Optimal and near-optimal frequency-hopping sequences based on Gaussian period

Yan Wang; Yanxi Fu; Nian Li; Huanyu Wang; Yan Wang; Yanxi Fu; Nian Li; Huanyu Wang

doi:10.3934/math.20231493

AIMS Mathematics

2023, Volume 8, Issue 12: 29158-29170. doi: 10.3934/math.20231493

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Research article Special Issues

Optimal and near-optimal frequency-hopping sequences based on Gaussian period

1.
School of Science, Xi'an University of Architecture and Technology, Caosi East Road, Xi'an, Shaanxi, 710311, China
2.
Hubei Key Laboratory of Applied Mathematics, School of Cyber Science and Technology, Hubei University, Wuhan, Hubei, 430062, China

Received: 13 August 2023 Revised: 09 October 2023 Accepted: 20 October 2023 Published: 26 October 2023
MSC : 11T22, 94A60

Frequency-hopping sequences (FHSs) have a decisive influence on the whole frequency-hopping communication system. The Hamming correlation function plays an important role in evaluating the performance of FHSs. Constructing FHS sets that meet the theoretical bounds is crucial for the research and development of frequency-hopping communication systems. In this paper, three new classes of optimal FHSs based on trace functions are constructed. Two of them are optimal FHSs and the corresponding periodic Hamming autocorrelation value is calculated by using the known Gaussian period. It is shown that the new FHSs are optimal according to the Lempel-Greenberger bound. The third class of FHSs is the near-optimal FHSs.

Keywords:

Citation: Yan Wang, Yanxi Fu, Nian Li, Huanyu Wang. Optimal and near-optimal frequency-hopping sequences based on Gaussian period[J]. AIMS Mathematics, 2023, 8(12): 29158-29170. doi: 10.3934/math.20231493

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Abstract

1. Introduction

Frequency-hopping multiple-access is widely used in military radio communication, satellite communications, fiber-optic communications, underwater communications, microwave, and radar systems. The user's frequency slots used are chosen pseudo-randomly through a code called frequency-hopping sequences (FHS). The theoretical bound of the FHS gives the constraint relations that should be satisfied between different parameters.

Lempel and Greenberger ^[1] established a theoretical lower bound on the maximum Hamming autocorrelation of FHS for a given length and frequency set size, which is called the Lempel and Greenberger bound, and the FHS satisfying the bound is called an optimal FHS. Constructing such optimal FHSs became a hot topic in FHS research ^[2,3].

Both algebraic and combinatorial constructions of optimal FHSs have been proposed in the literature (see ^{[4,5,6,7,8,9,10,11]}) and the references therein. Among all known constructions, cyclotomy ^[12] is one of the most useful techniques for coding theory and cryptography. Chung et al. ^[13] constructed several optimal FHSs of length from $k$ -fold cyclotomic classes for distinct odd primes. A class of FHSs with flexible parameters was given based on the cyclotomic division of rings by Zeng ^[14]. In ^[15], Xu et al. constructed a family of FHSs based on the Zeng-Cai-Tang-Yang cyclotomy and the Chinese remainder theorem.

For a given sequence period and frequency set size, the optimal FHS does not always exist. Therefore, in the absence of the optimal parameters, the near-optimal FHS is a substitution of an optimal FHS. It is also important to construct a more near-optimal FHS with new parameters.

At present, the construction of near-optimal FHSs can be referred to in literature ^{[16,17,18,19]}. In 2008, Han et al. ^[20] first proposed the concept of near-optimal FHSs. In 2010, Chung et al. ^[21] generated two kinds of near-optimal FHSs by using the cyclotomic coset over finite fields. In 2014, Ren et al. ^[22] proposed a class of constructions of near-optimal FHSs by means of the Chinese remainder theorem and cyclotomic over finite fields. See Table 1 for more near-optimal FHSs.

Table 1. Parameters of known near-optimal frequency sequences.

References	$\left({n, l, \lambda } \right)$	Constraints	Lempel-Greenberger bound
^[13]	$({p^2}, p + 1, p)$		near-optimal
^[13]	$({p^n}, \frac{{{p^n} - 1}}{f}, k)$	$p = kf + 1, f \text{ is even}.$	near-optimal
^[16]	$(q - 1, e, f + 1)$	$q = ef + 1 \text{ is an odd prime} \\\text{ power}, f \text{ is odd}.$	near-optimal
^[17]	$(\frac{{q + 1}}{k}, \frac{{q + 2k + 1}}{{2k}}, 2)$	$q \text{ is is odd prime power}, \\k\mid(q + 1), \frac{{q + 1}}{k} \text{ is even}.$	near-optimal
^[18]	$(pq, m, \frac{{pq - 1}}{m} + 1)$	$p \text{ and}{\text{ }}q \text{ are distinct odd primes }\\\text{ satisfying }p \equiv m + 1(\bmod 2m) \text{ and} \\q \equiv 1(\bmod 2m), \text{ and } m \text{ is}\\\text{ even common divisor of } p - 1 \text{ and } {\text{ }}q - 1$	near-optimal
^[19]	$(q, e, f + 1)$	$q = ef + 1 \text{ is a prime power }, \\f \text{ is even }.$	near-optimal
Theorem 3.3	$(\frac{{q + 1}}{{{k^2}}}, \frac{{q + 2{k^2} + 1}}{{2{k^2}}}, 2)$	$q \text{ is an odd prime power}, \\k^2\mid(q + 1), \frac{{q + 1}}{k^2} \text{ is even}.$	near-optimal

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Our purpose is to construct new optimal FHSs for some cases that are not covered in the literature. In this paper, we present three constructions for FHSs with optimal Hamming autocorrelation. The parameters of the optimal FHSs obtained in this paper are listed in Table 2, which gives a comparison of our constructions.

Table 2. Parameters of known optimal frequency sequences.

References	$\left({n, l, \lambda } \right)$	Constraints	Lempel-Greenberger bound
^[3]	$({p^2}, p, p)$	$p$ is a prime.	optimal
^[5]	$(\frac{{q + 1}}{k}, \frac{{q + k + 1}}{2k}, 1)$	$k\mid(q + 1), \text{ and } \frac{{q + 1}}{k} \text{ is odd}.$	optimal
^[6]	$(p, M, f)$	$p = Mf + 1 \text{ is a prime }, \\f \text{ is even }, p \equiv 3\, \bmod \, 4,$	optimal
^[8]	$(\frac{{{q^n} - 1}}{e}, q, \frac{{{q^{n - 1}} - 1}}{e})$	$q \text{ is a prime power }, \\e \mid(q - 1), \gcd (e, n) = 1$	optimal
^[9]	$(\frac{{{q^m} - 1}}{e}, {q^k}, \frac{{{q^{m - 1}} - 1}}{e})$	$1 \leqslant k \leqslant m, \\e\mid(q - 1), \gcd (e, m) = 1$	optimal
^[10]	$(q - 1, e + 1, f - 1)$	$q = ef + 1 \text{ is a prime power}.$	optimal
^[12]	$(p, e + 1, f + 1)$	$p = ef + 1, \\e \geqslant 3f, f \geqslant 2$	optimal
Theorem 3.1	$\left({\frac{{4(q + 1)}}{5}, \frac{{4q + 9}}{{10}}, 1} \right)$	${k^2}\mid(q + 1), \text{ and} \frac{{q + 1}}{{{k^2}}} \text{ is odd}.$	optimal
Theorem 3.2	$(\frac{{q + 1}}{{{k^2}}}, \frac{{q + {k^2} + 1}}{{2{k^2}}}, 1)$	${k^2}\mid(q + 1), \text{ and}{\text{ }}\frac{{q + 1}}{{{k^2}}} \text{ is odd}.$	optimal

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The rest of this paper is organized as follows. In section two, we present some notations and definitions about FHSs, as well as the cyclotomic class and Gaussian period. In section three, we propose two classes of optimal FHSs and prove they are optimal. In section four, we construct a class of near-optimal FHSs. The conclusions are provided in section five.

2. Preliminaries

For any positive integer $l \geqslant 2$ , let $\mathbb{F} = \{ {f_0}, {f_1}, \cdots, {f_{l - 1}}\}$ be a set of $l$ available frequencies, called an alphabet. A sequence $X = \left\{ {x(t)} \right\}_{t = 0}^{n - 1}$ is called an FHS of length $n$ over $\mathbb{F}$ if $x(t) \in \mathbb{F}$ for $0 \leqslant t \leqslant n - 1$ . For any FHS $X = \left\{ {x(t)} \right\}_{t = 0}^{n - 1}$ of length $n$ over $\mathbb{F}$ , its Hamming autocorrelation ${H_X}$ is defined by

$\begin{equation} {H_X}(\tau ) = \sum\limits_{t = 0}^{n - 1} {h\left[ {x(t),x(t + \tau )} \right]},\ \ 0 \leqslant \tau < n. \end{equation}$

(2.1)

Where $h\left[ {a, b} \right] = 1$ if $a = b$ and zero, the addition is performed modulo $n$ . The maximum out-of-phase Hamming autocorrelation of $X$ is defined as

$H(X) = \mathop {\max }\limits_{1 \leqslant \tau < n} \{ {H_X}\left( \tau \right)\} .$

Throughout this paper, let $\left({n, l, \lambda } \right)$ denote an FHS $X$ of length $n$ over an alphabet with size $l$ with $\lambda = H(X)$ . For a real number $a$ , let $\left\lceil a \right\rceil$ denote the least integer no less than $a$ and let $\left\lfloor a \right\rfloor$ denote the integer a part of $a$ . A lower bound of $H(X)$ was established by Lempel and Greenberger as follows.

Lemma 2.1. (Lempel-Greenberger bound ^[1], Lemma 4) For every FHS $X$ of length $n$ over an alphabet with size $l$ ,

$\begin{equation} H(X) \geqslant \left\lceil {\frac{{\left( {n - \epsilon } \right)\left( {n + \epsilon - l} \right)}}{{l(n - 1)}}} \right\rceil , \end{equation}$

(2.2)

where $\epsilon$ is the least nonnegative residue of $n$ modulo $l$ .

Lemma 2.2. (^[23], Corollary 1.2) Let $X$ be any FHS of period $n$ on a frequency set with size $l$ ,

$\begin{equation} H(X) = \left\{ \begin{array}{l} 0,\;\;\;if\;n = l, \\ \left\lfloor {n/l} \right\rfloor ,\;if\;n > l. \\ \end{array} \right. \end{equation}$

(2.3)

We denote ${\lambda _{opt}}$ as the righthand side in (2.2); that is, the value given by the Lempel-Greenberger bound. The following definitions will be used in this paper.

Definition 2.1. An FHS X is optimal if $H(X) = {\lambda _{opt}}$ , i.e. X is optimal with respect to the Lempel-Greenberger bound; an FHS X is near-optimal if $H(X) = {\lambda _{opt}+1}$ , i.e. X is near-optimal with respect to the Lempel-Greenberger bound.

Let $h$ be a positive integer, $p$ be a prime number and $q = {p^h}$ . Let $n$ be a positive integer, $r = {q^n}$ , $\mathbb{F}_{r}$ be a finite field containing $r$ elements, and $\theta$ be the generator of the multiplicative group $\mathbb{F}^{*}_{q^m}$ . Trace function $Tr_q^r$ from finite field $\mathbb{F}_{r}$ to finite field $\mathbb{F}_{q}$ is defined as

$Tr_q^r(x) = x + {x^q} + {x^{{q^2}}} + \cdots + {x^{{q^{n - 1}}}},{\text{ }}x \in \mathbb{F}_{r}.$

Let $r - 1 = nN$ , where $n$ and $N$ are positive integers greater than two. The $Nth$ order of cyclotomic class $C_i^{\left({N, r} \right)}$ of $\mathbb{F}_{r}$ is defined as

$C_i^{(N,r)} = \{ {\alpha ^{Nt + i}}:0 \leqslant t < N\} ,{\text{ 0}} \leqslant i < N.$

Let $\; {\zeta _p} = {e^{\frac{{2\pi \sqrt { - 1} }}{p}}}$ be the root of the primitive unit to the $pth$ degree. The canonical addition feature $\chi$ over $\mathbb{F}_{r}$ is defined as

$\chi (x) = \zeta _p^{Tr_p^r(x)}, \ \ x \in \mathbb{F}_{r}.$

The orthogonal relation of addition characteristic is

$\begin{equation} \sum\limits_{x \in \mathbb{F}_{r}} {\chi (ax) = \left\{ \begin{gathered} r,{\text{ }}if{\text{ }}a = 0, \\ 0,{\text{ }}if{\text{ }}a \in \mathbb{F}^{*}_{r}. \\ \end{gathered} \right.} \end{equation}$

(2.4)

The Gaussian period $\eta _i^{(N)}$ of order $N$ over $\mathbb{F}_{r}$ is defined as

$\eta _i^{(N)} = \sum\limits_{x \in C_i^{(N)}} {\chi \left( x \right)} ,{\text{ }}0 \leqslant i < N.$

Here's the convention:If $i \geqslant N$ , then $\eta _i^{(N)} = \eta _{i(\bmod N)}^{(N)}$ .

The following Gaussian period is from the conjugate case.

Lemma 2.3. ^[24] Suppose $j$ is the smallest positive integer such that ${p^j} \equiv - 1\ (\ mod\ N)$ . Let $r = {p^{2j\gamma }}$ and $\gamma$ be a positive integer, then the $Nth$ order Gaussian period $\eta _i^{(N)}$ over $\mathbb{F}_{r}$ satisfies

1) when $\gamma$ , $p$ and $\frac{{{p^j} + 1}}{N}$ are all odd,

$\begin{equation} {\;\eta _i^{(N)} = \left\{ \begin{gathered} \frac{{(N - 1)\sqrt r - 1}}{N},\;if\;i = \frac{N}{2}, \\ \frac{{ - \sqrt r - 1}}{N},\;otherwise. \\ \end{gathered} \right.} \end{equation}$

(2.5)

2) otherwise,

$\begin{equation} {\;\eta _i^{(N)} = \left\{ \begin{gathered} \frac{{{{( - 1)}^{\gamma + 1}}(N - 1)\sqrt r - 1}}{N},\;if\;i = 0, \\ \frac{{{{( - 1)}^\gamma }\sqrt r - 1}}{N},\;otherwise. \\ \end{gathered} \right.} \end{equation}$

(2.6)

3. Results

Construction A. Let $q$ be a power of an odd prime $p$ and $r = {q^2}$ . An FHS $X = \left({{x_0}, {x_1}, {x_2}, \cdots {x_{\frac{{4(q + 1)}}{5} - 1}}} \right)$ of period ${\frac{{4(q + 1)}}{5}}$ is defined as follows

$\begin{equation} {X_t} = Tr_q^{{q^2}}\left( {{\alpha ^{\frac{{5\left( {q - 1} \right)}}{4}t}}} \right),{\text{ }}1 \leqslant t < \frac{{4(q + 1)}}{5}. \end{equation}$

(3.1)

Lemma 3.1. For

$0 \leqslant {t_1} \leqslant {t_2} < \frac{{4(q + 1)}}{5},$

we have

${x_{{t_1}}} = {x_{{t_2}}} \Leftrightarrow {t_1} + {t_2} = \frac{{4(q + 1)}}{5}.$

Proof. According to Eq (3.1),

$\begin{align*} {x_t} & = {\alpha ^{\frac{5}{4}\left( {q - 1} \right)t}} + {\left( {{\alpha ^{\frac{5}{4}\left( {q - 1} \right)t}}} \right)^q} \\ {\text{ }} & = {\alpha ^{\frac{5}{4}\left( {q - 1} \right)t}} + {\alpha ^{ - \frac{5}{4}\left( {q - 1} \right)t}}, \end{align*}$

thus

$\begin{array}{l} \ \ \ \ \ {x_{{t_1}}} = {x_{{t_2}}}{\text{ }} \\ \Leftrightarrow {\alpha ^{\frac{5}{4}\left( {q - 1} \right){t_1}}} + {\alpha ^{ - \frac{5}{4}\left( {q - 1} \right){t_1}}} = {\alpha ^{\frac{5}{4}\left( {q - 1} \right){t_2}}} + {\alpha ^{ - \frac{5}{4}\left( {q - 1} \right){t_2}}} \\ \Leftrightarrow {\alpha ^{\frac{5}{4}\left( {q - 1} \right){t_1}}} - {\alpha ^{\frac{5}{4}\left( {q - 1} \right){t_2}}} = {\alpha ^{ - \frac{5}{4}\left( {q - 1} \right){t_2}}} - {\alpha ^{ - \frac{5}{4}\left( {q - 1} \right){t_1}}} \\ \Leftrightarrow {\alpha ^{\frac{5}{4}\left( {q - 1} \right)\left( {{t_1} + {t_2}} \right)}} = 1 \\ \Leftrightarrow {t_1} + {t_2} = \frac{{4(q + 1)}}{5}. \\ \end{array}$

□

Theorem 3.1. Let the FHS $X$ be given by Eq (3.1), then $X$ has parameters $\left({\frac{{4(q + 1)}}{5}, \frac{{4q + 9}}{{10}}, 1} \right)$ , which is optimal with respect to the Lempel-Greenberger bound.

Proof. First, from Lemma 3.1 we know that the frequency set size of the sequence $X$ is $\frac{{\frac{{4(q + 1)}}{5} - 1}}{2} + 1 = \frac{{4q + 9}}{{10}}$ , then for $1 \leqslant \tau < \frac{{4(q + 1)}}{{5}}$ we have

$\begin{eqnarray*} {H_X}(\tau ) & = & \mid {\left\{ {0 \leqslant t < \frac{{4(q + 1)}}{5}:Tr_q^{{q^2}}\left( {{\alpha ^{\frac{{5(q - 1)t}}{4}}}} \right) = Tr_q^{{q^2}}\left( {{\alpha ^{\frac{{5(q - 1)(t + \tau )}}{4}}}} \right)} \right\}} \mid \\ & = &\frac{1}{q}\sum\limits_{x \in {F_q}} {\sum\limits_{t = 0}^{\frac{{4(q + 1)}}{5} - 1} {\varsigma _p^{Tr_p^q\left[ {x \cdot Tr_q^{{q^2}}\left( {\left( {{\alpha ^{\frac{{5\left( {q - 1} \right)}}{4}\tau }} - 1} \right){\alpha ^{\frac{{5(q - 1)t}}{4}}}} \right)} \right]}} } \\ & = & \frac{{4(q + 1)}}{{5q}} + \frac{1}{q}\sum\limits_{x \in F_q^*} {\sum\limits_{t = 0}^{\frac{{4(q + 1)}}{5} - 1} {\varsigma _p^{Tr_p^q\left[ {Tr_q^{{q^2}}\left( {x \cdot \left( {{\alpha ^{\frac{{5\left( {q - 1} \right)}}{4}\tau }} - 1} \right){\alpha ^{\frac{{5(q - 1)t}}{4}}}} \right)} \right]}} } \\ & = & \frac{{4(q + 1)}}{{5q}} + \frac{1}{q}\sum\limits_{t = 0}^{\frac{{4(q + 1)}}{5} - 1} {\sum\limits_{i = 0}^{q - 2} {\varsigma _p^{Tr_p^q\left[ {Tr_q^{{q^2}}\left( {\left( {{\alpha ^{\frac{{5\left( {q - 1} \right)}}{4}\tau }} - 1} \right){\alpha ^{\frac{{5(q - 1)t + (q + 1)i}}{4}}}} \right)} \right]}} } \\ & = & \frac{{4(q + 1)}}{{5q}} + \frac{1}{q}\sum\limits_{t = 0}^{\frac{{4(q + 1)}}{5} - 1} {\sum\limits_{i = 0}^{q - 2} {\chi \left( {\left( {{\alpha ^{\frac{{5(q - 1)\tau }}{4}}} - 1} \right){\alpha ^{\frac{{5\left( {q - 1} \right)t + \left( {q + 1} \right)i}}{4}}}} \right)} } \\ & = & \frac{{4(q + 1)}}{{5q}} + \frac{1}{q}\sum\limits_{x \in C_0^{\left( {\frac{5}{4}} \right)}} { \chi\left( {\left( {{\alpha ^{\frac{{5\left( {q - 1} \right)\tau }}{4}}} - 1} \right)x} \right)} \\ & = & \frac{{4(q + 1)}}{{5q}} + \frac{1}{q}\sum\limits_{x \in C_j^{\left( {\frac{5}{4}} \right)}} {\chi \left( x \right)} \\ & = & \frac{{4(q + 1)}}{{5q}} + \frac{1}{q}\eta _j^{\left( {\frac{5}{4}} \right)}.\\ \end{eqnarray*}$

From Lemma 2.3, the minimum $j$ is $h$ while $\gamma = 1$ . When $p = 2$ , according to Eq (2.6),

$H(X) \leqslant \frac{{4(q + 1)}}{{5q}} + \frac{1}{q}\mathop {\max }\limits_{0 \leqslant j < \frac{5}{4}} \left\{ {\eta _j^{\left( {\frac{5}{4}} \right)}} \right\} = \frac{{4(q + 1)}}{{5q}} + \frac{1}{q}\frac{{{{( - 1)}^2}4(\frac{5}{4} - 1)q - 1}}{{{5}}} = 1.$

Similarly, when $p$ is an odd prime number, it can be known from Eq (2.5) that

$H(X) \leqslant \frac{{4(q + 1)}}{{5q}} + \frac{1}{q}\mathop {\max }\limits_{0 \leqslant j < \frac{5}{4}} \left\{ {\eta _j^{\left( {{\frac{5}{4}}} \right)}} \right\} = \frac{{4(q + 1)}}{{5q}} + \frac{1}{q}\frac{{4(\frac{5}{4} - 1)q - 1}}{{5}} = 1.$

Thus, $H(X) \leqslant 1$ for all $\gamma$ and $p$ .

However,

$H(X) \geqslant \left\lceil {\frac{{\left( {\frac{{4\left( {q + 1} \right)}}{5} - \frac{{4q - 1}}{{10}}} \right)\left( {\frac{{4(q + 1)}}{5} + \frac{{4q - 1}}{{10}} - \frac{{4q + 9}}{{10}}} \right)}}{{\frac{{4q + 9}}{{10}}\left( {\frac{{4(q + 1)}}{5} - 1} \right)}}} \right\rceil = 1.$

Therefore, $H\left(X \right) = 1$ , which is the Lempel-Greenberger bound. □

Construction B. Let $q = {p^h}$ , $p$ be a prime number and $h$ be a positive integer. Let $\theta$ be the generator of the multiplication group $\mathbb{F}^{*}_{q^m}$ and $m$ is even. The positive integer $k$ is a factor of $q+1$ , and ${\frac{{q + 1}}{{{k^2}}}}$ is odd. An FHS $X = ({x_0}, {x_1}, \cdots, {x_{\frac{{q + 1}}{{{k^2}}} - 1}})$ of period ${\frac{{q + 1}}{{{k^2}}}}$ is defined as follows

$\begin{equation} {x_t} = Tr_q^{{q^m}}({\theta ^{{k^2}(q - 1)t}}),\ \ 1 \leqslant t < \frac{{q + 1}}{{{k^2}}}. \end{equation}$

(3.2)

Lemma 3.2. For

$0 \leqslant {t_1} \leqslant {t_2} < \frac{{q + 1}}{{{k^2}}},$

we have

${x_{{t_1}}} = {x_{{t_2}}} \Leftrightarrow {t_1} + {t_2} = \frac{{q + 1}}{{{k^2}}}.$

Proof. According to Eq (3.2),

$\begin{array}{l} {x_t} = \left( {{\theta ^{{k^2}\left( {q - 1} \right)t}} + {{({\theta ^{{k^2}(q - 1)t}})}^q} + \cdots + {{\left( {{\theta ^{{k^2}\left( {q - 1} \right)t}}} \right)}^{{q^m}}}} \right) \\ \ \ = \left( {{\theta ^{{k^2}(q - 1)t}} + ({\theta ^{{k^2}q(q - 1)t}}) + \cdots + \left( {{\theta ^{{k^2}{q^m}(q - 1)}}^t} \right)} \right) \\ \ \ = \frac{m}{2}\left( {{\theta ^{{k^2}(q - 1)t}} + {\theta ^{ - {k^2}\left( {q - 1} \right)t}}} \right). \\ \end{array}$

Thus,

$\begin{array}{l} \ \ \ \ \ \ {x_{{t_1}}} = {x_{{t_2}}} \\ {\text{ }} \Leftrightarrow \frac{m}{2}\left( {{\theta ^{{k^2}\left( {q - 1} \right){t_1}}} + \left( {{\theta ^{ - {k^2}\left( {q - 1} \right){t_1}}}} \right)} \right) = {\text{ }}\frac{m}{2}\left( {{\theta ^{{k^2}\left( {q - 1} \right){t_2}}} + \left( {{\theta ^{ - {k^2}\left( {q - 1} \right){t_2}}}} \right)} \right) \\ {\text{ }} \Leftrightarrow {\theta ^{{k^2}\left( {q - 1} \right){t_1}}} + {\theta ^{ - {k^2}\left( {q - 1} \right){t_1}}} = {\theta ^{{k^2}\left( {q - 1} \right){t_2}}} + {\theta ^{ - {k^2}\left( {q - 1} \right){t_2}}} \\ {\text{ }} \Leftrightarrow {\theta ^{{k^2}\left( {q - 1} \right){t_1}}} - {\theta ^{{k^2}\left( {q - 1} \right){t_2}}} = {\theta ^{ - {k^2}\left( {q - 1} \right){t_2}}} - {\theta ^{ - {k^2}\left( {q - 1} \right){t_1}}} \\ {\text{ }} \Leftrightarrow {\theta ^{{k^2}\left( {q - 1} \right)({t_1} + {t_2})}} = 1 \\ {\text{ }} \Leftrightarrow {t_1} + {t_2} = \frac{{q + 1}}{{{k^2}}}. \\ \end{array}$

□

Theorem 3.2. Let the FHS $X$ be given by Eq (3.2), then $X$ has parameters $\left({\frac{{q + 1}}{{{k^2}}}, \frac{{q + {k^2} + 1}}{{2{k^2}}}, 1} \right)$ , which is optimal with respect to the Lempel-Greenberger bound, where ${k^2}\mid(q + 1)$ and $\frac{{q + 1}}{{{k^2}}}$ is odd.

Proof. First, from Lemma 3.2 we know that the frequency set size of the sequence $X$ is $\frac{{\frac{{q + 1}}{{{k^2}}} - 1}}{2} + 1 = \frac{{q + {k^2} + 1}}{{2{k^2}}}$ , then for $1 \leqslant \tau < \frac{{q + 1}}{{{k^2}}}$ we have

$\begin{eqnarray*} {H_X}(\tau ) & = & \lvert {\left\{ {0 \leqslant t < \frac{{q + 1}}{{{k^2}}}:Tr_q^{{q^m}}({\theta ^{{k^2}(q - 1)t}}) = Tr_q^{{q^m}}({\theta ^{{k^2}(q - 1)(t + \tau )}})} \right\}} \rvert \nonumber \\ & = & \frac{1}{q}\sum\limits_{x \in \mathbb{F} _q} {\sum\limits_{t = 0}^{\frac{{q + 1}}{{{k^2}}} - 1} {\zeta _p^{Tr_p^q[x \cdot Tr_q^{{q^m}}(({\theta ^{{k^2}(q - 1)\tau }} - 1){\theta ^{{k^2}(q - 1)t}})]}} } \\ & = & \frac{{q + 1}}{{{k^2}q}} + \frac{1}{q}\sum\limits_{x \in \mathbb{F}^{*}_{q}} {\sum\limits_{t = 0}^{\frac{{q + 1}}{{{k^2}}} - 1} {\zeta _p^{Tr_p^q[Tr_q^{{q^m}}(x({\theta ^{{k^2}(q - 1)\tau }} - 1){\theta ^{{k^2}(q - 1)t}})]}} } \\ & = & \frac{{q + 1}}{{{k^2}q}} + \frac{1}{q}\sum\limits_{t = 0}^{\frac{{q + 1}}{{{k^2}}} - 1} {\sum\limits_{i = 0}^{q - 2} {\zeta _p^{Tr_p^q[Tr_q^{{q^m}}(({\theta ^{{k^2}(q - 1)\tau }} - 1){\theta ^{{k^2}(q - 1)t + (q + 1)i}})]}} } \\ & = & \frac{{q + 1}}{{{k^2}q}} + \frac{1}{q}\sum\limits_{t = 0}^{\frac{{q + 1}}{{{k^2}}} - 1} {\sum\limits_{i = 0}^{q - 2} {\chi ((} } {\theta ^{{k^2}(q - 1)\tau }} - 1){\theta ^{{k^2}(q - 1)t + (q + 1)i}}) \\ & = & \frac{{q + 1}}{{{k^2}q}} + \frac{1}{q}\sum\limits_{x \in C_0^{({k^2})}} {\chi (({\theta ^{{k^2}(q - 1)\tau }} - 1)x)} \\ & = & \frac{{q + 1}}{{{k^2}q}} + \frac{1}{q}\sum\limits_{x \in C_j^{({k^2})}} {\chi (x)} = \frac{{q + 1}}{{{k^2}q}} + \frac{1}{q}\eta _j^{({k^2})}. \end{eqnarray*}$

From Lemma 2.3, the minimum $j$ is $h$ while $\gamma = 1$ . When $p = 2$ , according to Eq (2.6) we have

$H(X) \leqslant \frac{{q + 1}}{{{k^2}q}} + \frac{1}{q}\mathop {\max }\limits_{0 \leqslant j < {k^2}} \left\{ {\eta _j^{\left( {{k^2}} \right)}} \right\} = \frac{{q + 1}}{{{k^2}q}} + \frac{1}{q}\frac{{{{( - 1)}^2}({k^2} - 1)q - 1}}{{{k^2}}} = 1.$

Similarly, when $p$ is an odd prime number, it can be known from Eq (2.5) that

Thus, $H(X) \leqslant 1$ for all $\gamma$ and $p$ .

However,

$H(X) \geqslant \left\lceil {\frac{{\left( {\frac{{q + 1}}{{{k^2}}} - \frac{{q - {k^2} + 1}}{{2{k^2}}}} \right)\left( {\frac{{q + 1}}{{{k^2}}} + \frac{{q - {k^2} + 1}}{{2{k^2}}} - \frac{{q + {k^2} + 1}}{{2{k^2}}}} \right)}}{{\frac{{q + {k^2} + 1}}{{2{k^2}}}\left( {\frac{{q + 1}}{{{k^2}}} - 1} \right)}}} \right\rceil = 1.$

Therefore, $H\left(X \right) = 1$ , which is the Lempel-Greenberger bound. □

Example 3.1. Let $p = 211, {\text{ }}h = 1, {\text{ }}k = 2$ and ${\text{ }}m = 2$ , thus $q = {p^h} = 211$ , ${k^2}\mid(q+1)$ and $\frac{{q + 1}}{{{k^2}}} = 53$ are odd. The FHS $X$ defined by Eq (3.2) is

$X = (2,99,93,35,207,202,168,183,14,148,79,77,$

$\ \ \ \ \ \ \ \ \ 159,50,149,142,194,74,169,199,120,76,19,$

$\ \ \ \ \ \ \ \ \ 117,170,44,177,177,44,170,117,19,76,120,$

$\ \ \ \ \ \ \ \ \ 199,169,74,194,142,149,50,159,77,79,148,$

$\ \ \ \ \ \ \ \ \ 14,183,168,202,207,35,93,99).$

It can be obtained by using Magma that the periodic Hamming autocorrelation ${H_X}(\tau)(1\leqslant\tau\leqslant52)$ of $X$ is all one. Hence, the FHS $X$ has parameters (53, 27, 1), and the Lempel-Greenberger bound is optimal. This is consistent with Theorem 3.2.

Example 3.2. Let $p = 239, {\text{ }}h = 1, {\text{ }}k = 4$ and ${\text{ }}m = 2$ , thus $q = {p^h} = 239$ , ${k^2}\mid(q+1)$ and $\frac{{q + 1}}{{{k^2}}} = 15$ are odd. The FHS $X$ defined by Eq (3.2) is

$X = (2,145,230,223,79,238,15,25,25,15,238,79,223,230,145).$

It can be obtained by using Magma that the periodic Hamming autocorrelation ${H_X}(\tau)(1 \leqslant \tau \leqslant 14)$ of $X$ is all one. Hence, the FHS $X$ has parameters (15, 8, 1), and the Lempel-Greenberger bound is optimal. This is consistent with Theorem 3.2.

Example 3.3. Let $p = 107, {\text{ }}h = 1, {\text{ }}k = 2$ and ${\text{ }}m = 2$ , thus $q = {p^h} = 107$ , ${k^2}\mid(q+1)$ and $\frac{{q + 1}}{{{k^2}}} = 27$ are odd. The FHS $X$ defined by Eq (3.2) is

$X = (2,84,99,100,62,79,47,17,97,106,33,98,67,73,$

$\ \ \ \ \ \ \ \ \ 73,67,98,33,106,97,17,47,79,62,100,99,84).$

It can be obtained by using Magma that the periodic Hamming autocorrelation ${H_X}(\tau)(1 \leqslant \tau \leqslant 26)$ of $X$ is all one. Hence, the FHS $X$ has parameters (27, 14, 1), and the Lempel-Greenberger bound is optimal. This is consistent with Theorem 3.2.

Construction C. Let $q = {p^h}$ , $p$ be an odd prime number and $h$ be a positive integer. Let $\theta$ be the generator of the multiplication group $\mathbb{F}^{*}_{q^m}$ , and $m$ is even. The positive integer $k$ is a factor of $q+1$ , and ${\frac{{q + 1}}{{{k^2}}}}$ is even. An FHS $X = ({x_0}, {x_1}, \cdots, {x_{\frac{{q + 1}}{{{k^2}}} - 1}})$ of period ${\frac{{q + 1}}{{{k^2}}}}$ is defined as follows

$\begin{equation} {x_t} = Tr_q^{{q^m}}({\theta ^{{k^2}(q - 1)t}}),\ \ 1 \leqslant t < \frac{{q + 1}}{{{k^2}}}. \end{equation}$

(3.3)

Lemma 3.3. For any $1 \leqslant \tau < \frac{{q + 1}}{{{k^2}}}$ , we have

${\theta ^{{k^2}(q - 1)\tau }-1} \in \left\{ \begin{array}{l} C_0^{(2{k^2},{q^2})},\;\;{\mathit{\text{if}}}\;\frac{{q + 1}}{{2{k^2}}}\;{\mathit{\text{and}}}\;\tau\; {\mathit{\text{are parity}}}, \\ C_{{k^2}}^{(2{k^2},{q^2})},{\mathit{\text{otherwise}}}. \\ \end{array} \right.$

Proof.

$\begin{eqnarray*} {({\theta ^{{k^2}(q - 1)\tau }} - 1)^{\frac{{{q^2} - 1}}{{2{k^2}}}}} & = & {(({\theta ^{{k^2}(q - 1)\tau }} - 1)^{q - 1})^{\frac{{q + 1}}{{2{k^2}}}}} \\ & = & {(\frac{{{{({\theta ^{{k^2}(q - 1)\tau }} - 1)}^q}}}{{{\theta ^{{k^2}(q - 1)\tau }} - 1}})^{\frac{{q + 1}}{{2{k^2}}}}} \\ & = & {(\frac{{{\theta ^{ - {k^2}(q - 1)\tau }} - 1}}{{{\theta ^{{k^2}(q - 1)\tau }} - 1}})^{\frac{{q + 1}}{{2{k^2}}}}} \\ & = & {( - 1)^{\frac{{q + 1}}{{2{k^2}}} - \tau }} \\ & = & \left\{ \begin{gathered} 1,\quad \;\;\;\,\;if{\text{ }}\frac{{q + 1}}{{2{k^2}}}{\text{ }}and{\text{ }}\tau {\text{ }}are{\text{ }}parity, \\ - 1,\quad otherwise. \end{gathered} \right. \end{eqnarray*}$

Consequently, the conclusion is proven. □

Lemma 3.4. If ${\frac{{q + 1}}{{2{k^2}}}}$ is odd, then

$\eta _0^{(2{k^2},{q^2})} = - \frac{{q + 1}}{{2{k^2}}},\;\eta _{{k^2}}^{(2{k^2},{q^2})} = q - \frac{{q + 1}}{{2{k^2}}};$

if ${\frac{{q + 1}}{{2{k^2}}}}$ is even, then

$\eta _0^{(2{k^2},{q^2})} = q - \frac{{q + 1}}{{2{k^2}}},\;\eta _{{k^2}}^{(2{k^2},{q^2})} = - \frac{{q + 1}}{{2{k^2}}}.$

Proof. If $\frac{{q + 1}}{{2{k^2}}}$ is odd, then the smallest positive integer $j$ satisfies ${p^j} \equiv - 1{\text{ }}(\bmod {\text{ }}2{k^2})$ for $h$ . For Lemma 3.2, $\Delta = 1$ and $\frac{{{p^j} + 1}}{{2{k^2}}} = \frac{{q + 1}}{{2{k^2}}}$ are odd. Therefore, $\eta _0^{(2{k^2}, {q^2})} = \frac{{ - \sqrt r - 1}}{N} = \frac{{ - q - 1}}{{2{k^2}}} = - \frac{{q + 1}}{{2{k^2}}}$ and $\eta _k^{(2{k^2}, {q^2})} = \frac{{(N - 1)\sqrt r - 1}}{N} = \frac{{(2{k^2} - 1)q - 1}}{{2{k^2}}} = q - \frac{{q + 1}}{{2{k^2}}}$ . If $\frac{{q + 1}}{{2{k^2}}}$ is even, the proof is similar to before. □

Theorem 3.3. Let the FHS $X$ be given by Eq (3.3), then $X$ has parameters $\left({\frac{{q + 1}}{{{k^2}}}, \frac{{q + {2k^2} + 1}}{{2{k^2}}}, 2} \right)$ , and the Lempel-Greenberger bound is near-optimal.

Proof. First, from Lemma 3.2, we know that the frequency set size of the sequence $X$ is $\frac{{\frac{{q + 1}}{{{k^2}}} - 2}}{2} + 2 = \frac{{q + {2k^2} + 1}}{{2{k^2}}}$ , then for $1 \leqslant \tau < \frac{{q + 1}}{{{k^2}}}$ we have

$\begin{eqnarray} {H_X}(\tau ) & = & \lvert{\left\{ {0 \leqslant t < \frac{{q + 1}}{{{k^2}}}:Tr_q^{{q^m}}({\theta ^{{k^2}(q - 1)t}}) = Tr_q^{{q^m}}({\theta ^{{k^2}(q - 1)(t + \tau )}})} \right\}} \rvert \\ & = & \frac{1}{q}\sum\limits_{x \in \mathbb{F} _q} {\sum\limits_{t = 0}^{\frac{{q + 1}}{{{k^2}}} - 1} {\zeta _p^{Tr_p^q[x \cdot Tr_q^{{q^m}}(({\theta ^{{k^2}(q - 1)\tau }} - 1){\theta ^{{k^2}(q - 1)t}})]}} } \\ & = & \frac{{q + 1}}{{{k^2}q}} + \frac{1}{q}\sum\limits_{x \in \mathbb{F}^{*}_{q}} {\sum\limits_{t = 0}^{\frac{{q + 1}}{{{k^2}}} - 1} {\zeta _p^{Tr_p^q[Tr_q^{{q^m}}(x({\theta ^{{k^2}(q - 1)\tau }} - 1){\theta ^{{k^2}(q - 1)t}})]}} } \\ & = & \frac{{q + 1}}{{{k^2}q}} + \frac{1}{q}\sum\limits_{t = 0}^{\frac{{q + 1}}{{{k^2}}} - 1} {\sum\limits_{i = 0}^{q - 2} {\zeta _p^{Tr_p^q[Tr_q^{{q^m}}(({\theta ^{{k^2}(q - 1)\tau }} - 1){\theta ^{{k^2}(q - 1)t + (q + 1)i}})]}} } \\ & = & \frac{{q + 1}}{{{k^2}q}} + \frac{1}{q}\sum\limits_{t = 0}^{\frac{{q + 1}}{{{k^2}}} - 1} {\sum\limits_{i = 0}^{q - 2} {\chi ((} } {\theta ^{{k^2}(q - 1)\tau }} - 1){\theta ^{{k^2}(q - 1)t + (q + 1)i}}). \end{eqnarray}$

(3.4)

Since

$\frac{{\frac{{q + 1}}{{{k^2}}} \times (q - 1) \times \gcd ({k^2}(q - 1),q + 1)}}{{{q^2} - 1}} = \frac{{\frac{{q + 1}}{{{k^2}}} \times (q - 1) \times 2{k^2}}}{{{q^2} - 1}} = 2,$

we have Eq (3.4) as

$\begin{eqnarray} & = & \frac{{q + 1}}{{{k^2}q}} + \frac{2}{q}\sum\limits_{x \in C_0^{(2{k^2},{q^2})}} {\chi (({\theta ^{{k^2}(q - 1)\tau }} - 1)x)} \\ & = & \left\{ \begin{gathered} \frac{{q + 1}}{{{k^2}q}} + \frac{2}{q}\sum\limits_{x \in C_0^{(2{k^2},{q^2})}} {\chi (x),{\text{ }}if{\text{ }}\frac{{q + 1}}{{2{k^2}}}{\text{ }}and{\text{ }}\tau {\text{ }}are{\text{ }}parity,} \\ \frac{{q + 1}}{{{k^2}q}} + \frac{2}{q}\sum\limits_{x \in C_{{k^2}}^{(2{k^2},{q^2})}} {\chi (x),{\text{ }}otherwise.} \\ \end{gathered} \right. \\ & = & \left\{ \begin{gathered} \frac{{q + 1}}{{{k^2}q}} + \frac{2}{q}\eta _0^{(2{k^2},{q^2})},{\text{ }}if{\text{ }}\frac{{q + 1}}{{2{k^2}}}{\text{ }}and{\text{ }}\tau {\text{ }}are{\text{ }}parity, \\ \frac{{q + 1}}{{{k^2}q}} + \frac{2}{q}\eta _{{k^2}}^{(2{k^2},{q^2})},{\text{ }}otherwise. \\ \end{gathered} \right. \\ & = & \left\{ \begin{gathered} 0,\quad if\;\,\tau \;\,is\;\,odd,\\2,\quad if\;\,\tau \;\,is\;\,even. \end{gathered} \right. \end{eqnarray}$

(3.5)

The penultimate row is derived from Lemma 3.3. Formula (3.5) is obtained from Lemma 3.4. Thus, $H(X) = 2$ and

$\left\lfloor {\frac{{\frac{{q + 1}}{{{k^2}}}}}{{\frac{{q + 2{k^2} + 1}}{{2{k^2}}}}}} \right\rfloor = \left\lfloor {1 + \frac{{\frac{{q + 1}}{{2{k^2}}}}}{{\frac{{q + 1}}{{2{k^2}}}}}} \right\rfloor = 1.$

Hence, $H(X) = 2 = \left\lfloor {\frac{n}{l}} \right\rfloor + 1$ . That is, the FHS $X$ is near-optimal with respect to the Lempel-Greenberger bound. □

Example 3.4. Let $p = 167, {\text{ }}h = 1, {\text{ }}k = 2$ and ${\text{ }}m = 2$ , thus $q = {p^h} = 167$ , ${k^2}\mid(q+1)$ and $\frac{{q + 1}}{{{k^2}}} = 42$ are even. The FHS $X$ defined by Eq (3.3) is

$X = (2,21,24,68,76,34,57,1,47,73,75,8,10,36,$

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 82,26,49,7,15,59,62,81,62,59,15,7,49,26,$

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 82,36,10,8,75,73,47,1,57,34,76,68,24,21 ).$

It can be obtained by using Magma that the periodic Hamming autocorrelation is

${H_X}(\tau ) = \left\{ \begin{gathered} 0,{\text{ }}if{\text{ }}\tau {\text{ }}is{\text{ }}an{\text{ }}odd, \\ 2,{\text{ }}if{\text{ }}\tau {\text{ }}is{\text{ }}an{\text{ }}even. \\ \end{gathered} \right.$

Therefore, the FHS $X$ has parameters (42, 22, 2), and the Lempel-Greenberger bound is near-optimal. This is consistent with Theorem 3.3.

Example 3.5. Let $p = 79, {\text{ }}h = 1, {\text{ }}k = 3$ and ${\text{ }}m = 2$ , thus $q = {p^h} = 79$ , ${k^2}\mid(q+1)$ and $\frac{{q + 1}}{{{k^2}}} = 10$ are even. The FHS $X$ defined by Eq (3.3) is

$X = (2,80,79,10,9,87,9,10,79,80).$

It can be obtained by using Magma that the periodic Hamming autocorrelation is

Therefore, the FHS $X$ has parameters (10, 6, 2), and the Lempel-Greenberger bound is near-optimal. This is consistent with Theorem 3.3.

Example 3.6. Let $p = 499, {\text{ }}h = 1, {\text{ }}k = 5$ and ${\text{ }}m = 2$ , thus $q = {p^h} = 49 9$ , ${k^2}\mid(q+1)$ and $\frac{{q + 1}}{{{k^2}}} = 20$ are even. The FHS $X$ defined by Eq (3.3) is

$X = (2,355,275,464,274,0,225,35,224,144,497,144,224,35,225,0,274,464,275,355).$

It can be obtained by using Magma that the periodic Hamming autocorrelation is

${H_X}(\tau ) = \left\{ \begin{gathered} 0,{\text{ }}\tau {\text{ }}is{\text{ }}an{\text{ }}odd, \\ 2,{\text{ }}if{\text{ }}\tau {\text{ }}is{\text{ }}an{\text{ }}even. \end{gathered} \right.$

Consequently, the FHS $X$ has parameters (20, 11, 2), and the Lempel-Greenberger bound is near-optimal. This is consistent with Theorem 3.3.

4. Conclusions

In this paper, we proposed three classes of FHSs based on trace function, and showed they are optimal and near-optimal respectively according to the Lempel-Greenberger bound. Our construction was a discussion in the case of even numbers, though it would be interesting to discuss in the case of odd numbers. We leave this problem for one of our further works.

Use of AI tools declaration

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

Acknowledgments

The work of Yan Wang was supported by the National Natural Science Foundation of China under Grant 61902304 and Natural Science Basic Research Plan in Shaanxi Province of China 2021JQ-495. The work of Nian Li was supported in part by the Natural Science Foundation of Hubei Province of China under Grant 2021CFA079 and the Knowledge Innovation Program of Wuhan-Basic Research under Grant 2022010801010319.

Conflict of interest

The authors declare there is no conflict of interest.

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Optimal and near-optimal frequency-hopping sequences based on Gaussian period

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Optimal and near-optimal frequency-hopping sequences based on Gaussian period

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3. Results

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References