ with base $p_{i}$, i.e., $c_{i}=\sum_{k=1}^{m_{i}}c_{i,k}p_{i}^{k-1}$ and $\mathbf{D}_{j}=(d_{j,1},d_{j,2},\ldots,d_{j,n_{j}})\in \mathbb{A}_{q_{j}}^{n_{j}}$ be the vector representation of  $d_{j}$  with base $q_{j}$,  i.e., $d_{j}=\sum_{l=1}^{n_{j}}d_{j,l}q_{j}^{l-1}$ where $1\leq i\leq a$, $1\leq j\leq b, 0\leq c_{i,k}\leq p_{i}-1,0\leq d_{j,l}\leq q_{j}-1 $. We define vectors associated with $c$ and $d$ as
We define an array associated with $f$ as $\psi({f})$, where

\begin{equation}
\psi({f})=\left(\begin{array}{cccc}
\omega_{\lambda}^{f_{0,0}} & \omega_{\lambda}^{f_{0,1}} & \cdots & \omega_{\lambda}^{f_{0,m}} \\
\omega_{\lambda}^{f_{1,0}} & \omega_{\lambda}^{f_{1,1}} & \cdots & \omega_{\lambda}^{f_{1,m}} \\
\vdots & \vdots & \ddots & \vdots \\
\omega_{\lambda}^{f_{n,0}} & \omega_{\lambda}^{f_{n,1}} & \cdots & \omega_{\lambda}^{f_{n,m}}
\end{array}\right)
\end{equation}

$m=p_{1}^{m_{1}}p_{2}^{m_{2}}\ldots p_{a}^{m_{a}}-1,n=q_{1}^{n_{1}}q_{2}^{n_{2}}\ldots q_{b}^{n_{b}}-1$ and the $(c,d)$-th component of $\psi({f})$ is given by $\omega_{\lambda}^{f_{c,d}}$, $\omega_{\lambda}=\exp\left(2\pi\sqrt{-1}/\lambda\right)$ and
$f_{c,d}=f\left(\phi(c),\phi(d)\right)$. For simplicity, from from hereon, we  denote $\left(\mathbb{A}_{p_1}^{m_1}\!\!\times \!\mathbb{A}_{p_2}^{m_2}\!\times \dots \times \mathbb{A}_{p_a}^{m_a}\right)\!\times\!$$(\mathbb{A}_{q_1}^{n_1}\!\!\times \mathbb{A}_{q_2}^{n_2} \times \dots \times \mathbb{A}_{q_b}^{n_b})$ by $\mathcal{C}$.

§.§ Peak-to-Mean Envelope Power Ratio (PMEPR)
[23]

Let $\lambda\geq 2$ and $\mathbb{Z}_{\lambda}$ denotes ring of integer modulo $\lambda$. Let $\mathbf{x}$ be a $\mathbb{Z}_{\lambda}$ valued vector of length $l$ given by $\mathbf{x}=\left(x_{0}, x_{1}, \ldots, x_{l-1}\right)$

. For the vector $\mathbf{x}$, the OFDM signal is the real part of the complex envelope

S_{\mathbf{x}}(t)=\sum_{i=1}^{l-1} \omega_{\lambda}^{x_{ i}+\lambda f_{i} t},

where $f_{i}=f+i \Delta f, f$ is a constant, $\Delta f$ is an integer multiple of OFDM symbol rate, and $0 \leq \Delta f t \leq 1$.
The PMEPR of the vector $\mathbf{x}$ is given by

PMEPR\left(\mathbf{x}\right)=\sup _{0 \leq \Delta f t \leq 1} \frac{\abs{S_{\mathbf{x}}(t)}^{2}}{l}.
The term $\frac{\abs{S_{\mathbf{x}}(t)}^{2}}{l}$ is called the instantaneous-to-average power ratio (IAPR).

§ PROPOSED CONSTRUCTION OF 2D-CCC

In this Section, we construct 2D-CCC and give an example. Let  $0\leq \gamma\leq p_{1}^{m_{1}}p_{2}^{m_{2}}\ldots p_{a}^{m_{a}}-1$ and $0\leq \mu\leq q_{1}^{n_{1}}q_{2}^{n_{2}}\ldots q_{b}^{n_{b}}-1$ be such that

\begin{equation*}
\begin{split}
\end{split}
\end{equation*}

where $0\leq \gamma_{i}\leq p_{i}^{m_{i}}-1$ and $0\leq \mu_{j}\leq q_{j}^{n_{j}}-1$, $p_i$ is a prime number or $1$ and $q_j$ is a prime number. Let $\boldsymbol{\gamma}_{i}=(\gamma_{i,1},\gamma_{i,2},\ldots,\gamma_{i,m_{i}})\in \mathbb{A}_{p_{i}}^{m_{i}}$ be the vector representation of $\gamma_{i}$ with base $p_{i}$, i.e, $\gamma_{i}=\sum_{k=1}^{m_{i}}\gamma_{i,k}p_{i}^{k-1}$. Similarly,
$\boldsymbol{\mu}_{j}=(\mu_{j,1},\mu_{j,2},\ldots,\mu_{j,n_{j}})\in \mathbb{A}_{q_{j}}^{n_{j}}$ is the vector representation of $\mu_{j}$ with base $q_{j}$. Let
$$\phi(\gamma)=\left(\boldsymbol{\gamma}_{1},\boldsymbol{\gamma}_{2},\ldots,\boldsymbol{\gamma}_{a}\right)\in \mathbb{A}_{p_1}^{m_1}\!\!\times \!\mathbb{A}_{p_2}^{m_2}\!\times \dots \times \mathbb{A}_{p_a}^{m_a}$$
be the vector associated with $\gamma$ and
$$\phi(\mu)=\left(\boldsymbol{\mu}_{1},\boldsymbol{\mu}_{2},\ldots,\boldsymbol{\mu}_{b}\right)\in \mathbb{A}_{q_1}^{n_1}\!\!\times \!\mathbb{A}_{q_2}^{n_2}\!\times \dots \times \mathbb{A}_{q_b}^{n_b}$$
be the vector associated with $\mu$.
$\pi_{i}$ and $\sigma_{j}$ be any permutations of the set  $\{1,2,\ldots,m_{i}\}$ and  $\{1,2,\ldots,n_{j}\}$, respectively.
Let us also define a MVF $f:\mathcal{C}\rightarrow\mathbb{Z}_\lambda,$ as

\begin{equation*}
\label{kanha}
\begin{split}
&f\big(\phi(\gamma),\phi(\mu)\big)=f\big((\boldsymbol{\gamma}_{{1}},\boldsymbol{\gamma}_{{2}}, \ldots, \boldsymbol{\gamma}_{{a}}),(\boldsymbol{\mu}_{{1}},\boldsymbol{\mu}_{{2}}, \ldots, \boldsymbol{\mu}_{b})\big)\\
&=\!\!\sum_{i=1}^{a}\!\frac{\lambda}{p_{i}}\!\!\sum_{e=1}^{m_{i}-1}\!\!\gamma_{i, \pi_{i}(e)} \gamma_{i, \pi_{i}(e+1)}+\!\!\sum_{j=1}^{b}\!\frac{\lambda}{q_{j}}\!\!\sum_{o=1}^{n_{j}-1} \mu_{j, \sigma_{j}(o)} \mu_{j, \sigma_{j}(o+1)}\\
&+\sum_{i=1}^{a}\!\sum_{e=1}^{m_{i}}\!d_{i,e} \gamma_{i, e}\!\!+\!\!\sum_{j=1}^{b}\!\sum_{o=1}^{n_{j}} c_{j,o} \mu_{j,o}~(\text{mod}~\lambda)
\end{split}
\end{equation*}

where $\lambda$ is divisible by l.c.m$(p_{1},\ldots,p_{a},q_{1},\ldots,q_{b})$, $d_{i,e},c_{j,o}\in \mathbb{A}_{\lambda}$ and

all the operations
between those integers, the modulus is performed. From now hereon, we will exclude the term $(\text{mod}~\lambda)$ from the function. Let us define sets $\Theta$ and $T$ as

\begin{equation*}
\label{kaka}
\begin{split}
&\Theta=\{\theta:\theta=(r_{{1}}, r_{{2}}, \ldots, r_{{a}},s_{{1}}, s_{{2}}, \ldots, s_{{b}})\},\\
&T=\{t:t=(x_{{1}}, x_{{2}}, \ldots, x_{{a}},y_{{1}}, y_{{2}}, \ldots, y_{{b}})\},
\end{split}
\end{equation*}

where $0\leq r_{i},x_{i}< p_{i}^{k_{i}}$ and $0\leq s_{j},y_{j}< q_{j}^{l_{j}}$ and $k_{i},l_{j}$ are positive integers.
Again, we define a function 	$a^{\theta}_{t}\!\!:\mathcal{C}\rightarrow\!\! \mathbb{Z}_\lambda,$ as

\begin{equation}{\label{5}}
\begin{split}
&a^{\theta}_{t}\big(\phi(\gamma),\phi(\mu)\big)=a^{\theta}_{t}\big((\boldsymbol{\gamma}_{{1}},\boldsymbol{\gamma}_{{2}}, \ldots, \boldsymbol{\gamma}_{{a}}),(\boldsymbol{\mu}_{{1}},\boldsymbol{\mu}_{{2}}, \ldots, \boldsymbol{\mu}_{b})\big)\\
&=\!\!f\big(\phi(\gamma),\phi(\mu)\big)\!\!+\!\!\sum_{i=1}^{a} \frac{\lambda}{p_{i}} \gamma_{i, \pi_{i}(1)}{r_{i}}+\!\sum_{j=1}^{b} \frac{\lambda}{q_{j}} \mu_{j, \sigma_{j}(1)}{s_{j}}\\
&+\sum_{i=1}^{a} \frac{\lambda}{p_{i}} \gamma_{i, \pi_{i}(m_{i})}{x_{i}}+\sum_{j=1}^{b} \frac{\lambda}{q_{j}} \mu_{j, \sigma_{j}(n_{j})}{y_{j}},
\end{split}
\end{equation}

where $d_{\theta}\in \mathbb{A_{\lambda}}$ be any integer for each value of $\theta$. For simplicity, we denote $a^{\theta}_{t}\big(\phi({\gamma}),\phi({\mu})\big)$ by $(a^{\theta}_{t})_{\gamma,\mu}$ and $f\big(\phi({\gamma}),\phi({\mu})\big)$ by $f_{\gamma,\mu}$.

We define $G_{t}=\{a^{\theta}_{t}:\theta\in \Theta\}$ and $\psi(G_{t})=\{\psi\left(a^{\theta}_{t}\right):\theta\in \Theta\}$. Then the set  $\{\psi(G_{t}):t\in T\}$ forms a 2D
$, where, $\alpha=\prod_{i=1}^{a}p^{k_{i}}_{i}\prod_{j=1}^{b}q^{l_{j}}_{j}$, $m=\prod_{i=1}^{a}p_{i}^{m_{i}}$ and $n=\prod_{j=1}^{b}q_{j}^{n_{j}}$.

The proof of this theorem is given in Appendix <ref>

Let $a=1,b=1,p_{1}=2,q_1=3,m_1=2,k_{1}=1,l_{1}=1$, and $n_1=2$. Consider $\pi_1$ and $\sigma_1$ are permutations of the set $\{1,2\}$  such that $\pi_{1}(1)=2, \pi_{1}(2)=1$ and  $\sigma_{1}(1)=2, \sigma_{1}(2)=1$.
Let $\boldsymbol{\gamma}_{1}=(\gamma_{11},\gamma_{12})$ be the vector associated with ${\gamma}_{1}$ where $0\leq \gamma_{1}\leq 3$  and $\boldsymbol{\mu}_{1}=(\mu_{11},\mu_{12})$ be the vector associated with $\mu_{1}$ where $0\leq\mu_{1}\leq 8$.
We define a MVF $f:\mathbb{A}_{2}^{2}\times \mathbb{A}_{3}^{2}\rightarrow \mathbb{Z}_{6}$ as

\begin{equation*}
\begin{split}
\end{split}
\end{equation*}

Consider $\Theta=\{\theta:\theta=(r_1,s_1)\}$, $T=\{t:t=(x_1,y_1)\}$ where $0\leq r_1,x_1\leq 1$,$0\leq s_1,y_1\leq 2$.
Let $c_{\theta}$ be any element in $\mathbb{A}_{6}$, then by the defination of $a_{t}^{\theta}$, we have

\begin{equation*}
\begin{split}
\end{split}
\end{equation*}

Consider a set $G_{t}=\{a^{\theta}_{t}:\theta\in \Theta\}$, then the set  $\mathcal{G}=\{G_{t}:t\in T\}$ forms a 2D
(6,6,4,9)-CCC$. The AACS of any code set from $\mathcal{G}$  is given in  Fig.<ref> and the ACCS between any two code sets from $\mathcal{G}$ is given in Fig. <ref>.

Auto-correlation result of any set of array from $\mathcal{G},$

Cross-correlation result of any two sets of array from $\mathcal{G}.$

§ BOUND FOR PMEPR

In this section we  derive the PMEPR bound of the $2$D-CCC arrays given in Theorem 1.

Given the 2D-CCC arrays as defined in Theorem <ref>, the PMEPR of the arrays can be characterized as follows: the PMEPR of the column sequences is bounded by $\max \{p_{i}^{k_{i}}:1\leq i\leq a\}$. Similarly, the PMEPR of the row sequences is bounded by $\max \{q_{j}^{l_{j}}:1\leq j\leq b\}$.

From the Theorem 1, we have
$\psi(G_{t})=\{\psi(a_{t}^{\theta}):\theta\in \Theta\}$.
Let $\left(\psi(a_{t}^{\theta})\right)_{d}$ denotes the $d$-th column of the matrix $\psi(a_{t}^{\theta})$ where $\theta=(r_{1},r_{2},\ldots,r_{a},s_{1},s_{2},\ldots,s_{b})$. From the definition of the complex envelope,  it is easy to observe that for any  sequence $\mathbf{x}$ of length $l$  it is easy to verify that

\begin{equation}
\abs{S_{\mathbf{x}}(t)}^{2}=l+\sum_{u_{1}\neq0}\boldsymbol{C}(\mathbf{x})(u_{1})\omega_{\lambda}^{-\lambda u_{1}ft}.
\end{equation}

Therefore, the complex envelope for the sequence $\left(\psi(a_{t}^{\theta})\right)_{d}$ is given by

\begin{equation}
\label{an}
\abs{S_{\left(\psi(a_{t}^{\theta})\right)_{d}}(t)}^{2}=m+\sum_{u_{1}\neq0}\boldsymbol{C}(\left(\psi(a_{t}^{\theta})\right)_{d})(u_{1})\omega_{\lambda}^{-\lambda u_{1}ft}.
\end{equation}

Now, we have

\begin{equation}
\label{a4}
\boldsymbol{C}[(\psi(a_{t}^{\theta}))_{d}](u_{1})=\sum_{\gamma=0}^{m-1-u_{1}}\omega_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,d}-(a_{t}^{\theta})_{\delta,d}\right)},
\end{equation}

$\delta=\gamma+u_{1}$, $0\leq {\delta},{\gamma}\leq {p_{1}^{m_{1}}p_{2}^{m_{2}}\ldots p_{a}^{m_{a}}-1}$ and $0\leq \abs{u_{1}}\leq p_{1}^{m_{1}}p_{2}^{m_{2}}\ldots p_{a}^{m_{a}}-1$. Let

\begin{equation}
\begin{split}
&\gamma=\gamma_{1}+\gamma_{2}p_{1}^{m_{1}}+\gamma_{3}p_{1}^{m_{1}}p_{2}^{m_{2}}+\ldots+\gamma_{a}p_{1}^{m_{1}}p_{2}^{m_{2}}\ldots p_{a-1}^{m_{a-1}},\\ &\delta=\delta_{1}+\delta_{2}p_{1}^{m_{1}}+\delta_{3}p_{1}^{m_{1}}p_{2}^{m_{2}}+\ldots+\delta_{a}p_{1}^{m_{1}}p_{2}^{m_{2}}\ldots p_{a-1}^{m_{a-1}}.
\end{split}
\end{equation}

Substracting $(a_{t}^{\theta})_{\gamma,d}$ and $(a_{t}^{\theta})_{\delta,d}$, we have

\begin{equation*}
\label{maha}
\begin{split}
\end{split}
\end{equation*}

\begin{equation}
\label{kalas}
\begin{split}
\end{split}
\end{equation}

$$\mathcal{W}_{\gamma}=\omega_{\lambda}^{\left(f_{\gamma,d}-f_{\delta,d}+\sum_{i=1}^{a}\frac{\lambda}{p_{i}}x_{i}\left(\gamma_{i,\pi_{i}(m_{i})}-\delta_{i,\pi_{i}(m_{i})}\right)+\sum_{i'=1,i'\neq i }^{a}\frac{\lambda}{p_{i'}}r_{i'}\left(\gamma_{i',\pi_{i'}(1)}-\delta_{i',\pi_{i'}(1)}\right)\right)}.$$

Consider the subset $\Theta_{1}$ of $\Theta$ as

\begin{equation}
\begin{split}
&\Theta_{1}=\{\theta_{1}=(r_{1},r_{2},\ldots,r_{i-1},r_{i},r_{i+1},\ldots,r_{a},s_{1},s_{2},\ldots,s_{b}):0\leq r_{i}\leq p^{k{i}}_{i}-1\}.
\end{split}
\end{equation}

Taking the summation both side of (<ref>), we obtain

\begin{equation}
\begin{split}
&\sum_{\theta_{1}\in \Theta_{1} }\sum_{\gamma=0}^{m-1-u_{1}}\omega_{\lambda}^{\left((a_{t}^{\theta_{1}})_{\gamma,d}-(a_{t}^{\theta_{1}})_{\delta,d}\right)}=
\sum_{r_{i}=0}^{p^{k_{i}}_{i}-1}\sum_{\gamma=0}^{m-1-u_{1}}\mathcal{W}_{\gamma}\omega_{p_{i}}^{r_{i}\left(\gamma_{i,\pi_{i}(1)}-\delta_{i,\pi_{i}(1)}\right)}\\
\end{split}
\end{equation}

To derive the PMEPR bound, we consider the two cases as follows.

$(\gamma_{i,\pi_{i}(1)}\neq\delta_{i,\pi_{i}(1)}$, for any $i\in\{1,2,\ldots,a\})$.

We have

Hence, for such $\gamma$, we have

\begin{equation}
\label{a1}
\sum_{\theta_{1}\in \Theta_{1} }\sum_{\gamma=0}^{m-1-u_{1}}\omega_{\lambda}^{\left((a_{t}^{\theta_{1}})_{\gamma,d}-(a_{t}^{\theta_{1}})_{\delta,d}\right)}=0.
\end{equation}

($\gamma_{i\pi_{i}(1)}=\delta_{i\pi_{i}(1)}$ $\forall i\in\{1,2,\ldots,a\}$).

Just as in the case of proof of Thm <ref>,   Subcase (ii)  of Case 1, it can be shown that

\begin{equation}
\begin{split}
}-(a_{t'}^{\theta_{1}})_{\delta,d}\right)}\!\!+\!\!\!\sum_{\theta_{1}\in \Theta_{1}}\!\!\!\!\!\sum_{k_{1}=1}^{\gamma_{1\pi_{1}(v-1)}}\!\!\!\!\omega_{\lambda}^{\left((a_{t}^{\theta_{1}})_{\gamma^{k_{1}},d}-(a_{t'}^{\theta_{1}})_{\delta^{k_{1}},d}\right)}\!\\
&+\!\!\!\sum_{\theta_{1}\in \Theta_{1}}\sum_{k_{2}=1+\gamma_{1\pi_{1}(v-1)}}^{p_{1}-1}\!\!\!\!\!\!\!\!\!\!\!\!\omega_{\lambda}^{\left((a_{t}^{\theta_{1}})_{\gamma^{k_{2}},d}-(a_{t'}^{\theta_{1}})_{\delta^{k_{2}},d}\right)}=0.
\end{split}
\end{equation}

Combining the above two cases and (<ref>),   we have

\begin{equation}
\label{kasa}
\sum_{\theta_{1}\in \Theta_{1}}\boldsymbol{C}\left[(\psi(a_{t}^{\theta_{1}}))_{d}\right](u_{1})=0.
\end{equation}

Taking summation on both sides of (<ref>), we have

\begin{equation}
\begin{split}
\sum_{\theta_{1}\in \Theta_{1}}\abs{S_{\left(\psi(a_{t}^{\theta})\right)_{d}}(t)}^{2}
&=\sum_{\theta_{1}\in \Theta_{1}}m+\sum_{\theta_{1}\in \Theta_{1}}\sum_{u_{1}\neq0}\boldsymbol{C}\left(\left(\psi(a_{t}^{\theta_{1}})\right)_{d}\right)(u_{1})\omega_{\lambda}^{-\lambda u_{1}ft}\\
&=\sum_{\theta_{1}\in \Theta_{1}}m+\sum_{u_{1}\neq0}\omega_{\lambda}^{-\lambda u_{1}ft}\sum_{\theta_{1}\in \Theta_{1}}\boldsymbol{C}\left(\left(\psi(a_{t}^{\theta_{1}})\right)_{d}\right)(u_{1})\\
\end{split}
\end{equation}

Given that $\theta$ belongs to $\Theta_{1}$, it follows that

\begin{equation}
\abs{S_{\left(\psi(a_{t}^{\theta})\right)_{d}}(t)}^{2}\leq \sum_{\theta_{1}\in \Theta_{1}}\abs{S_{\left(\psi(a_{t}^{\theta_{1}})\right)_{d}}(t)}^{2}=mp_{i}^{k_{i}}.
\end{equation}

Hence the column sequence PMEPR is bound by max $\{p_{i}^{k_{i}}:1\leq i\leq a\}$. Similarly, the row sequence PMEPR is also bounded by max $\{q_{j}^{l_{j}}:1\leq j\leq b\}$.

Since 2D-CCC is a collection of 2D-GCAS, the proposed work also produces 2D-GCAS with a set size. $\prod_{i=1}^{a}p^{k_{i}}_{i}\prod_{j=1}^{b}q^{l_{j}}_{j}$, row size
$\prod_{i=1}^{a}p^{m_{i}}_{i}$ and coloumn size $\prod_{j=1}^{b}q^{n_{j}}_{j}$.

When $a=b=1$, $p_1=q_1= 2$, and $k_1=l_1=1$, Theorem 1 provides a 2D-GCAS of array size $2^{m_1}\times 2^{n_1}$. In this case, our row and column sequence PMEPR is bounded by $2$ which is comparable to the 2D-GCAS of array size $2^{n}\times 2^{n}$ in <cit.>. However, in <cit.>, the PMEPRs of the row and column sequences are upper bound by $2^v$, where $v$ is greater than or equal to $1$ and it depends on the choice of permutation functions $\pi_s$.
Therefore, the proposed 2D-CCC is a promising solution for reducing PMEPR in comparison to [23].

In this example, we derive  2D-GCAS from 2D-CCC and look at the row and column sequence PMEPR. We have taken $a=1,b=1,p_{1}=2,q_1=2,m_1=3,k_{1}=1,l_{1}=1$, and $n_1=4$ in Theorem 1, and get  2D-GCAS with array size  $8\times 16$. The row and column sequence PMEPRs are computed as 1.8093 and 2, respectively. In contrast, for a similar 2D-CCC in [28], the PMEPR of the row sequence is 3.1237 and that of the column sequence is 2.3621, indicating a notable reduction in our presented approach. In Fig. <ref>, we have shown the
corresponding plot of row and column sequence IAPR and PMEPR bound.

Row and column sequence

§.§ Generalization of the proposed work

This section illustrates the generalizability of the work.
In Theorem <ref> , if we take $a=1,b=1,p_{1}=p,q_{1}=q$, then we have $p^{k_{1}}q^{k_{2}}$ number GCAS with array size $p^{m_{1}}\times q^{n_{1}}$ and set size $p^{k_{1}}q^{k_{2}}$, where $m_{1}>0$ and $n_{1}>0$. Therefore, this paper covers the GCAS construction given in [22].
In Theorem <ref> , by considering $m_{i}=2k_{i}$ and $n_{j}=2t_{j}$,  we have 2D $(MK,MK,M^2,K^2)-CCC$. In [33], Farkas et al. proposed 2D-CCC  with parameter $(MK,MK,M^2,K^2)-CCC$ and the method depends on the initial choice of 1D sequences. Therefore, the proposed work covers 2D-CCC given in [33] without depending on 1D sequences.
In Theorem <ref> , if we take  $m=1,b=k,q_{1}=q_{2}=\cdots=q_{k}=2$ and $n_{1}+n_{2}+\hdots+n_{k}=n$, then we can cover the CCC given in [37, 12].
In Theorem <ref> , if we consider $a=1,b=1,p_{1}=2,q_{1}=2,m_{1}=m,n_{1}=n$, then we have $2^{k}$ number of GCAS with array size $2^{m}\times 2^{n}$ and set size $2^{k}$ where $k=k_{1}+t_{1}$ and $m_{1},n_{1},k_{1},t_{1}\geq 1$. Therefore, the GCAS proposed in [23, 38] can be constructed by the proposed method except for GCAP.
In Theorem <ref>, if we consider $a=1,b=1,p_{1}=p,q_{1}=p,m_{1}=m,n_{1}=n$, then we have $p^{k}$ number of GCAS with array size $p^{m}\times p^{n}$ and set size $p^{k}$, where $k=k_{1}+t_{1}$ and $m_{1},n_{1},k_{1},t_{1}\geq 1$. Therefore, the GCAS proposed in [22] can be constructed by the proposed method except, GCAP.
In Theorem <ref>, by considering $p_{1}=2$, $k_{1}=k$, $m_{1}=kr$ and $n_{1}=ks$,  we have $2^{k}$ GCAS with array size $2^{kr}\times2^{ks}$ and set size $2^{k}$ where $k,r,s$ are positive integer.  Therefore, the GCAS proposed in wang2021new <cit.> can be constructed by the proposed method except $k=1$.
In [15], Sarkar et al. constructed 1D-CCC with all possible lengths and set size $\prod_{i=1}^{k}p_{i}^{k_{i}}$. In Theorem <ref>, if we take $m=1$, then our proposed construction covers the CCC in [15]. Since CCC is just a group of GCS, our work also makes GCS with any length of sequence.

§ COMPARISON WITH PREVIOUS WORKS

We also compare the proposed constructions with previous indirect constructions given in [35, 34, 36] and direct construction given in [38] and provide these in TABLE I.

Comparision of proposed 2D-CCC with existing works

Source      Set Size     Array Size     Based on     Constraints

[35]      $MK$     $M^{2}\times K^{2}$     $1 \mathrm{D}-\mathrm{CCCs}$     $M, K \geq 2$

[34]      $M^{2}$     $2^{m}M\times2^{m}M$     $1 \mathrm{D}-\mathrm{CCCs}$     $m \geq 1$,$M\geq 2$

[36]      $M$     $M\times M$     BH matrices     $M \geq 2$

[38]      $2^{k}$     $2^{n}\times2^{m}$     GBF     $m,n \geq k$, $m,n,k\in\mathbb{Z}^{+}$

Theorem <ref>      $x$     $s_{1}\times s_{2}$     MVF     $x=\prod_{i=1}^{a}p^{k_{i}}_{i}\prod_{j=1}^{b}q^{l_{j}}_{j},$

The constructions given in  [35, 34, 36]  depend on the initial choices of some sequence and matrices, whereas the construction given in [38] is obtained by GBF. The array size and the set size in [38] is of power-of-two. Our construction is based on MVF, which does not need any sequences or matrices initially. Also, it produces all possible array size and flexible set size.

§ CONCLUSION

This paper presents new results in the design of 2D-CCC, which is important for omnidirectional transmission in massive MIMO. We have introduced an MVF-based direct 2D-CCC construction. The set size of the proposed 2D-CCC is of the form $\prod_{i=1}^{a}p^{k_{i}}_{i}\prod_{j=1}^{b}q^{l_{j}}_{j}$ which is more flexible and has not been reported in the existing literature. The suggested 2D-CCC can handle all array sizes, which is also not reported.
As a special case of the design, we have come up with 2D-GCAS with any array size and  flexible set size of the form $\prod_{i=1}^{a}p_{i}\prod_{j=1}^{b}q_{j}$. We have also successfully used the proposed 2D-CCC as precoding matrices in OP-based massive MIMO URA systems. We have shown that the BER performance of the proposed scheme. Moreover, the PMEPR of row and column sequences have been examined. The 2D-CCC given in [33],
GCP given in <cit.>, CCC given in[37, 12, 15], and GCAS given in
<cit.> appears as a special case of the proposed construction.

§ APPENDIX: PROOF OF THEOREM <REF>

Let $\gamma,\delta \in \{0,1,\ldots,p_{1}^{m_{1}}p_{2}^{m_{2}}\ldots p_{a}^{m_{a}}-1\}$ and $\mu,\rho \in \{0,1,\ldots,q_{1}^{n_{1}}q_{2}^{n_{2}}\ldots q_{b}^{n_{b}}-1\}$, be such that

\begin{equation*}
\begin{split}
\mu=u_{1}+\displaystyle\sum_{j=2}^{b}\mu_{j}\left(\prod_{j_{1}=1}^{j-1}q_{j_{1}}^{n_{j_{1}}}\right),\\
\rho=\rho_{1}+\displaystyle\sum_{j=2}^{b}\rho_{j}\left(\prod_{j_{1}=1}^{j-1}q_{j_{1}}^{n_{j_{1}}}\right),
\end{split}
\end{equation*}

where, $0\leq \gamma_{i},\delta_{i}\leq p_{i}^{m_{i}}-1$ and $0\leq \mu_{j},\rho_{j}\leq q_{j}^{n_{j}}-1$.
We choose $u_{1}$ and $u_{2}$ such that $\delta=\gamma+u_{1}$, $\rho=\mu+u_{2}$, where $0\leq |u_{1}|\leq p_{1}^{m_{1}}p_{2}^{m_{2}}\ldots p_{a}^{m_{a}}-1$ and $0\leq |u_{2}|< q_{1}^{n_{1}}q_{2}^{n_{2}}\ldots q_{b}^{n_{b}}-1$. The ACCS between $G_{t}$ and $G_{t'}$ can be expressed as

\begin{equation}
\label{g}
\begin{split}
\boldsymbol{C}(\psi(G_{t}),\psi(G_{t'}))(u_{1},u_{2})
&=\sum_{\theta\in \Theta}\boldsymbol{C}\big(\psi(a_{t}^{\theta}),\psi(a_{t'}^{\theta})\big)(u_{1},u_{2})\\
&=\sum_{\theta\in \Theta}\sum_{\gamma=0}^{m-1-u_{1}}\!\sum_{\mu=0}^{n-1-u_{2}}\omega_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,\mu}-(a_{t'}^{\theta})_{\delta,\rho}\right)}\\
&=\sum_{\gamma=0}^{m-1-u_{1}}\sum_{\mu=0}^{n-1-u_{2}}\sum_{\theta\in \Theta} \omega_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,\mu}-(a_{t'}^{\theta})_{\delta,\rho}\right)}.
\end{split}
\end{equation}

By subtracting $(a_{t}^{\theta})_{\gamma,\mu}$ and $(a_{t'}^{\theta})_{\delta,\rho}$, we get

\begin{equation}
\label{f}
\begin{split}
\end{split}
\end{equation}

\begin{equation*}
\begin{aligned}
\theta&=(r_{1},r_{2},\ldots,r_{a},s_{1},s_{2},\ldots,s_{b}),\\
D_{\theta}&=\sum_{i=1}^{a}\frac{\lambda}{p_{i}}r_{i}(\gamma_{i,\pi_{i}(1)}- \delta_{i,\pi_{i}(1)}),\\
D_{\theta}'&=\sum_{j=1}^{b}\frac{\lambda}{q_{j}}s_{j}(\mu_{j,\sigma_{j}(1)}- \rho_{j,\sigma_{j}(1)}),\\
\end{aligned}
\end{equation*}

\begin{equation*}
\end{equation*}

\begin{equation}
\label{z1}
\sum_{\theta\in \Theta} w_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,\mu}-(a_{t'}^{\theta})_{\delta,\rho}\right)}=\omega_{\lambda}^{\left(f_{\gamma,\mu}-f_{\delta,\rho}+E+E'\right)}FG,
\end{equation}

\begin{equation}
F=\sum_{\theta\in\Theta}\omega_{\lambda}^{D_{\theta}}=\left(\prod_{j=1}^{b}q^{l_{j}}_{j}\right)\prod_{i=1}^{a}\left(\sum_{r_{i}=0}^{p^{k_{i}}-1}\omega_{p_{i}}^{r_{{i}}(\gamma_{i,\pi_{i}(1)}- \delta_{i,\pi_{i}(1)})}\right)
\end{equation}

\begin{equation}
G=\sum_{\theta\in\Theta}\omega_{\lambda}^{D'_{\theta}}=\left(\prod_{i=1}^{a}p^{k_{i}}_{i}\right)\prod_{j=1}^{b}\left(\sum_{s_{j}=0}^{q^{l_{j}}_{j}-1}\omega_{q_{j}}^{s_{{j}}(\mu_{j,\sigma_{j}(1)}- \rho_{j,\sigma_{j}(1)})}\right).
\end{equation}

The rest of the proof is split into the following cases

$\left(\text{For some}~j\in \{1,2,\ldots,b\}, \mu_{j,\sigma_{j}(1)}\neq \rho_{j,\sigma_{j}(1)}\right)$.

Therefore, we have

\begin{equation}
\sum_{s_{j}=0}^{q^{l_{j}}_{j}-1}\omega_{q_{j}}^{s_{{j}}(\mu_{j,\sigma_{j}(1)}- \rho_{j,\sigma_{j}(1)})}=0.
\end{equation}

As a result, $G=0$. Hence from (<ref>), we have
$$\sum_{\theta\in \Theta} w_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,\mu}-(a_{t'}^{\theta})_{\delta,\rho}\right)}=0.$$

$\left(\textit{For all}~j\in \{1,2,\ldots,b\}, \mu_{j,\sigma_{j}(1)}=\rho_{j,\sigma_{j}(1)}\right)$.

Let us assume that $v$ is the smallest integer for which $\mu_{1,\sigma_{1}(v)}\neq \rho_{1,\sigma_{1}(v)}$.
Let us consider $\mu^{k_{1}}$ and $\rho^{k_{1}}$ be the integers and corresponding  associated vectors are $\phi\left({\mu}^{k_{1}}\right)=(\boldsymbol{\mu}_{1}^{k_{1}},\boldsymbol{\mu}_{2},\ldots,\boldsymbol{\mu}_{b})$ and $\phi\left({\rho}^{k_{1}}\right)=(\boldsymbol{\rho}_{1}^{k_{1}},\boldsymbol{\rho}_{2},\ldots,\boldsymbol{\rho}_{b})$, respectively, where, $\boldsymbol{\mu_{1}^{k_{1}}}$ and  $\boldsymbol{\rho_{1}^{k_{1}}}$  are the vectors which
differ from $\boldsymbol{\mu_{1}}$ and $\boldsymbol{\rho_{1}}$ only at the position $\sigma_{1}(v-1)$,  i.e.,

\begin{equation}
\begin{split}
\end{split}
\end{equation}

where $1\leq k_{1}\leq \mu_{1,\sigma_{1}(v-1)}$.
Similarly, let us consider $\mu^{k_{2}}$ and $\rho^{k_{2}}$ be the integers and whose associated vectors are $\phi\left({\mu}^{k_{2}}\right)=(\boldsymbol{\mu}_{1}^{k_{2}},\boldsymbol{\mu}_{2},\ldots,\boldsymbol{\mu}_{b})$ and $\phi\left({\rho}^{k_{2}}\right)=(\boldsymbol{\rho}_{1}^{k_{2}},\boldsymbol{\rho}_{2},\ldots,\boldsymbol{\rho}_{b})$ respectively, where $\boldsymbol\mu_{1}^{k_{2}}$ and  $\boldsymbol\rho_{1}^{k_{2}}$  are the vectors differ from $\boldsymbol\mu_{1}$ and $\boldsymbol\rho_{1}$ only at the position $\sigma_{1}(v-1)$,  i.e.,

\begin{equation}
\begin{split}
\end{split}
\end{equation}

where $\mu_{1,\sigma_{1}(v-1)}+1\leq k_{2}\leq q_{1}-1$. As $\delta=\gamma+u_{1}$ and $\rho=\mu+u_{2}$,
therefore, $(\delta,\rho^{k _{i}})=(\gamma,\mu^{k _{i}})+(u_{1},u_{2})$ for $1\leq i \leq 2$. By subtracting $(a_{t}^{\theta})_{\gamma,\mu^{k_{i}}}$ and $(a_{t'}^{\theta})_{\gamma,\rho^{k_{i}}}$,  we get,

\begin{equation}
\label{h1}
\begin{split}
%\sum_{i=1}^{a}\frac{\lambda}{p_{i}}r_{i}(\gamma_{i,\pi_{i}(1)}- \delta_{i,\pi_{i}(1)})\\
%&+\sum_{j=1}^{b}\frac{\lambda}{q_{j}}s_{j}(\mu^{k_{i}}_{j,\sigma_{j}(1)}- \rho^{k_{i}}_{j,\sigma_{j}(1)})+o+p,
\end{split}
\end{equation}

By subtracting (<ref>) and (<ref>), we get

\begin{equation}
\label{l}
\begin{split}
\end{split}
\end{equation}

By subtracting $f_{\gamma,\mu}$ with $f_{\gamma,\mu^{k_{1}}}$  and $f_{\delta,\rho}$ with $f_{\delta,\rho^{k_{1}}}$, we get

\begin{equation}
\label{i}
\begin{split}
\end{split}
\end{equation}

By subtracting $\left(f_{\gamma,\mu^{k_{1}}}-f_{\delta,\rho^{k_{1}}}\right)$ with $\left(f_{\gamma,\mu}-f_{\delta,\rho}\right)$,  we get
\begin{equation}
\label{j}
\begin{split}
\left(f_{\gamma,\mu^{k_{1}}}-f_{\delta,\rho^{k_{1}}}\right)-\left(f_{\gamma,\mu}-f_{\delta,\rho}\right)&=(f_{\gamma,\mu^{k_{1}}}-f_{\gamma,\mu})-(f_{\delta,\rho^{k_{1}}}-f_{\delta,\rho})\\
\end{split}
\end{equation}

From (<ref>) and (<ref>), we have

\begin{equation}
\label{das}
\begin{split}
\end{split}
\end{equation}

Similarly, by subtracting $f_{\gamma,\mu}$ with $f_{\gamma,\mu^{k_{2}}}$  and $f_{\delta,\rho}$ with $f_{\delta,\rho^{k_{2}}}$, we get

\begin{equation*}
\label{m}
\begin{split}
\end{split}
\end{equation*}

where ${\kappa}=(k_{2}-q_{1})c_{1,\sigma(v-1)}$. Again, by subtracting $ \left(f_{\gamma,\mu^{k_{2}}}-f_{\delta,\rho^{k_{2}}}\right)$ with $\left(f_{\gamma,\mu}-f_{\delta,\rho}\right)$,
we get
\begin{equation}
\label{n}
\begin{split}
\left(f_{\gamma,\mu^{k_{2}}}-f_{\delta,\rho^{k_{2}}}\right)-(f_{\gamma,\mu}-f_{\delta,\rho})&=\left(f_{\gamma,\mu^{k_{2}}}-f_{\gamma,\mu}\right)-\left(f_{\delta,\rho^{k_{2}}}-f_{\delta,\rho}\right)\\
\end{split}
\end{equation}

From (<ref>) and (<ref>), we have,

\begin{equation}
\label{narottam}
\begin{split}
\end{split}
\end{equation}

Adding (<ref>) and (<ref>), we have

\begin{equation}
\label{35}
\begin{split}
&\omega_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,\mu}-(a_{t'}^{\theta})_{ \delta,\rho}\right)}+\!\!\!\sum_{k_{1}=1}^{\mu_{1,\sigma_{1}(v-1)}}\omega_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,\mu^{k_{1}}}-(a_{t'}^{\theta})_{\delta,\rho^{k_{1}}}\right)}\\
\end{split}
\end{equation}

Since (<ref>) holds for all $\theta\in \Theta$, therefore,  taking summation on both side of (<ref>), we get

\begin{equation*}
\begin{split}
}-(a_{t'}^{\theta})_{\delta,\rho}\right)}\!+\!\!\sum_{\theta\in \Theta}\!\!\!\!\!\sum_{k_{1}=1}^{\mu_{1,\sigma_{1}(v-1)}}\omega_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,\mu^{k_{1}}}-(a_{t'}^{\theta})_{\delta,\rho^{k_{1}}}\right)}\\
&+\sum_{\theta\in \Theta}\sum_{k_{2}=1+\mu_{1,\sigma_{1}(v-1)}}^{q_{1}-1}\!\!\!\!\!\!\!\!\!\!\!\omega_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,\mu^{k_{2}}}-(a_{t'}^{\theta})_{\delta,\rho^{k_{2}}}\right)}=0.
\end{split}
\end{equation*}

Combining the above two subcases, we get $\boldsymbol{C}(\psi(G_{t}),\psi(G_{t'}))(u_{1},u_{2})=0$.
($t\neq t'$).

Without loss of generality, let us assume that $y_{j}\neq y'_{j}$. Now, we have

\begin{equation}
\label{17}
\begin{split}
&\sum_{\gamma=0}^{m-1}\sum_{\mu=0}^{n-1} w_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,\mu}-(a_{t'}^{\theta})_{\gamma,\mu}\right)}=AB.
\end{split}
\end{equation}

\begin{equation}
\begin{split}
\end{split}
\end{equation}

We have $\mu=\mu_{1}+\displaystyle\sum_{j=2}^{b}\mu_{j}\left(\prod_{j_{1}=1}^{j-1}q_{j_{1}}^{n_{j_{1}}}\right)$ where $0\leq \mu_{j}\leq q_{j}^{n_{j}}-1$ and the vector representation of $\mu_{j}$ with base $q_{j}$ and length $n_{j}$ is $\boldsymbol{\mu}_{j}$ where $\boldsymbol{\mu}_{j}=(\mu_{j,1},\mu_{j,2},\ldots,\mu_{j,n_{j}})\in \mathbb{A}_{q_{j}}^{n_{j}}$.

We define a mapping $h:\mathbb{A}_{q_{j}^{n_{j}}}\rightarrow \mathbb{A}_{q_{j}}$ given by $h(\mu_{j})=\mu_{j,n_{j}}$. The mapping is well defined beacuse $\boldsymbol{\mu}_{j}$ is the vector representation of $\mu_{j}$, and $\mu_{j,n_{j}}$ is the $n_{j}$th element of the vector  $\boldsymbol{\mu}_{j}$. Now with the help of $h$ and the permutation $\sigma_{j}$ we define a composite mapping $h'=\sigma \circ h:\mathbb{A}_{q_{j}^{n_{j}}}\rightarrow \mathbb{A}_{q_{j}}$ given by $h'(\mu_{j})=\sigma \circ h(\mu_{j})=\sigma(h(\mu_{j}))=\sigma(\mu_{j,n_{j}})=\mu_{j,\sigma_{j}(n_{j})}$. The mapping $h'$ is well defined because both the mapping $h$ and $\sigma_{j}$ are well defined. Let $\mathcal{F}=\{\boldsymbol{\mu}=\left(\mu_{1},\mu_{2},\ldots,\mu_{b}\right):0\leq\mu_{j}\leq q_{j}^{n_{j}}-1\}$ then the sum $B$ can be expressed as

\begin{equation}
B=\sum_{\left(\mu_{1},\mu_{2},\ldots,\mu_{b}\right)\in \mathcal{F}}\omega_{\lambda}^{\sum_{j=1}^{b}\frac{\lambda}{q_{j}}h'(\mu_{j})(y_{j}-y'_{j})},
\end{equation}

\begin{equation}
\label{18}
\begin{split}
B&=\sum_{\mu_{1}=0}^{q_{1}^{n_{1}}-1}\sum_{\mu_{2}=0}^{q_{2}^{n_{2}}-1}\ldots\sum_{\mu_{b}=0}^{q_{b}^{n_{b}}-1} \omega_{\lambda}^{\sum_{j=1}^{b}\frac{\lambda}{q_{j}}h'(\mu_{j})(y_{j}-y'_{j})}
\end{split}
\end{equation}

Now, we have

\begin{equation}
\label{19}
\begin{split}
\sum_{\mu_{j}=0}^{q_{j}^{n_{j}}-1}\!\!\!\omega_{q_{j}}^{h'(\mu_{j})(y_j-y'_j)}=q_{j}^{n_{j}-1}\!\!\!\!\!\!\!\sum_{\mu_{j,\sigma_{j}(n_{j})}=0}^{q_{j}-1}\!\!\!\omega_{q_{j}}^{\mu_{j,\sigma_{j}(n_{j})}(y_j-y'_j)}=0.
\end{split}
\end{equation}

Therefore, from (<ref>), (<ref>) and  (<ref>), we have

$$\sum_{\gamma=0}^{m}\sum_{\mu=0}^{n} w_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,\mu}-(a_{t'}^{\theta})_{\gamma,\mu}\right)}=0.$$

\begin{equation}
\label{2.d}
\begin{split}
\boldsymbol{C}(G_{t},G_{t})(0,0)
&=  \sum_{\theta\in \Theta}\sum_{\gamma=0}^{p_{1}^{m_1}p_{2}^{m_2}\ldots p_{a}^{m_{a}}-1}\sum_{\mu=0}^{q_{1}^{n_1}q_{2}^{n_2}\ldots q_{b}^{n_{b}}-1}w_{\lambda}^{\left((a_{t}^{\theta})_{\gamma,\mu}-(a_{t}^{\theta})_{\gamma,\mu}\right)}\\
&=p_{1}^{m_{1}+k_{1}}p_{2}^{m_{2}+k_{2}}\ldots p_{a}^{m_{a}+k_{a}}q_{1}^{n_{1}+l_{1}}q_{2}^{n_{2}+l_{2}}\ldots q_{b}^{n_{b}+l_{b}}.
\end{split}
\end{equation}

Combining the Subcase (i) and Subcase (ii), we have

\begin{equation}
\begin{split}
\boldsymbol{C}(G_{t},G_{t'})(0,0)
\begin{cases}
\prod_{i=1}^{a}p_{i}^{m_{i}+k_{i}}\prod_{j=1}^{b}q_{j}^{n_{j}+l_{j}},t= t':\\
0, t\neq t'.
\end{cases}
\end{split}
\end{equation}

For all the other cases, the proof follows exactly the same as Case 1. Hence,
combining all the above cases, we have

\begin{equation}
\begin{split}
\boldsymbol{C}(G_{t},G_{t'})(u_{1},u_{2})
\begin{cases}
\prod_{i=1}^{a}p_{i}^{m_{i}+k_{i}}\prod_{j=1}^{b}q_{j}^{n_{j}+l_{j}},
~~~ (u_{1},u_{2})=(0,0), t=t';\\
~~~ (u_{1},u_{2})\neq(0,0), t=t' ;\\
~~~~ t\neq t'.
\end{cases}
\end{split}
\end{equation}

Hence, the proof follows.

#1ISBN #1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1<https://doi.org/#1>et al.#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1<><#>1#1#1#1#1#1#1#1#1#1#1#1#1#1PreBibitemsHook

[1]

Golay, M.J.:
Multi-slit spectrometry.

[2]

Popovic, B.M.:
Synthesis of power efficient multitone signals with flat amplitude
IEEE Trans. Commun.

[3]

Davis, J.A.,
Jedwab, J.:
Peak-to-mean power control in OFDM, Golay complementary
sequences, and Reed-Muller codes.
IEEE Trans. Inf. Theory

[4]

Wang, Z.,
Parker, M.G.,
Gong, G.,
Wu, G.:
On the PMEPR of binary Golay sequences of length
IEEE Trans. Inf. Theory

[5]

Spasojevic, P.,
Georghiades, C.N.:
Complementary sequences for ISI channel estimation.
IEEE Trans. Inf. Theory

[6]

Pezeshki, A.,
Calderbank, A.R.,
Moran, W.,
Howard, S.D.:
Doppler resilient Golay complementary waveforms.
IEEE Trans. Inf. Theory

[7]

Tseng, C.-C.,
Liu, C.:
Complementary sets of sequences.
IEEE Trans. Inf. Theory

[8]

Schmidt, K.-U.:
Complementary sets, generalized Reed-Muller codes, and
power control for OFDM.
IEEE Trans. Inf. Theory

[9]

Chen, C.-Y.,
Wang, C.-H.,
Chao, C.-C., :
Complementary sets and Reed-Muller codes for
peak-to-average power ratio reduction in OFDM.
Lecture notes in computer science

[10]

Paterson, K.G.:
Generalized Reed-Muller codes and power control in OFDM
IEEE Trans. Inf. Theory

[11]

Suehiro, N.,
Hatori, M.:
N-shift cross-orthogonal sequences.
IEEE Trans. Inf. Theory

[12]

Rathinakumar, A.,
Chaturvedi, A.K.:
Complete mutually orthogonal Golay complementary sets from
Reed-Muller codes.
IEEE Trans. Inf. Theory

[13]

Das, S.,
Majhi, S.,
Budišin, S.,
Liu, Z.:
A new construction framework for polyphase complete complementary
codes with various lengths.
IEEE Trans. Signal Process.

[14]

Das, S.,
Majhi, S.,
Liu, Z.:
A novel class of complete complementary codes and their applications
for APU matrices.
IEEE Trans. Signal Process.

[15]

Sarkar, P.,
Liu, Z.,
Majhi, S.:
Multivariable function for new complete complementary codes with arbitrary
arXiv preprint arXiv:2102.10517

[16]

Das, S.,
Budišin, S.,
Majhi, S.,
Liu, Z.,
Guan, Y.L.:
A multiplier-free generator for polyphase complete complementary
IEEE Trans. Signal Process.

[17]

De Marziani, C.,
Urena, J.,
Hernández, Á.,
Mazo, M.,
Álvarez, F.J.,
Garcia, J.J.,
Donato, P.:
Modular architecture for efficient generation and correlation of
complementary set of sequences.
IEEE Trans. Signal Process.

[18]

Chen, H.-H.,
Yeh, J.-F.,
Suehiro, N.:
A multicarrier CDMA architecture based on orthogonal
complementary codes for new generations of wideband wireless communications.
IEEE Commun. Mag.

[19]

Liu, Z.,
Guan, Y.L.,
Chen, H.-H.:
Fractional-delay-resilient receiver design for interference-free
MC-CDMA communications based on complete complementary codes.
IEEE Trans. Wirel. Commun.

[20]

Kojima, T.,
Oizumi, A.,
Okayasu, K.,
Parampalli, U.:
An audio data hiding based on complete complementary codes and its
application to an evacuation guiding system.
In: IEEE Sixth International Workshop on Signal Design and Its
Applications in Communications,
pp. 118–121

[21]

Kojima, T.,
Tachikawa, T.,
Oizumi, A.,
Yamaguchi, Y.,
Parampalli, U.:
A disaster prevention broadcasting based on data hiding scheme using
complete complementary codes.
In: International Symposium on Information Theory and Its
pp. 45–49

[22]

Liu, T.,
Men, X.,
Li, Y.,
Chen, X.:
Constructions of 2-D Golay complementary array sets for MIMO
omnidirectional transmission.
IEEE Commun. Lett.

[23]

Pai, C.-Y.,
Chen, C.-Y.:
Two-dimensional Golay complementary array pairs/sets with bounded row
and column sequence PAPRs.
IEEE Trans. Commun.

[24]

Wang, Z.,
Ma, D.,
Gong, G.,
Xue, E.:
New construction of complementary sequence (or array) sets and
complete complementary codes.
IEEE Trans. Inf. Theory

[25]

Wang, Z.,
Gong, G.:
Constructions of complementary sequence sets and complete complementary codes
by 2-level autocorrelation sequences and permutation polynomials.
arXiv preprint arXiv:2005.05825

[26]

Zhang, C.,
Lin, X.,
Hatori, M.:
Novel two dimensional complementary sequences in ultra wideband
wireless communications.
In: IEEE Conference on Ultra Wideband Systems and Technologies, 2003,
pp. 398–402

[27]

Meng, X.,
Gao, X.,
Xia, X.-G.:
Omnidirectional precoding based transmission in massive MIMO
IEEE Trans. Commun.

[28]

Pai, C.-Y.,
Liu, Z.,
Zhao, Y.-Q.,
Huang, Z.-M.,
Chen, C.-Y.:
Designing two-dimensional complete complementary codes for
omnidirectional transmission in massive MIMO systems.
In: 2022 IEEE International Symposium on Information Theory (ISIT),
pp. 2285–2290

[29]

Zhao, Y.-Q.,
Pai, C.-Y.,
Huang, Z.-M.,
Liu, Z.:
Two-dimensional Golay complementary array sets with arbitrary lengths
for omnidirectional MIMO transmission.
arXiv preprint arXiv:2301.01225

[30]

Lu, A.-A.,
Gao, X.,
Meng, X.,
Xia, X.-G.:
Omnidirectional precoding for 3D massive MIMO with
uniform planar arrays.
IEEE Trans. Wirel. Commun.

[31]

Turcsany, M.,
Farkas, P.,
Duda, P.,
Kralovic, J.:
Performance evaluation of two-dimensional quasi orthogonal complete
complementary codes in fading channels.
In: IEEE Joint IST Workshop on Mobile Future, and the Symposium
on Trends in Communications. (SympoTIC)'.,
pp. 84–87

[32]

El-Khamy, S.E.,
Mokhtar, M.A.,
El-Ganainy, N.:
New techniques for image change and motion detection based on complete
complementary code arrays.
In: Proceedings of the IEEE Twenty-First National Radio Science
Conference (NRSC),
pp. 35–1

[33]

Farkas, P.,
Turcsány, M.:
Two-dimensional orthogonal complete complementary codes.
In: Joint IEEE 1st Workshop on Mobile Future and Symposium on
Trends in Communications (sympoTIC),
pp. 21–24

[34]

Dávideková, M.,
Farkaŝ, P.,
Ružickỳ, E.:
Generalized construction of two-dimensional complete complementary
In: 36th IEEE International Conference on Telecommunications
and Signal Processing (TSP),
pp. 747–750

[35]

Turcsany, M.,
Farkas, P.:
Two-dimensional quasi orthogonal complete complementary codes.
In: Joint 1st IEEE Workshop on Mobile Future and Symposium on
Trends in Communications (SympoTIC),
pp. 37–40

[36]

Das, S.,
Majhi, S.:
Two-dimensional Z-complementary array code sets based on
matrices of generating polynomials.
IEEE Trans. Signal Process.

[37]

Chen, C.-Y.,
Wang, C.-H.,
Chao, C.-C.:
Complete complementary codes and generalized Reed-Muller
IEEE Commun. Lett.

[38]

Pai, C.-Y.,
Chen, C.-Y.,
Liu, Z.:
Designing two-dimensional complete complementary codes for omnidirectional
transmission in massive MIMO systems.
arXiv preprint arXiv:2108.06863

[39]

Wang, Z.,
Ma, D.,
Gong, G.,
Xue, E.:
New construction of complementary sequence (or array) sets and
complete complementary codes.
IEEE Trans. Inf. Theory

[40]

Ghosh, G.,
Majhi, S.,
Paul, S.:
Construction of optimal binary z-complementary code sets with new
lengths using generalized boolean function.
Cryptogr. Commun.

[41]

Shen, B.,
Meng, H.,
Yang, Y.,
Zhou, Z.:
New constructions of z-complementary code sets and mutually orthogonal
complementary sequence sets.
Des. Codes Cryptogr.

[42]

Golay, M.:
Complementary series.
IRE transactions on information theory

[43]

Ghosh, G.,
Majhi, S.,
Sarkar, P.,
Upadhyay, A.K.:
Direct construction of optimal Z-complementary code sets with even
lengths by using generalized Boolean functions.
IEEE Signal Process. Lett.

[44]

Ghosh, G.,
Majhi, S.,
Paul, S.:
A direct construction of optimal 2d-zcacs with flexible array size and
large set size.
Cryptogr. Commun.


#  Exploration of Hyperdimensional Computing Strategies
for Enhanced Learning on Epileptic Seizure Detection ††thanks: This work has
been partially supported by the ML-Edge Swiss National Science Foundation
(NSF) Research project (GA No. 200020182009/1), and the PEDESITE Swiss NSF
Sinergia project (GA No. SCRSII5 193813/1). ††thanks: 1 U. Pale, T. Teijeiro,
and D. Atienza are with the Embedded Systems Laboratory (ESL) of Swiss Federal
Institute of Technology Lausanne (EPFL), Switzerland.
{una.pale, tomas.teijeiro<EMAIL_ADDRESS>
Una Pale, Tomas Teijeiro, and David Atienza1

###### Abstract

Wearable and unobtrusive monitoring and prediction of epileptic seizures has
the potential to significantly increase the life quality of patients, but is
still an unreached goal due to challenges of real-time detection and wearable
devices design. Hyperdimensional (HD) computing has evolved in recent years as
a new promising machine learning approach, especially when talking about
wearable applications. But in the case of epilepsy detection, standard HD
computing is not performing at the level of other state-of-the-art algorithms.
This could be due to the inherent complexity of the seizures and their
signatures in different biosignals, such as the electroencephalogram (EEG),
the highly personalized nature, and the disbalance of seizure and non-seizure
instances. In the literature, different strategies for improved learning of HD
computing have been proposed, such as iterative (multi-pass) learning, multi-
centroid learning and learning with sample weight ("OnlineHD"). Yet, most of
them have not been tested on the challenging task of epileptic seizure
detection, and it stays unclear whether they can increase the HD computing
performance to the level of the current state-of-the-art algorithms, such as
random forests. Thus, in this paper, we implement different learning
strategies and assess their performance on an individual basis, or in
combination, regarding detection performance and memory and computational
requirements. Results show that the best-performing algorithm, which is a
combination of multi-centroid and multi-pass, can indeed reach the performance
of the random forest model on a highly unbalanced dataset imitating a real-
life epileptic seizure detection application.

##  I Introduction

Hyperdimensional (HD) computing is a novel machine learning paradigm inspired
by neuroscience that has attracted lots of interest in the last years in many
different domains. For example, in the biomedical domain it has been used for
emotion recognition from GSR (galvanic-skin response), electrocardiogram (ECG)
and electroencephalogram (EEG) [1], EEG error-related potentials detection
[2], electromyogram (EMG), gesture recognition [3], epileptic seizure
detection from EEG [4], etc. HD computing is an interesting alternative to
standard machine learning approaches and it offers opportunities for
continuous online learning [5, 6], semi-supervised [7], distributed [8] or
multi-centroid learning [9]. Furthermore, it is highly efficient due to its
lower energy and memory requirements [10, 4, 11], and offers opportunities for
design novel algorithms [7, 12, 9], which makes it interesting for wearable
devices and applications. Thus, hardware implementations and optimizations for
HD computing are a common topic of recent works, and show promising results
[13, 14].

In recent years, a lot of effort has been put into designing wearable devices
for patient monitoring, with detection and prediction capabilities. Epilepsy
monitoring and real-time seizures detection are one of such applications.
Epilepsy is a chronic neurological disorder characterized by the unpredictable
occurrence of seizures, affecting a significant portion of the world
population (0.6 to 0.8%) [15]. It attracts a lot of medical attention as,
despite pharmacological treatments, one-third of patients still suffer from
seizures [16]. The unexpected occurrence of seizures imposes serious health
risks and many restrictions on daily life. As such, solutions that would allow
continuous unobtrusive monitoring and a reliable detection (and ideally
prediction) of seizures will be of high importance. Such technology will also
be instrumental in deepening knowledge and designing novel treatments.

Figure 1: Schematic of the HD computation workflow. Adapted from [9].

Epilepsy is not only challenging from a medical perspective related to its
origins, treatment, and prevention but also from the engineering perspective
of detection from the recorded physiological signals. Even from multi-channel
EEG recordings, it is a big challenge to design a lightweight, real-time
detection system with (almost) no false positive alarms. Reasons are multi-
fold: huge disbalance in data distribution (i.e., the amount of seizure vs.
non-seizure data), scarcity of training data available, highly variable and
personalized morphologies of signals, as well as computational and memory
requirements of algorithms used.

HD computing comes as an interesting option due to the aforementioned positive
aspects. Despite initial optimistic performance reported in the literature
[10, 4], HD computing is not performing so well when taking into account more
realistic data distributions [9]. As will be shown in this paper, when using
ten times more non-seizure data, performance is significantly reduced with the
HD computing approach when compared to the Random Forest model.

At the same time, various strategies have been proposed in the literature to
improve learning and detection with HD computing. For example, the iterative
learning approach [17] has been proposed instead of single-pass learning, but
it has not been fully explored yet for epilepsy. Further, the multi-centroid
approach [9], which is appropriate for the high variability of seizure and
non-seizure signals, also showed significant improvements when compared to the
standard single-centroid approach. In the end, the "OnlineHD" approach [12]
also works in a single-pass manner but multiplies vectors with a weight factor
depending on the novelty each data window brings. It aims, as well as both
iterative and multi-centroid approaches, to overcome the dominance problem of
more common patterns in the final prototype vectors. This approach also has
not yet been tested for the challenging task of epileptic seizure detection.

Thus, following the challenges epileptic seizure detection poses, we aim to
test and compare various algorithms for HD computing improvement on this
problem, hoping to achieve a performance that would be acceptable by users. In
this work, we contribute to the state of the art in the following manner:

  * •

We systematically compare the performance of the standard single-pass one-
centroid HD computing approach with the classical random forest learning model
on epileptic seizure detection with more realistic data distributions and show
that HD computing still has a performance gap to cross.

  * •

We then implement several existing proposals for improved HD computing
learning: iterative (multi-pass) one-centroid learning, multi-centroid single-
pass as well as "OnlineHD". Not all of them have been yet tested for epileptic
seizure detection.

  * •

We combine two strategies to test whether multi-centroid with multi-pass
approach can bring additional improvements to the performance.

  * •

Finally, we compare the mentioned strategies in terms of their performance and
their memory and computational requirements with a wearable implementation in
mind.

##  II HD computing learning strategies

###  II-A Traditional HD computing workflow

HD computing is based on computations with very long and redundant (mostly
binary) vectors (usually >10000 dimensions), which represent information in a
condensed and distributed way. Calculating and learning with vectors is based
on a few specific algebraic properties. The two most important ones are: 1)
any randomly chosen pair of vectors are nearly orthogonal and 2) when summing
two or more vectors, the result will be with high probability more similar to
the added vectors than to any other randomly chosen vector. These properties
are crucial for learning and inference.

Figure 2: Schematic of different improvements on HD computing learning: A.
Standard single-pass single-centroid HD learning, B. Multi-pass (iterative)
single-centroid learning, C. Multi-centroid single-pass learning and D.
OnlineHD single-pass learning.

More specifically, HD learning starts by encoding data and its relations to HD
vectors. As illustrated in Fig. 1, baseline vectors representing different
scalar values, features and channels are combined during the encoding stage to
get one vector. This vector represents that specific data sample (window),
instead of a feature set, as in other ML approaches. The training phase is
relatively simple and consists of summing (bundling) all vectors from the same
class to one prototype vector representing each class. Summation of the
vectors is usually done by bit-wise summation, followed by majority voting
normalization. In the end, for inference, a vector representing the current
data sample is compared with prototype vectors of all classes, and the label
of the most similar one is given as output. The most common measure of
similarity is the Hamming distance in the case of binary HD vectors (elements
are 0 or 1), but cosine or dot products can also be used in the case of
integer or float HD vectors.

Traditionally, HD computing classifiers have been based on a single-pass
learning and a single-centroid model vector per class as an outcome of
learning, as illustrated in a simplified way in Fig. 2A. However, many current
machine learning problems are challenging due to the immense complexity and
variability of patterns in data, especially when compared to the amount of
training data available. Epilepsy is one such case, where electroencephalogram
(EEG) signatures of epileptic seizures are highly unique and variable among
people, brain states, and time instances, especially if they are grouped under
only two given labels (seizure and non-seizure). Non-seizure data represents
many different brain states, such as awake, sleeping, physical or mental
effort conditions, which can significantly increase data variability grouped
under one class. But, all of these states have their own brain signatures that
need to be learned. In the standard single-pass one-centroid HD approach, all
data samples are equally important during learning, leading to more common
patterns dominating the prototype vectors. This means that less common
patterns could be potentially under-represented and wrongly predicted even on
the same training data.

Thus, in this work, we explore and compare different proposed strategies that
can tackle this problem: iterative (multi-pass) learning, multi-centroid,
multi-centroid with iterative learning, and "OnlineHD". We find epileptic
seizure detection a perfect test case for this due to the variability of
patterns in both seizure and non-seizure classes, the inherent disbalance in
the amount of seizure/non-seizure recordings, and a generally relatively small
size of training data.

###  II-B Multi-pass learning approach

The first approach aimed at overcoming the problem of dominance of common
patterns is iterative learning [17]. We call it also multi-pass learning to
clarify its relation to other approaches. In the first pass of learning, all
samples are added to the corresponding data class, the same as in traditional
single-pass learning. Next, an iterative process of multiple passes starts
wherein each pass for each sample prediction is given based on the learned
prototype vectors from the previous pass. In case of a wrong prediction, this
sample is added again to the correct class. In this way, less common patterns
that got under-represented are strengthened by adding them multiple times.
After each pass, performance is evaluated on the same training set. The multi-
pass process stops once there is no more significant improvement in
performance. The workflow of iterative learning is shown in Fig. 2B.

This approach can be improved if the vector of a wrongly classified sample is
added to the correct class prototype vector and also subtracted from the other
ones. Further, before adding/subtracting vectors to prototype vectors, it can
be multiplied with a factor (learning rate) to increase its weight. In [17],
authors test different values of multiplication factors (learning rate) and
their influence on the number of iterations needed to achieve a stable
performance and the final performance itself. They show that small factors
lead to clear performance increase but require many iterations. On the other
side, too big multiplication factors lead to fluctuations in performance, thus
potentially never converging and never finishing the training procedure.
Hence, they also tested an adaptable threshold strategy to exploit the
advantages of a large learning rate at the beginning and a small learning rate
in the end for fine-tuning the performance. They tested these approaches on
four different classification applications and showed performance improvement
as well as improved energy and computational complexity.

Unfortunately, none of the applications used for testing was epileptic seizure
detection, and one could argue that the chosen problems were less complex and
challenging than the epileptic seizure one. Further, in order to be able to
compare properly with other approaches (explained below), we implemented this
approach as well.

###  II-C Multi-centroid learning approach

Another way to not let the frequency of certain patterns make them dominate
the model, leading to under-representation and worse classification of less
common patterns, is to treat them as separate sub-classes (centroids) of the
same class. In this way, they are not all accumulated to the same prototype
vectors, but it is possible to have multiple vectors representing the same
class. This approach was explored in [9] and has shown to significantly
improve epileptic seizure detection. Balanced datasets, with the same amount
of seizure and non-seizure data, are commonly used, but are not realistic in
real-life applications. Thus, authors compared performance also on unbalanced
datasets with more non-seizure than seizure data. The more unbalanced the
dataset was, the bigger the performance improvement when compared to the
single-centroid approach.

The workflow of the approach is illustrated in Fig. 2C. This approach is
interesting as it allows the semi-supervised creation of an unlimited number
of centroids (sub-classes) of each class. Namely, if the current sample vector
is more similar to the wrong class than any correct sub-class (meaning that it
would be wrongly classified), a new subclass of the correct class is created.
This process can sometimes lead to many sub-classes where some have only a few
samples that contributed (were added) to it. Hence, in the last step of the
workflow, the number of sub-classes is reduced by either removing the less
common sub-classes or clustering them with the most similar sub-classes
sharing the same label. This step significantly reduces the number of sub-
classes and thus the memory required to store them while still achieving
higher performance by allowing several centroids. When compared to the
iterative multi-pass approach, this approach is more memory-consuming but
should be faster due to the single-pass approach. This is the motivation to
compare those two approaches in more detail.

Furthermore, it is interesting to test whether combining iterative and multi-
centroid approaches could further improve performance. Namely, the multi-
centroid and multi-pass approach would consist of the first step with one pass
of multi-centroid training and then passing several times through the training
data to fine-tune the exact centroid values.

###  II-D Weighted learning approach

Following the data-driven learning rate idea from [17], in [12], the multi-
pass approach is replaced with single-pass to reduce the training costs. More
specifically, as shown in Fig. 2D., a naive accumulation of equally important
samples is replaced by using the weighting approach before adding the current
vector to the prototype vectors. The weight is defined by the similarity of
the current vector to the current prototype vectors; the higher the
similarity, the lower the weight. This approach identifies the most dominating
patterns and lowers model saturation by them. In [12], authors compared this
so called "OnlineHD" approach in performance to the previously described
iterative approach. Comparable accuracy was achieved on several different
classification problems.

This approach should be as memory-consuming as the traditional one-centroid
single-pass, but might be more time-consuming. This is due to the need to
update continuously and normalize the prototype vector after adding each
training sample (and not at the end of the pass as in other approaches). It is
interesting to analyze the performance of this approach and compare it to the
two previously proposed methods for preventing the under-representation of
less common patterns.

##  III Experimental Setup

Epilepsy is a highly challenging and relevant problem for which science and
technology have not yet proposed a good wearable solution for monitoring,
detecting, and even predicting seizures to increase patients’ quality of life.
Thus, we focus on testing HD learning strategies on this use-case with the
final goal of reaching a performance that could be acceptable by users.

###  III-A Databases

CHB-MIT [18, 19] is a widely known and publicly available database that
contains long EEG (electrocardiogram) recordings that can be used to test
algorithms not only in a balanced manner (same amount of seizure and non-
seizure) but also in more realistic data balance. It contains scalp EEG
recordings from 24 subjects, and includes 183 seizures, with an average of 7.6
$\pm$ 5.8 seizures per subject. To standardize the experiment, we use the 18
channels from the international 10-20 bipolar montage that are common to all
patients.

From the raw database, we prepared a dataset that contains ten times more non-
seizure data than seizure data to be closer to a more real-life data balance.
The main reason to avoid a balanced scenario, which is common in many works in
the literature, is that it can lead to a highly overestimated performance, not
achievable during the continuous monitoring with a wearable device [9].

Non-seizure segments were chosen randomly from available non-seizure data, but
excluding data 1 min before and 15 min after a seizure. This data might
contain ictal patterns and thus make classes less separable. In reality,
performance on this data segment would not be very relevant for the subject as
seizures would be either detected slightly earlier (which could be interpreted
as prediction) or seizure detection would last longer, which is not critical
for the patient either. Moreover, neurologists often have difficulties in
defining the exact end of a seizure, and thus many databases do not even have
labeled end of the seizure.

###  III-B Feature extraction and mapping to HD vectors

In this work, we use 46 features as used in [9] to make it comparable, as we
use the same database and share one of the approaches. Most of the features
were based on [20] which contains eight frequency spectrum features, 27
entropy-based features and mean amplitude value. Entropy-based features
contain sample, permutation, Renyi, Shannon, and Tsallis entropies. Frequency-
domain features present power spectral density and the relative power in the
five common brain wave frequency bands; delta: [0.5-4] Hz, theta: [4-8] Hz,
alpha: [8-12] Hz, beta: [12-30] Hz, gamma: [30-45] Hz, and a low-frequency
component ([0-0.5] Hz). These features are considered medically relevant for
detecting seizures [21].

###  III-C Validation

Training and evaluation are performed in a personalized manner. This is due to
the subject-specific nature of epileptic seizures and their signal patterns.
More specifically, for each subject and seizure, data is preprocessed to
contain 10 times more randomly selected non-seizure data and saved to an
individual file. Then leave-one-seizure-out cross-validation is performed,
where HD models are trained on all but one file (containing one seizure each).
Final performances reported per subject are the average of all cross-
validation iterations.

####  III-C1 Performance Evaluation

We quantify prediction performance with two measures to increase the
interpretability of performance, as proposed in [22]. This is an ongoing
discussion that started with [23] and [24]. Here, more specifically, we
measure performance on the 1) episode level and on the 2) seizure duration
level. The episode level performance cares if seizure episodes are correctly
detected, but not necessarily that the full duration of seizure and the start
and end time points are correctly classified. It treats predictions and true
labels as blocks of 1’s and 0’s and evaluates on a level of these blocks. The
duration level performance cares about the correct prediction of each time-
point, also meaning that seizures need to be correctly classified during their
whole duration. This metric can sometimes be challenging to use due to the
difficulty in labeling the true start and end of a seizure. Indeed, it might
not be possible to achieve 100% accuracy. Thus, we use both metrics to give us
a better insight into the operation of the proposed algorithms.

For both levels, sensitivity, precision, and F1 score are calculated.
Sensitivity or true positive rate $TPR$ is calculated as $TP/(TP+FN)$, while
precision or positive predictive value or $PPV$ is calculated as $TP/(TP+FP)$.
F1 score is calculated as the harmonic mean of sensitivity and precision:
$2*TPR*PPV/(TPR+PPV)$. Finally, to have a single measure for easier comparison
of methods, we calculate the mean value of the F1 score for episodes ($F1E$)
and duration($F1D$) as $F1DEmean$.

####  III-C2 Label post-processing

The raw predictions from the classifier can show unrealistic behavior for
dynamics of epileptic seizures (e.g., seizures lasting only a few seconds or
separate seizures that are a few seconds apart). Thus, we utilize time
information to perform label post-processing, which consists of two steps. In
the first step, time information of the signal is exploited to smooth the
predictions by going through the predicted labels with a moving average window
of a certain size $SW_{len}$ (5s) and then performing majority voting. In the
second step, seizures closer than 30s are merged together into one seizure.

####  III-C3 Statistical analysis

Due to the high performance variability between subjects, we perform
statistical analyses to compare different strategies. We compare each learning
approach with the traditional single-pass one-centroid approach using the
Wilcoxon statistical test. It compares the performance of two paired groups,
and we report the p-value.

Finally, the code and data required to reproduce the presented results are
available online as open-
source111https://c4science.ch/source/LearningImprovForHDcomputingOnEpilepsy/.

Figure 3: Performance of random forest as a benchmark for HD computing
performance. Performance is shown for both episode and duration level,
including sensitivity, precision and F1 score values. In the end, the mean
value of F1 for episodes and duration is calculated.

##  IV Experimental Results

###  IV-A Standard HD computing learning

Standard, traditional HD computing learning is single-pass single-centroid per
class, where each sample is equally important. As explained previously, it
consists of simple summing up (and normalizing in the end) all vectors coming
from the same class to get prototype vectors of the class.

Fig. 3 shows a performance comparison of the standard HD computing approach
with Random Forest performance as a benchmark. The random forest contained up
to 100 trees and was trained on the same dataset, with identical preparation,
train-test split, and post-processing as the HD computing approach. Here we
see that HD computing is performing significantly worse than the random forest
approach. TPR or sensitivity stays equally good with the HD approach, meaning
that all seizures are usually detected, but a significant drop is perceived in
PPV (precision), which means that many false positives occur when using HD
computing. For F1 score for episodes the drop in the mean performance of all
subjects is 25.7%, while for F1 duration score is 16.9%. Thus, in the next
experiments, we test if improved learning strategies of HD learning can help
resolve this problem and reach the performance of random forest.

Figure 4: Iterative (multi-pass) approach compared to single-pass approach.
The average number of iterations, percentage of data that had to be re-added
and performances on the test set are shown.

###  IV-B Multi-pass (iterative) learning

In Fig. 4, two versions of the multi-pass approach are compared with the
standard single-pass approach. In one approach, when the current data window
would be wrongly classified, it was just added to the correct class again
($2C_{+}$), whereas in another approach, it was also subtracted from the wrong
class ($2C_{+-}$). The average number of iterations, percentage of re-added
data, and prediction performance are analyzed and shown as a distribution over
all subjects. It can be observed that on average, it took around 13 passes for
$2C_{+}$ and 12 passes for $2C_{+-}$ to get a stable performance value. The
amount of data that had to be re-added on average was 24% for $2C_{+}$ and 16%
for $2C_{+-}$. Interestingly, for $2C_{+-}$ model, the variability between
patients is smaller than for the $2C_{+}$ model. Concerning the performance,
the F1 score for episodes ($F1_{E}$) and for duration ($F1_{D}$) as well as
their mean ($F1_{DE}$) is shown. Both multi-pass ($2C_{+}$ and $2C_{+-}$)
approaches have significantly higher performance when compared to single-pass
($2C$) approach ($p=1.63e^{-5}$ for $2C_{+}$ and $p=2.6e^{-2}$ for $2C_{+-}$
for mean of $F1_{E}$ and $F1_{D}$). There is no significant difference in
$F1_{DE}$ performance between between using either the $2C_{+}$ or $2C_{+-}$
approach ($p=0.74$).

###  IV-C Multi-centroid learning

Fig. 5 shows the performance of several versions of a multi-centroid approach
when compared to a traditional single-centroid one. As described in [9] first
step of the multi-centroid approach is to allow an unlimited creation of sub-
classes ($MC$), after which the essential step of removing unnecessary sub-
classes follows. This can be done either by simply removing the least common
centroids of each class ($MCr$) or clustering ($MCc$) least common centroids
with the closest same class ones. Here, results for only $MCr$ approach are
showed as they were slightly better and it is lighter to implement.

Results show that the first step of $MC$ approach creates, in average for all
subjects, 18 sub-classes for seizure and 22 for non-seizure. The approach of
removing least populated classes ($MCr$) results in, on average, five seizure
and six non-seizure sub-classes. Variability is very high between subjects and
is within [6-117] sub-classes (summed for seizure and non-seizure) after the
first $MC$ step, and is reduced to range [3-26] after $MCr$ step. Going from
single-centroid to multi-centroid significantly improves the performance on
both train and test sets. More specifically, for the test set the average
performance of all subjects is increased from 61% to 76% for mean value of F1
score for duration and episodes. It is important to note that after the second
step of reducing the number of sub-classes, despite a significant reduction in
the number of sub-classes, the performance is not significantly reduced for
the $MCr$ approach.

Next, we tested whether performing additional steps of iterative learning on
centroids decided after $MCr$ can lead to additional performance improvements.
Thus, the number of centroids (and their initial structure) was fixed based on
the previous step, but then by passing several times through the training
dataset, we allowed slight fine-tuning of the centroids. Although additional
performance improvement can be obtained in the training set, but not in the
test set. We believe this is due to overfitting.

Figure 5: Multi-centroid learning approach is compared to traditional single-
centroid approach. The average number of centroids per seizure and non-seizure
as well as performance is shown.

###  IV-D Weighted learning

The weighted approach, also called "OnlineHD" [12], is an alternative to the
multi-pass and multi-centroid approaches. In Fig. 6, the distribution of the
weights for seizure and non-seizure is shown. The average values of weights
are between 0.1 and 0.2, meaning that between 80% and 90% of the bits in
vectors were identical when adding a new sample. For some subjects, such as
subjects 5, 9, 10, 14, 19 and 24, values are similar for both seizure and non-
seizure, while for some subjects, seizure values are larger than for non-
seizure, meaning that they were bringing more novelty. For a few subjects
(e.g., for subj 1 and 23), it was the opposite, i.e., non-seizure data brought
more novelty.

Finally, when comparing performance, we compared two versions of weighted
learning. The first one where the weighed sample is added only to the correct
class ($On_{+}$) and and a second one, similarly to multi-pass, where it is
also subtracted from the wrong class if it would be wrongly predicted
($On_{+-}$). Performance results in Fig. 6 show that just weighted adding
($On_{+}$) does not significantly improve results on the test set, but that
adding and subtracting does help significantly.

Figure 6: Weighted learning approach is compared to the traditional single-
centroid approach. The distribution of the weights for seizure and non-seizure
is shown for each subject on the upper plot. The lower plot shows the
performance when comparing two weighted approaches with the traditional (non-
weighted) approach.  Figure 7: All learning approaches tested in this work are
compared together regarding seizure detection performance. Only one measure
combining all together is shown (mean of F1 score for episodes and duration).
Results on the test set are shown, where boxplots represent a distribution of
performance over all subjects.

###  IV-E Comparison of all tested learning strategies

Finally, in Fig. 7, we compare the seizure detection performance for all the
studied strategies. The mean of F1 score for duration and episodes is shown as
a distribution over all subjects. Again, the performance of standard HD
computing ($2C$) with single-pass and single-centroid leads to a significantly
lower performance and higher inter-subject variability than the random forest
($RF$) approach. Performing iterative (multi-pass) learning improves the
performance both for training and test sets, but a larger improvement is
possible when both adding the current data vector to the correct class and
subtracting it from the wrong class ($2C_{+-}$) than only adding it
($2C_{+}$), as shown previously in the literature.

The multi-centroid approach ($MC$, $MCr$) also leads to a significantly
improved performance (with respect to $2C$) but not yet achieving the level of
random forest. If centroids are then fine-tuned through several passes of
multi-pass learning ($MCri$), performance is improved even more. In that case,
there is no significant difference between multi-centroid multi-pass approach
with removal of less common classes ($MCri$) with respect to random forest
($p=0.18$).

The "OnlineHD" approach also leads to a significant improvement in
performance, but only when using the approach of adding and subtracting
vectors from correct and wrong classes, respectively. However, it did not
reach the level of no significant difference with random forest ($p=0.013$).

These results show that with additional improvements to standard HD computing,
it is possible to achieve a performance as good as with random forests for the
seizure detection problem. Nevertheless, Fig. 7 highlights that variance
between subjects is high, especially for HD, and there is space for further
improvements of HD performance.

###  IV-F Memory and computational requirements

Figure 8: Comparison of different learning approaches with respect to the time
needed for calculation of prototype vectors and memory required for storing
them.

Finally, we analyse also the memory and computational complexity of the
different learning strategies. Memory is calculated as the memory needed to
store all the prototype vectors. In Fig. 8, the relative amount of memory is
shown to compare different approaches. Multi-centroid learning ($MC$)
obviously requires the largest amount of memory, but this is reduced to less
than half after optimization steps of removing unnecessary centroids in the
second step of the algorithm ($MCr$). This is still up to 5 times more than
the standard single-centroid "onlineHD" ($On+$ or $On_{+-}$) approach.

In order to quantify computational complexity we analysed the time needed to
perform each type of training using the python implementationrunning on a
single-core. We evaluate the relative time between learning strategies. Time
was measured as an average over all subjects and all cross-validation
iterations. As expected, iterative, multi-pass learning ($2C_{+}$ or
$2C_{+-}$) is significantly more complex (>30x time for $2C$), due to the need
for evaluation of performance after each pass of training. Multi-centroid
approach in its first step ($MC$) is not extremely time consuming ($\sim$5x
time for $2C$), but the second step ($Mrc$ ) of optimizing the number of
centroids increases the computational complexity (>35x time for $2C$). In the
end, multi-centroid and multi-pass ($MCri$ ) is understandably the most time
and computationally intensive ($\sim$85x time for $2C$). Interestingly,
"onlineHD" ($On_{+}$ or $On_{+-}$) learning, even though more complex than
single-pass, is significantly more lightweight than the multi-pass or multi-
centroid approaches (only $\sim$2x time for $2C$).

##  V Conclusion

In this work, we have investigated and compared the characteristics of
different HD computing strategies in the context of epileptic seizure
detection with respect to more established approaches such as random forest.
In particular, our results have shown a significantly lower performance of the
standard HD approach compared to the random forest one, mainly due to many
false-positive seizure detections. Moreover, enhancing this standard HD
approach by using multi-pass, multi-centroid learning and their combination
improves this performance. However, only their combination reaches a
performance level comparable to the random forest. Then, we tested a single-
pass training approach with sample importance ("OnlineHD") and it
significantly improved the performance. Furthermore, this approach is much
more memory and computationally friendly when compared to a multi-pass
approach, which takes more time to train, or when compared to a multi-centroid
approach, which requires more memory. Our analyses and results have proven the
applicability of HD computing for real-life epileptic seizure detection.

## References

  * [1] En-Jui Chang et al.  “Hyperdimensional Computing-based Multimodality Emotion Recognition with Physiological Signals”  In _AICAS_ , 2019  DOI: 10.1109/AICAS.2019.8771622
  * [2] Abbas Rahimi et al.  “Hyperdimensional Computing for Blind and One-Shot Classification of EEG Error-Related Potentials”  In _MNET_ 25.5, 2020, pp. 1958–1969  DOI: 10.1007/s11036-017-0942-6
  * [3] A. Rahimi et al.  “Hyperdimensional biosignal processing: A case study for EMG-based hand gesture recognition”  In _ICRC_ , 2016, pp. 1–8  DOI: 10.1109/ICRC.2016.7738683
  * [4] F. Asgarinejad et al.  “Detection of Epileptic Seizures from Surface EEG Using Hyperdimensional Computing” ISSN: 2694-0604  In _EMBC_ , 2020, pp. 536–540  DOI: 10.1109/EMBC44109.2020.9175328
  * [5] Ali Moin et al.  “A wearable biosensing system with in-sensor adaptive machine learning for hand gesture recognition”  In _Nature Electronics_ 4.1, 2021, pp. 54–63  DOI: 10.1038/s41928-020-00510-8
  * [6] Simone Benatti et al.  “Online Learning and Classification of EMG-Based Gestures on a Parallel Ultra-Low Power Platform Using Hyperdimensional Computing”  In _IEEE T-BioCAS_ 13.3, 2019, pp. 516–528  DOI: 10.1109/TBCAS.2019.2914476
  * [7] Mohsen Imani et al.  “SemiHD: Semi-Supervised Learning Using Hyperdimensional Computing” ISSN: 1558-2434  In _ICCAD_ , 2019, pp. 1–8  DOI: 10.1109/ICCAD45719.2019.8942165
  * [8] Mohsen Imani et al.  “A Framework for Collaborative Learning in Secure High-Dimensional Space” ISSN: 2159-6190  In _CLOUD_ , 2019, pp. 435–446  DOI: 10.1109/CLOUD.2019.00076
  * [9] Una Pale et al.  “Multi-Centroid Hyperdimensional Computing Approach for Epileptic Seizure Detection” arXiv: 2111.08463  In _arXiv:2111.08463 [cs, eess]_ , 2021  URL: http://arxiv.org/abs/2111.08463
  * [10] Alessio Burrello et al.  “An Ensemble of Hyperdimensional Classifiers: Hardware-Friendly Short-Latency Seizure Detection With Automatic iEEG Electrode Selection” Conference Name: IEEE Journal of Biomedical and Health Informatics  In _IEEE JBHI_ 25.4, 2021, pp. 935–946  DOI: 10.1109/JBHI.2020.3022211
  * [11] Mohsen Imani et al.  “SparseHD: Algorithm-Hardware Co-optimization for Efficient High-Dimensional Computing” ISSN: 2576-2621  In _2019 IEEE FCCM_ , 2019, pp. 190–198  DOI: 10.1109/FCCM.2019.00034
  * [12] Alejandro Hernández-Cano et al.  “Real-Time and Robust Hyperdimensional Classification”  In _GLSVLSI 2021_ , pp. 397–402  DOI: 10.1145/3453688.3461749
  * [13] Sahand Salamat et al.  “F5-HD: Fast Flexible FPGA-based Framework for Refreshing Hyperdimensional Computing”  In _2019 ACM SIGDA_ , 2019, pp. 53–62  DOI: 10.1145/3289602.3293913
  * [14] Saransh Gupta et al.  “FELIX: Fast and Energy-Efficient Logic in Memory” ISSN: 1558-2434  In _2018 ICCAD_ , 2018, pp. 1–7  DOI: 10.1145/3240765.3240811
  * [15] Florian Mormann et al.  “Seizure prediction: the long and winding road”  In _Brain_ 130.Pt 2, 2007, pp. 314–333  DOI: 10.1093/brain/awl241
  * [16] Dieter Schmidt and Matti Sillanpää  “Evidence-based review on the natural history of the epilepsies”  In _Curr. Opin. Neurol._ 25.2, 2012, pp. 159–163  DOI: 10.1097/WCO.0b013e3283507e73
  * [17] M. Imani et al.  “AdaptHD: Adaptive Efficient Training for Brain-Inspired Hyperdimensional Computing”  In _BioCAS_ , 2019  DOI: 10.1109/BIOCAS.2019.8918974
  * [18] Ali Hossam Shoeb  “Application of machine learning to epileptic seizure onset detection and treatment”, 2009  URL: https://dspace.mit.edu/handle/1721.1/54669
  * [19] Goldberger Ary L. et al.  “PhysioBank, PhysioToolkit, and PhysioNet” Publisher: American Heart Association  In _Circulation_ 101.23, 2000, pp. e215–e220  DOI: 10.1161/01.CIR.101.23.e215
  * [20] Renato Zanetti et al.  “Robust Epileptic Seizure Detection on Wearable Systems with Reduced False-Alarm Rate”  In _EMBC_ , 2020  DOI: 10.1109/EMBC44109.2020.9175339
  * [21] Michal Teplan  “Fundamental of EEG Measurement”  In _Meas. Sci. Rev._ 2, 2002
  * [22] Una Pale et al.  “Systematic Assessment of Hyperdimensional Computing for Epileptic Seizure Detection” arXiv: 2105.00934  In _in EMBC_ , 2021  URL: http://arxiv.org/abs/2105.00934
  * [23] Saeedeh Ziyabari et al.  “Objective evaluation metrics for automatic classification of EEG events” arXiv: 1712.10107  In _arXiv:1712.10107_ , 2019  URL: http://arxiv.org/abs/1712.10107
  * [24] V. Shah et al.  “Validation of Temporal Scoring Metrics for Automatic Seizure Detection” ISSN: 2473-716X  In _SPMB_ , 2020, pp. 1–5  DOI: 10.1109/SPMB50085.2020.9353631