Optical fractals and Hump soliton structures in integrable Kuralay-Ⅱ system

Azzh Saad Alshehry; Safyan Mukhtar; Ali M. Mahnashi; Azzh Saad Alshehry; Safyan Mukhtar; Ali M. Mahnashi

doi:10.3934/math.20241361

AIMS Mathematics

2024, Volume 9, Issue 10: 28058-28078. doi: 10.3934/math.20241361

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Research article Special Issues

Optical fractals and Hump soliton structures in integrable Kuralay-Ⅱ system

1.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Basic Sciences, General Administration of Preparatory Year, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
3.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Kingdom of Saudi Arabia

Received: 08 August 2024 Revised: 30 August 2024 Accepted: 09 September 2024 Published: 27 September 2024
MSC : 34G20, 35A20, 35A22, 35R11

The integrable Kuralay-Ⅱ system (K-IIS) plays a significant role in discovering unique complex nonlinear wave phenomena that are particularly useful in optics. This system enhances our understanding of the intricate dynamics involved in wave interactions, solitons, and nonlinear effects in optical phenomena. Using the Riccati modified extended simple equation method (RMESEM), the primary objective of this research project was to analytically find and analyze a wide range of new soliton solutions, particularly fractal soliton solutions, in trigonometric, exponential, rational, hyperbolic, and rational-hyperbolic expressions for K-IIS. Some of these solutions displayed a combination of contour, two-dimensional, and three-dimensional visualizations. This clearly demonstrates that the generated solitons solutions are fractals due to the instability produced by periodic-axial perturbation in complex solutions. In contrast, the genuine solutions, within the framework of K-IIS, take the form of hump solitons. This work demonstrates the adaptability of the K-IIS for studying intricate nonlinear phenomena in a wide range of scientific and practical disciplines. The results of this work will eventually significantly influence our comprehension and analysis of nonlinear wave dynamics in related physical systems.

Keywords:

Citation: Azzh Saad Alshehry, Safyan Mukhtar, Ali M. Mahnashi. Optical fractals and Hump soliton structures in integrable Kuralay-Ⅱ system[J]. AIMS Mathematics, 2024, 9(10): 28058-28078. doi: 10.3934/math.20241361

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Abstract

1. Introduction

Throughout this paper, let $p$ be an odd prime and $q = p^n$ for some positive integer $n$ . Codebooks (also known as signal sets) with low coherence are typically used to distinguish signals of different users in code division multiple access (CDMA) systems. An $(N, K)$ codebook $\mathcal{C}$ is a finite set $\{\mathbf{c}_0, \mathbf{c}_1, \ldots, \mathbf{c}_{N-1}\}$ , where the codeword $\mathbf{c}_i$ , $0\leq i \leq N-1$ , is a unit norm $1\times K$ complex vector over an alphabet $\mathcal{A}$ . The maximum inner-product correlation $I_{\max}(\mathcal{C})$ of $\mathcal{C}$ is defined by

$\begin{align*} I_{\max}(\mathcal{C}) = \max\limits_{0\leq i\neq j\leq N-1}|\mathbf{c}_i\mathbf{c}_j^{H}|, \end{align*}$

where $\mathbf{c}_j^{H}$ denotes the conjugate transpose of $\mathbf{c}_j$ . The maximal cross-correlation amplitude $I_{\max}(\mathcal{C})$ of $\mathcal{C}$ is an important index of $\mathcal{C}$ , as it can approximately optimize many performance metrics such as outage probability and average signal-to-noise ratio. For a fixed $K$ , researchers are highly interested in designing a codebook $\mathcal{C}$ with the parameter $N$ being as large as possible and $I_{\max}(\mathcal{C})$ being as small as possible simultaneously. Unfortunately, there exists a bound between the parameters $N$ , $K$ and $I_{\max}(\mathcal{C})$ .

Lemma 1. (^[1]) For any $(N, K)$ codebook $\mathcal{C}$ with $N\geq K$ ,

$\begin{align} I_{\max}(\mathcal{C})\geq \sqrt{\frac{N-K}{(N-1)K}}. \end{align}$

(1.1)

The bound in (1.1) is called the Welch bound of $\mathcal{C}$ and is denoted by $I_{w}(\mathcal{C})$ . If the codebook $\mathcal{C}$ achieves $I_{w}(\mathcal{C})$ , then $\mathcal{C}$ is said to be optimal with respect to the Welch bound. However, constructing codebooks achieving the Welch bound is extremely difficult. Hence, many researchers have focused their main energy on constructing asymptotically optimal codebooks, i.e., $I_{\max}(\mathcal{C})$ asymptotically meets the Welch bound $I_{w}(\mathcal{C})$ for sufficiently large $N$ ^{[2,3,4,5,6,7]}.

The objective of this paper is to construct a class of complex codebooks and investigate their maximum inner-product correlation. Results show that these constructed complex codebooks are nearly optimal with respect to the Welch bound, i.e., the ratio of their maximal cross-correlation amplitude to the Welch bound approaches $1$ . These codebooks may have applications in strongly regular graphs ^[8], combinatorial designs ^[9,10], and compressed sensing ^[11,12].

This paper is organized as follows. In Section 2, we review some essential mathematical concepts regarding characters and Gauss sums over finite fields. In Section 3, we present a class of asymptotically optimal codebooks using the trace functions and multiplicative characters over finite fields. Finally, we make a conclusion in Section 4.

2. Preliminaries

In this section, we review some essential mathematical concepts regarding characters and Gauss sums over finite fields. These concepts will play significant roles in proving the main results of this paper.

Let $n$ be a positive integer and $p$ an odd prime. Denote the finite field with $p^n$ elements by $\mathbb{F}_{p^n}$ . The trace function ${\operatorname{Tr}}_n$ from $\mathbb{F}_{p^n}$ to $\mathbb{F}_p$ is defined by

$\begin{align*} {\operatorname{Tr}}_n(x) = \sum\limits_{i = 0}^{n-1}x^{p^i}. \end{align*}$

Let $\zeta_p$ denote a primitive $p$ -th root of complex unity and ${\operatorname{Tr}}_n$ denote the trace function from $\mathbb{F}_{p^n}$ to $\mathbb{F}_p$ . For $x\in \mathbb{F}_{p^n}$ , it can be checked that $\chi_n$ given by $\chi_n(x) = \zeta_p^{{\operatorname{Tr}}_n(x)}$ is an additive character of $\mathbb{F}_{p^n}$ , and $\chi_n$ is called the canonical additive character of $\mathbb{F}_{p^n}$ . Assume $a\in \mathbb{F}_{p^n}$ , then every additive character of $\mathbb{F}_{p^n}$ can be obtained by $\mu_a(x) = \chi_n(ax)$ where $\ x\in \mathbb{F}_{p^n}$ . The orthogonality relation of $\mu_a$ is given by

$\begin{align} \sum\limits_{x\in \mathbb{F}_{p^n}}\mu_a(x) = \left\{\begin{array}{ll} p^n, & \text{if}\ a = 0, \\ 0, & \text{otherwise}. \end{array} \right. \end{align}$

(2.1)

Let $q = p^n$ and $\alpha$ be a primitive element of $\mathbb{F}_{q}^{*}$ , then all multiplicative characters of $\mathbb{F}_{q}$ are given by $\varphi_j(\alpha^i) = \zeta_{q-1}^{ij}$ , where $\zeta_{q-1}$ denotes a primitive $(q-1)$ -th root of unity and $0 \leq i, j\leq q-2$ . The quadratic character of $\mathbb{F}_{q}$ is the character $\varphi_{(q-1)/2}$ , which will be denoted by $\eta_n$ in the sequel, and $\eta_n$ is extended by setting $\eta_n(0) = 0$ . For $\varphi_j$ , its orthogonality relation is given by

$\begin{align*} \sum\limits_{x\in \mathbb{F}_{p^n}}\varphi_j(x) = \left\{\begin{array}{ll} q-1, & \text{if}\ j = 0, \\ 0, & \text{otherwise}. \end{array} \right. \end{align*}$

The Gauss sum $G(\eta_n)$ over $\mathbb{F}_{p^n}$ is defined by

$G(\eta_n) = \sum\limits_{x\in \mathbb{F}_{p^n}^{*}} \eta_n(x)\chi_n(x).$

The explicit value of $G(\eta_n)$ is given in the following lemma.

Lemma 2 (^[13], Theorem 5.15). With symbols and notations above, we have

$\begin{align*} G(\eta_n) = (-1)^{n-1}(-1)^{\frac{(p-1)n}{4}}q^{\frac{1}{2}}. \end{align*}$

The following results on exponential sums will play an important role in proving the main results of this paper.

Lemma 3 (^[13], p.195). With symbols and notations above, we have

$\begin{align*} \eta_1(x) = \frac{1}{p}\sum\limits_{a\in\mathbb{F}_{p}}G(\eta_1)\eta_1(-a)\chi_1(ax), \end{align*}$

where $\eta_1$ denotes the quadratic character and $\chi_1$ the canonical additive character of $\mathbb{F}_p$ .

Lemma 4 (^[13], Theorem 5.33). If $f(x) = a_2x^2+a_1x+a_0\in \mathbb{F}_{p^n}[x]$ with $a_2\neq0$ , then

$\begin{align*} \sum\limits_{x\in \mathbb{F}_{p^n}}\zeta_p^{{\operatorname{Tr}}_n(f(x))} = \eta_n(a_2)G(\eta_n)\zeta_p^{{\operatorname{Tr}}_n\left(a_0-a_1^2(4a_2)^{-1}\right)}. \end{align*}$

Lemma 5 (^[14], Theorem 2). Let $n = 2m$ be an even integer and $z\in \mathbb{F}_{p}^{*}$ , then

$\begin{align*} \sum\limits_{x\in \mathbb{F}_{p^n}}\zeta_p^{z{\operatorname{Tr}}_n\left(x^{p^m+1}\right)} = -p^m. \end{align*}$

Lemma 6 (^[15]). Let $n = 2m$ be an even integer, $a\in \mathbb{F}_{p^m}^{*}$ , and $b\in \mathbb{F}_{p^n}$ , then

$\begin{align*} \sum\limits_{x\in \mathbb{F}_{p^{n}}}\zeta_p^{{\operatorname{Tr}}_m\left(ax^{p^m+1}\right)+{\operatorname{Tr}}_n(bx)} = -p^m\zeta_p^{-{\operatorname{Tr}}_m\left(\frac{b^{p^m+1}}{a}\right)}. \end{align*}$

Lemma 7 (^[16], Lemma 3.12). If $A$ and $B$ are finite abelian groups, then there is an isomorphism

$\begin{align*} \widehat{A\times B}\cong \widehat{A} \times \widehat{B}, \end{align*}$

where $\widehat{A}$ consists of all characters of $A$ .

By this lemma, we know that

$\widehat{\mathbb{F}_{p^n}^{+}\times \mathbb{F}_{p^n}^{+}} = \{\mu_{a, b}: a, b\in \mathbb{F}_{p^n}\}$

where

$\mu_{a, b}(x, y) = \zeta_p^{{\operatorname{Tr}}_n(ax+by)}$

for $x, y\in \mathbb{F}_{p^n}$ .

3. Proofs and main results

In this section, we always suppose that $n = 2m$ is an even integer and $p$ is an odd prime. The set $D$ is defined as follows:

$\begin{align*} D = \left\{(x, y)\in \mathbb{F}_{p^n}\times \mathbb{F}_{p^n}:\eta_1\left({\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\right) = 1\right\}, \end{align*}$

where $\eta_1$ is the quadratic character of $\mathbb{F}_p$ . A codebook $\mathcal{C}$ is constructed by

$\begin{align} \mathcal{C} = \left\{\mathbf{c}_{a, b}:a, b\in \mathbb{F}_{p^n}\right\}, \end{align}$

(3.1)

where $\mathbf{c}_{a, b} = \frac{1}{\sqrt{|D|}}\left(\mu_{a, b}(x, y)\right)_{(x, y)\in D}$ , $\mu_{a, b}(x, y) = \zeta_p^{{\operatorname{Tr}}_n(ax+by)}$ for $(x, y)\in D$ and $|D|$ denotes the cardinality of the set $D$ .

Lemma 8. With symbols and notations as above, we have

$\begin{align*} |D| = \frac{p-1}{2}\left(p^{2n-1}-(-1)^{\frac{n(p-1)}{4}}p^{n-1}\right). \end{align*}$

Proof. Let

$\begin{align*} A_1 = \sum\limits_{x, y\in \mathbb{F}_{p^n} \atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right) = 0}1, \ A_2 = \sum\limits_{x, y\in \mathbb{F}_{p^n}\atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\neq0}\eta_1\left({\operatorname{Tr}}_n(x^2+y^{p^m+1})\right). \end{align*}$

Note that

$\begin{align*} \sum\limits_{x, y\in \mathbb{F}_{p^n} \atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right) = 0}1+\sum\limits_{x, y\in \mathbb{F}_{p^n}\atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\neq0}1 = p^{2n}. \end{align*}$

Together with the definition of $D$ , we have

$\begin{align} |D|& = \sum\limits_{x, y\in\mathbb{F}_{p^n}\atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\neq0}\frac{\eta_1\left(\mathrm{Tr}_n\left(x^2+y^{p^m+1}\right)\right)+1}2 \\ & = \frac12\sum\limits_{x, y\in\mathbb{F}_{p^n}\atop{\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\neq0}\eta_1\left(\mathrm{Tr}_n\left(x^2+y^{p^m+1}\right)\right) +\frac12\sum\limits_{x, y\in\mathbb{F}_{p^n}\atop{\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\neq0}1 \\ & = \frac{A_{2}}{2}+\frac{p^{2n}}{2}-\frac{1}{2}\sum\limits_{x, y\in\mathbb{F}_{p^n}\atop{\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right) = 0}1 \\ & = \frac12\left(p^{2n}-A_1+A_2\right). \end{align}$

(3.2)

By definition, we have

$\begin{align} A_1& = \frac{1}{p}\sum\limits_{x, y\in \mathbb{F}_{p^n}}\sum\limits_{z\in \mathbb{F}_p}\zeta_p^{z{\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)} \\ & = p^{2n-1}+\frac{1}{p}\sum\limits_{z\in \mathbb{F}_{p}^{*}}\sum\limits_{x, y\in \mathbb{F}_{p^n}}\zeta_p^{{\operatorname{Tr}}_n\left(zx^2\right)+{\operatorname{Tr}}_n\left(zy^{p^m+1}\right)} \\ & = p^{2n-1}+(-1)^{\frac{n(p-1)}{4}}p^{n-1}(p-1). \end{align}$

(3.3)

where the last equality follows from Lemmas 2, 4, and 5. Note that $\eta_n(z) = 1$ for $z\in \mathbb{F}_{p}^{*}$ if $n$ is even. By Lemma 3, we obtain

$\begin{align*} A_2 = \frac{G(\eta_n)}{p}\sum\limits_{a\in \mathbb{F}_{p}^{*}}\eta_1(-a)\sum\limits_{x\in \mathbb{F}_{p^n}}\zeta_p^{{\operatorname{Tr}}_n\left(ax^2\right)}\sum\limits_{y\in \mathbb{F}_{p^n}}\zeta_p^{{\operatorname{Tr}}_n\left(ay^{p^m+1}\right)} \end{align*}$

Using Lemmas 4 and 5, we get

$A_2 = -p^{m-1}G^2(\eta_n)\sum\limits_{a\in \mathbb{F}_{p}^{*}}\eta_1(-a) = 0.$

The desired conclusion follows from (3.2) and (3.3). □

Example 1. Let $p = 5$ and $n = 2$ . By the Magma program, we know that $|D| = 240$ , which is consistent with Lemma 8.

Theorem 9. Let symbols and notations be the same as before, then the codebook $\mathcal{C}$ defined in (3.1) has parameters $[p^{2n}, K]$ ,

$\begin{align*} K = \frac{p-1}{2}\left(p^{2n-1}-(-1)^{\frac{n(p-1)}{4}}p^{n-1}\right), \end{align*}$

and

$I_{\max}(\mathcal{C}) = (p+1)p^{n-1}/(2K).$

Proof. By the definition of the set $\mathcal{C}$ and Lemma 8, we deduce that $\mathcal{C}$ is a $[p^{2n}, K]$ codebook. If $a, b\in \mathbb{F}_{p^n}$ and $(a, b)\neq (0, 0)$ , then we have

$\begin{align*} \sum\limits_{x, y\in \mathbb{F}_{p^n} }\zeta_p^{{\operatorname{Tr}}_n(ax+by)} = \sum\limits_{x, y\in \mathbb{F}_{p^n} \atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right) = 0}\zeta_p^{{\operatorname{Tr}}_n(ax+by)}+\sum\limits_{x, y\in \mathbb{F}_{p^n} \atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\neq0}\zeta_p^{{\operatorname{Tr}}_n(ax+by)} = 0 \end{align*}$

This implies that

$\begin{align*} \sum\limits_{x, y\in \mathbb{F}_{p^n} \atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right) = 0}\zeta_p^{{\operatorname{Tr}}_n(ax+by)} = -\sum\limits_{x, y\in \mathbb{F}_{p^n} \atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\neq0}\zeta_p^{{\operatorname{Tr}}_n(ax+by)}. \end{align*}$

For $a, b\in \mathbb{F}_{p^n}$ and $(a, b)\neq (0, 0)$ , we have that

$\begin{align} \sum\limits_{(x, y)\in D}\mu_{a, b}(x, y) = &\sum\limits_{x, y\in \mathbb{F}_{p^n} \atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\neq0}\zeta_p^{{\operatorname{Tr}}_n(ax+by)}\frac{\eta_1\left({\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\right)+1}{2} \\ = &-\frac{1}{2}\sum\limits_{x, y\in \mathbb{F}_{p^n} \atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right) = 0}\zeta_p^{{\operatorname{Tr}}_n(ax+by)}+\frac{1}{2}\sum\limits_{x, y\in \mathbb{F}_{p^n} \atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\neq0}\zeta_p^{{\operatorname{Tr}}_n(ax+by)}\eta_1\left({\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\right) \\ = &\frac{1}{2}(-B_1+B_2), \end{align}$

(3.4)

where

$\begin{align*} B_1 = \sum\limits_{x, y\in \mathbb{F}_{p^n} \atop {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right) = 0}\zeta_p^{{\operatorname{Tr}}_n(ax+by)}, \ B_2 = \sum\limits_{x, y\in \mathbb{F}_{p^n} }\zeta_p^{{\operatorname{Tr}}_n(ax+by)}\eta_1\left({\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)\right). \end{align*}$

By (2.1), we derive that

$\begin{align*} \sum\limits_{z\in \mathbb{F}_p}\zeta_p^{z{\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)} = \left\{\begin{array}{ll} p, & \text{if}\ {\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right) = 0, \\ 0, & \text{otherwise}. \end{array} \right. \end{align*}$

Combining Lemmas 4 and 6, we get that

$\begin{align} B_1 = &\frac{1}{p}\sum\limits_{x, y\in \mathbb{F}_{p^n} }\sum\limits_{z\in \mathbb{F}_p} \zeta_p^{{\operatorname{Tr}}_n(ax+by)} \zeta_p^{z{\operatorname{Tr}}_n\left(x^2+y^{p^m+1}\right)}\\ = &\frac{1}{p}\sum\limits_{z\in \mathbb{F}_p^{*}}\sum\limits_{x, y\in \mathbb{F}_{p^n} }\zeta_p^{{\operatorname{Tr}}_n\left(zx^2+ax\right)}\zeta_p^{{\operatorname{Tr}}_n\left(zy^{p^m+1}+by\right)}\\ = &-p^{m-1}G(\eta_n)\sum\limits_{z\in \mathbb{F}_{p}^{*}}\eta_n(z)\zeta_p^{{\operatorname{Tr}}_n\left(a^2+b^{p^m+1}\right)z} \\ = &\left\{\begin{array}{ll} -p^{m-1}(p-1)G(\eta_n), & \text{if}\ {\operatorname{Tr}}_n\left(a^2+b^{p^m+1}\right) = 0, \\ p^{m-1}G(\eta_n), & \text{if}\ {\operatorname{Tr}}_n\left(a^2+b^{p^m+1}\right)\neq0, \end{array} \right. \end{align}$

(3.5)

where $G(\eta_n)$ is given in Lemma 2. By Lemma 3, we have that

$\begin{align*} B_2 = \frac{G(\eta_1)}{p}\sum\limits_{z\in \mathbb{F}_{p}^{*}}\eta_1(-z)\sum\limits_{x\in \mathbb{F}_{p^n}}\zeta_p^{{\operatorname{Tr}}_n\left(zx^2+ax\right)}\sum\limits_{y\in \mathbb{F}_{p^n}}\zeta_p^{{\operatorname{Tr}}_n\left(zy^{p^m+1}+by\right)}. \end{align*}$

Moreover, by Lemmas 4 and 6, we obtain

$\begin{align} B_2& = -p^{m-1}G(\eta_1)G(\eta_n)\sum\limits_{z\in \mathbb{F}_{p}^{*}}\eta_1(z)\zeta_p^{z{\operatorname{Tr}}_n\left(a^2+b^{p^m+1}\right)} \\ & = \left\{\begin{array}{ll} 0, & \text{if}\ {\operatorname{Tr}}_n\left(a^2+b^{p^m+1}\right) = 0, \\ -p^{m}\left(\frac{-1}{p}\right)G(\eta_n)\eta_1\left({\operatorname{Tr}}_n\left(a^2+b^{p^m+1}\right)\right), & \text{if}\ {\operatorname{Tr}}_n\left(a^2+b^{p^m+1}\right)\neq0. \end{array} \right. \end{align}$

(3.6)

It follows from Lemma 2 and (3.4) that

$\begin{align} \sum\limits_{(x, y)\in D}\mu_{a, b}(x, y)\in \left\{\frac{1}{2}(p-1)p^{m-1}G(\eta_n), -\frac{1}{2}(p+1)p^{m-1}G(\eta_n)\right\}. \end{align}$

(3.7)

For any two distinct codewords $\mathbf{c}_{z_1, z_2}$ , $\mathbf{c}_{z_1^{'}, z_2^{'}}\in \mathcal{C}$ , i.e., $(z_1, z_2)\neq (z_1^{'}, z_2^{'})$ , it is easy to check that

$\begin{align} \left|\mathbf{c}_{z_1, z_2}\mathbf{c}_{z_1^{'}, z_2^{'}}^{H}\right| = \frac{1}{K}\left|\sum\limits_{(x, y)\in D}\mu_{z_1-z_1^{'}, z_2-z_2^{'}}(x, y)\right|. \end{align}$

(3.8)

Combining (3.7) and (3.8), we get that $I_{\max}(\mathcal{C}) = (p+1)p^{n-1}/(2K).$ □

Example 2. Let $f(x)$ be an irreducible polynomial over the field $\mathbb{F}_3$ and $f(x) = x^2+x+2$ in $\mathbb{F}_3[x]$ . Suppose that $p = 3$ , $n = 2$ , and $\alpha$ is a root of $f(x)$ over $\mathbb{F}_3$ , then $m = 1$ , $q = 3^{2}$ , and $\mathbb{F}_{9} = \mathbb{F}_{3}(\alpha)$ . It can be verified that the set $D$ consists of the following $30$ elements:

$\begin{align*} D& = \left\{(x, y)\in\mathbb{F}_{9}\times\mathbb{F}_{9}:\mathrm{Tr}_{2}\left(x^{2}+y^{4}\right) = 1\right\} \\ & = \left\{ \begin{array}{lllll} (1+2\alpha, 0), &(2+\alpha, 0), &(1, 1), &(1, 2), &(1, 1+2\alpha), \\ (1, 2+\alpha), &(2, 1), &(2, 2), &(2, 1+2\alpha), &(2, 2+\alpha), \\ (\alpha, \alpha), &(\alpha, 2\alpha), &(\alpha, 1+\alpha), &(\alpha, 2+2\alpha), &(2\alpha, \alpha), \\ (2\alpha, 2\alpha), &(2\alpha, 1+\alpha), &(2\alpha, 2+2\alpha), &(0, \alpha), &(0, 2\alpha), \\ (0, 1+\alpha), &(0, 2+2\alpha), &(1+\alpha, \alpha), &(1+\alpha, 2\alpha), &(1+\alpha, 1+\alpha), \\ (1+\alpha, 2+2\alpha), &(2+2\alpha, \alpha), &(2+2\alpha, 2\alpha), &(2+2\alpha, 1+\alpha), &(2+2\alpha, 2+2\alpha) \end{array} \right\}. \end{align*}$

The corresponding codebook $\mathcal{C}$ is given by

$\begin{align*} \mathcal{C} = \left\{\frac{1}{\sqrt{30}}\left(\zeta_3^{{\operatorname{Tr}}_2(ax+by)}\right)_{(x, y)\in D}:a, b\in \mathbb{F}_9\right\}, \end{align*}$

where $\zeta_3 = e^{\frac{2\pi\sqrt{-1}}{3}}$ and ${\operatorname{Tr}}_2$ denotes the trace function from $\mathbb{F}_9$ to $\mathbb{F}_3$ .

Corollary 10. The codebook $\mathcal{C}$ constructed in (3.1) is asymptotically optimal with respect to the Welch bound.

Proof. The corresponding Welch bound is

$\begin{align*} I_{w}(\mathcal{C}) = \sqrt{\frac{p^{2n-1}(p+1)+(-1)^{\frac{n(p-1)}{4}}p^{n-1}(p-1)}{\left(p^{2n}-1\right)(p-1)\left(p^{2n-1}-(-1)^{\frac{n(p-1)}{4}}p^{n-1}\right)}} . \end{align*}$

We deduce that

$\begin{align*} \lim\limits_{p^n\rightarrow +\infty} \frac{I_{w}(\mathcal{C})}{I_{\max}(\mathcal{C})} = \lim\limits_{p^n\rightarrow +\infty}\sqrt{\frac{4K\left(p^{2n}-K\right)}{\left(p^{2n}-1\right)(p+1)^2p^{2n-2}}} = 1, \end{align*}$

which implies that $\mathcal{C}$ asymptotically meets the Welch bound. □

In , we show some parameters of some specific codebooks defined in (3.1). From this table, we conclude that $I_{\max}(\mathcal{C})$ is very close to $I_w(\mathcal{C})$ for largely enough $p$ , which ensures the correctness of Theorem 9 and Corollary 1.

Table 1. The parameters of the codebook

$\mathcal{C}$ in (3.1) for

$n = 4$ .

$p$	$N$	$K$	$I_{\max}(\mathcal{C})$	$I_w(\mathcal{C})$	$I_{\max}(\mathcal{C})/I_w(\mathcal{C})$
3	$3^8$	$2160$	1/40	$1.762 \times 10^{-2}$	1.4185
7	$7^8$	$2469600$	1/1800	$4.811\times 10^{-4}$	1.1548
11	$11^8$	$97429200$	1/12200	$7.483 \times 10^{-5}$	1.0955
13	$13^8$	$376477920$	1/24480	$3.782 \times 10^{-5}$	1.0801
17	$17^8$	$3282670080$	1/74240	$1.2699 \times 10^{-5}$	1.0607

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4. Concluding remarks

This paper presented a family of codebooks by the combination of additive characters and multiplicative characters over finite fields. Results show that the constructed codebooks are asymptotically optimal in the sense that the maximum cross correlation amplitude of the codebooks asymptotically achieves the Welch bound. As a comparison, parameters of some known nearly optimal codebooks and the constructed ones are listed in . From this table, we can conclude that the parameters of $\mathcal{C}$ are not covered by those in ^{[2,3,4,5,6,17,18,19]}. This means the presented codebooks have new parameters.

Table 2. The parameters of codebooks asymptotically meeting the Welch bound.

Ref.	Parameters $(N, K)$	Constraints
^[2]	$\left((q-1)^\ell+M, M \right)$ , $M=\frac{(q-1)^\ell+(-1)^{\ell+1}}{q}$ .	$q$ is a prime power, $\ell>2$ .
^[3]	$\left(2K+(-1)^{ln}, K\right)$ , $K=\frac{(q_1-1)^n\cdots (q_l-1)^n-(-1)^{ln}}{2}$ .	$1\leq i\leq l$ , $s_i>1$ , $q_i=2^{s_i},$ $l>1$ and $n>1$ .
^[4]	$\left((q^s-1)^m+q^{sm-1}, q^{sm-1}\right)$	$s>1$ , $m>1$ , $q$ is a prime power.
^[5]	$\left((p_{{\min}}+1)Q^2, Q^2\right)$	$Q>1$ is an integer, $p_{{\min}}$ is the smallest prime factor of $Q$ .
^[6]	$\left(p_{\min}N_1N_2, N_1N_2\right)$	$N_1\geq1$ , $N_2=N_1+o(N_1)$ , $p_{{\min}}$ is the smallest prime factor of $N_2$ .
^[6]	$\left(p_{\min}N_1N_2, N_1N_2\right)$	$N_1\geq1$ , $N_2=N_1+o(N_1)$ , $p_{{\min}}$ is the smallest prime factor of $N_2$ .
^[17]	$\left(q_1q_2\cdots q_l, \left(q_1q_2\cdots q_l-1\right)/2\right)$	$1\leq i\leq l$ , $q_i$ is a prime power, $q_i\equiv3\pmod{4}$
^[18]	$\left(q, \frac{q+1}{2}\right)$	$q$ is a prime power.
^[19]	$\left(q^3+q^2, q^2\right)$ or $\left(q^3+q^2-q, q^2-q\right)$	$q$ is a prime power.
Thm. 3.1	$\left(p^{2n}, \frac{p-1}{2}\left(p^{2n-1}-(-1)^{\frac{n(p-1)}{4}}p^{n-1}\right)\right)$	$p$ is an odd prime, $n=2m$ , $m$ is a positive integer.

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Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the Innovation Project of Engineering Research Center of Integration and Application of Digital Learning Technology (No.1221003), Humanities and Social Sciences Youth Foundation of Ministry of Education of China (No. 22YJC870018), the Science and Technology Development Fund of Tianjin Education Commission for Higher Education (No. 2020KJ112, 2022KJ075, KYQD1817), the National Natural Science Foundation of China (Grant No. 12301670), the Natural Science Foundation of Tianjin (Grant No. 23JCQNJC00050), Haihe Lab. of Information Technology Application Innovation (No. 22HHXCJC00002), Fundamental Research Funds for the Central Universities, China (Grant No. ZY2301, BH2316), the Open Project of Tianjin Key Laboratory of Autonomous Intelligence Technology and Systems (No. TJKL-AITS-20241004, No. TJKL-AITS-20241006).

Conflict of interest

The authors declare no conflicts of interest.

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