Construction of three classes of Strictly Optimal Frequency-Hopping Sequence Sets

TL;DR

This paper introduces three new classes of strictly optimal frequency-hopping sequence sets that optimize partial Hamming correlation and family size, using generic and trace map-based constructions.

Contribution

It presents novel constructions of FHS sets that are strictly optimal, improving design options for frequency-hopping communication systems.

Findings

Three classes of strictly optimal FHS sets are constructed.

The constructions achieve optimal partial Hamming correlation.

The methods include a generic approach and trace map-based techniques.

Abstract

In this paper, we construct three classes of strictly optimal frequency-hopping sequence (FHS) sets with respect to partial Hamming correlation and family size. The first class is based on a generic construction, the second and third classes are based from the trace map.

Tables

Table 1. Table 1. Known And Strictly Optimal FHS Sets

Length	Alphabet	$ℳ(ℱ;L)$	Family	Constraints	Reference
	Size		Size
$\frac{p^m-1}{r}$	$p^{m-1}$	$\lceil \frac{L}{T}\rceil$	$r$	$\psi \left(x\right)$ is identity,	[16]
				$r|p-1$, $\gcd (r,m)=1$
$p^{2m}-1$	$p^m$	$\lceil \frac{L}{p^m+1}\rceil$	1		[17]
$p\left(p^m-1\right)$	$p^m$	$\lceil \frac{L}{p^m-1}\rceil$	$p^{m-1}$		[18]
$p^k\left(p^m-1\right)$	$p^m$	$\lceil \frac{L}{p^m-1}\rceil$	$p^{m-k}$	$k\mid m$	Theorem 2 here
$\frac{p^m-1}{r}$	$p^{m-1}$	$\lceil \frac{L}{T}\rceil$	$r$	$\psi \left(x\right)$,	Theorem 3 here
				$r|p-1$, $\gcd (r,m)=1$
$\frac{p^{2m}-1}{r}$	$p^m$	$\lceil \frac{L}{p^m+1}\rceil$	$r$	$f\left(x\right)$,	Theorem 4 here
				$r|p-1$, $\gcd (r,2m)=1$

Equations

$$H_{X,Y}(\tau ;j\mid L):=\sum _{t=j}^{j+L-1}h[x_t,y_{t+\tau }](\tau \in \{0,\cdots ,N-1\}),$$

$$H_{X,Y}(\tau ;j\mid L)=\#\{t:j\le t\le j+L-1,x_t=y_{t+\tau }\}.$$

$$H(X;L):=\underset{\begin{array}{c}0\le j<N\\ 1\le \tau <N\end{array}}{\max }{H}_{X,X}(\tau ;j\mid L)(X\in \mathcal{F}),$$

$$H(X,Y;L):=\underset{\begin{array}{c}0\le j<N\\ 0\le \tau <N\end{array}}{\max }{H}_{X,Y}(\tau ;j\mid L)(X\ne Y\in \mathcal{F}),$$

$$\mathcal{M}(\mathcal{F};L):=\underset{\begin{array}{c}X,Y\in \mathcal{F}\\ X\ne Y\end{array}}{\max }\{H(X;L),H(X,Y;L)\}.$$

$$\mathcal{M}(\mathcal{F};L)\ge \lceil \frac{L}{N}\lceil \frac{(NM-d^{\prime })N}{(MN-1)d^{\prime }}\rceil \rceil ,$$

$$\mathcal{M}(\mathcal{F};L)\ge \lceil \frac{[2IMN-(I+1)I{d}^{\prime }]L}{(MN-1)MN}\rceil ,$$

$$M\le \underset{2\le L\le N}{\min }\left\{\lfloor \frac{1}{N}\lfloor \frac{(L-\mathcal{M}(\mathcal{F};L))d^{\prime }}{L-d^{\prime }\mathcal{M}(\mathcal{F};L)}\rfloor \rfloor :L>d^{\prime }\mathcal{M}(\mathcal{F};L)\right\},$$

$$M\le \underset{2\le L\le N}{\min }\left\{\lfloor \frac{d^{\prime \mathcal{M}(\mathcal{F};L)+1}}{N}\rfloor :L>\mathcal{M}(\mathcal{F};L)\right\}.$$

$$f(x)={\mathrm{Tr}}_{q^m/q}(x^d),$$

$$f(x)={\mathrm{Tr}}_{3^m/3}(x+x^s),$$

$$f(yx)=y^{d}f(x)$$

$$\varphi (u_{t+\tau }^i)-\varphi (u_t^j)=\text{constant}c(i,j,\tau )\in {\mathbb{F}}_q^{\ast }\text{for all}0\le t\le n-1.$$

$$\underset{0\le t\le n^{\prime }-1}{\max }\#\{t:\psi (v_{t+\tau })-\psi (v_t)=b,1\le \tau \le n^{\prime }-1\}=1.$$

$$s_t^i=\varphi (u_t^i)+\psi (v_t),0\le t\le N-1,$$

$$\mathcal{M}(\mathcal{S};L)\le \lceil \frac{L}{n^{\prime }}\rceil .$$

$$H_{S^i,S^j}(\tau ;l\mid L)$$

$$=\sum _{t=l}^{L+l-1}h[\varphi (u_t^i)+\psi (v_t),\varphi (u_{t+\tau }^j)+\psi (v_{t+\tau })]$$

$$=\sum _{t=l}^{L+l-1}h[\varphi (u_t^i)-\varphi (u_{t+\tau }^j),\psi (v_{t+\tau })-\psi (v_t)].$$

$$\varphi (u_t^i)-\varphi (u_{t+\tau }^j)=b\in {\mathbb{F}}_q\text{for all}t\in \mathbb{Z},$$

$$\sum _{t=l+n^{\prime }z}^{l+n^{\prime }z+n^{\prime }-1}h[s_t^i,s_{t+\tau }^j]=\sum _{t=0}^{n^{\prime }-1}h[b,\psi (v_{t+\tau })-\psi (v_t)]\le 1$$

$$H_{S^i,S^j}(\tau ;l\mid L)=\sum _{t=l}^{l+L-1}h[s_t^i,s_{t+\tau }^j]=\sum _{z=0}^{m_1-1}\sum _{t=l+n^{\prime }z}^{l+n^{\prime }z+n^{\prime }-1}h[s_t^i,s_{t+\tau }^j]\le m_1,$$

$$H_{S^i,S^j}(\tau ;l\mid L)=\sum _{z=0}^{m_1-1}\sum _{t=l+n^{\prime }z}^{l+n^{\prime }z+n^{\prime }-1}h[s_t^i,s_{t+\tau }^j]+\sum _{t=l+m_1{n}^{\prime }}^{l+m_1{n}^{\prime }+m_2-1}h[s_t^i,s_{t+\tau }^j]\le m_1+1.$$

$$\mathcal{M}(\mathcal{S};L)\le \lceil \frac{L}{n^{\prime }}\rceil ,$$

$$u_t^i=a_i+\sigma (t).$$

$${\varphi }_P(u_{t+\tau }^i)-{\varphi }_P(u_t^j)={\varphi }_P(a_i-a_j+\sigma (\tau ))\in {\mathbb{F}}_q^{\ast }.$$

$$\mathcal{M}(\mathcal{S};L)=\lceil \frac{L}{p^m-1}\rceil .$$

$$\lceil \frac{(NM-d^{\prime })N}{(NM-1)d^{\prime }}\rceil$$

$$=\lceil \frac{(p^m-2)p^k(p^m-1)}{p^m(p^m-1)-1}\rceil =p^k.$$

$$\mathcal{M}(\mathcal{F};L)\ge \lceil \frac{L}{N}\lceil \frac{(NM-d^{\prime })N}{(NM-1)d^{\prime }}\rceil \rceil =\lceil \frac{L}{p^m-1}\rceil .$$

$$M\le \lfloor \frac{p^{2m}}{p^k(p^m-1)}\rfloor =\lfloor \frac{p^{m-k}{p}^k(p^m-1)+p^m}{p^k(p^m-1)}\rfloor =p^{m-k}.$$

$${\varphi }_P(x)=x,\psi (x)=x^3,$$

$$\begin{array}{rl}{S}^0=& (1,{\alpha }^5,{\alpha }^7,\alpha ,0,\alpha ,{\alpha }^2,{\alpha }^2,0,{\alpha }^3,\alpha ,{\alpha }^6,{\alpha }^4,\\ & {\alpha }^6,{\alpha }^5,{\alpha }^5,{\alpha }^4,{\alpha }^2,{\alpha }^6,{\alpha }^7,1,{\alpha }^7,{\alpha }^3,{\alpha }^3);\\ S^1=& ({\alpha }^7,0,{\alpha }^2,{\alpha }^5,\alpha ,{\alpha }^5,1,1,\alpha ,{\alpha }^4,{\alpha }^5,{\alpha }^3,\\ & {\alpha }^6,{\alpha }^3,0,0,{\alpha }^6,1,{\alpha }^3,{\alpha }^2,{\alpha }^7,{\alpha }^2,{\alpha }^4,{\alpha }^4);\\ S^2=& ({\alpha }^2,\alpha ,1,0,{\alpha }^5,0,{\alpha }^7,{\alpha }^7,{\alpha }^5,{\alpha }^6,0,2,\\ & {\alpha }^3,2,\alpha ,\alpha ,{\alpha }^3,{\alpha }^7,2,1,{\alpha }^2,1,{\alpha }^6,{\alpha }^6).\end{array}$$

Full text

Keywords Frequency-hopping sequences, partial Hamming correlation, Difference-balanced functions.

Research is partially supported by Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200) and NSFC (Grant No. 11571328).

1. Introduction

With advantages such as secure properties, multiple-access, anti-jamming, and anti-fading, frequency-hopping multiple-access (FHMA) is now widely used in modern communication systems such as military communications, bluetooth, sonar echolocation systems and so on [1]. To reduce the multi-access interference, in those systems, the maximum of Hamming out-of-phase autocorrelation and cross correlation of the set of frequency-hopping sequences (FHSs) must be minimized. Thus, it is very interesting to design FHS sets with low Hamming correlation, large size, long period, and small available frequencies simultaneously. In fact, the parameters are not independent with each other, and they are subjected to limitation of some theoretic bounds, for example, the Lempel-Greenberger bound [2], the Peng-Fan bound [3], or the coding theory bound [4]. Therefore, it has received a lot of attention about constructing optimal FHSs with respect to the bounds and much progress have been made (see [5]-[8], [9]-[15] and the references therein).

The Hamming correlation of FHSs can be divided into three types in general: the periodic Hamming correlation, the aperiodic Hamming correlation, and the partial Hamming correlation. Compared with the first type, the results are relatively little known about the aperiodic and partial ones. However, in the practical application scenarios where the synchronization time is limited or the hardware is complex, the length of a correlation window is usually shorter than the period of the chosen FHSs [17], i.e., the correlation window length may vary from time to time according to the channel conditions. Consequently, the partial Hamming correlation begin to attract attention. Eun et al. [17] obtained a class of FHSs with optimal partial Hamming correlation from the $m$-sequence and GMW sequences over polynomial residue class ring. In 2012, Zhou et al. [16] derived FHS sets with optimal partial Hamming correlation from trace functions, and generalized the Peng-Fan bounds on the periodic Hamming correlation based on the array structure.

In 2014, Cai et al. [18] presented FHS sets with optimal partial Hamming correlation from generalized cyclotomy. Later, the authors gave some theoretic bounds of the size of FHS sets and presented a new class of FHSs with optimal partial Hamming correlation in [19]. Very recently, combinatorial constructions of FHSs with optimal partial Hamming correlation have been reported, see [20, 21, 22].

The purpose of this paper is to construct three classes of strictly optimal FHS sets with optimal partial Hamming correlation and optimal family sizes. We list the parameters of our construction and related known ones in Table 1, which gives a comparison of our construction and the constructions before us.

Our first construction generates optimal $(p^{k}(p^{m}-1),p^{m-k},\left\lceil\frac{L}{p^{m}-1}\right\rceil;p^{k})$ FHS sets with $k\mid m$. It can be viewed as a generalization of the construction [18]. However, under our generalization we can present the more strictly optimal FHS sets that can’t be obtained by [18] (see Theorem 2). Although our second and third constructions does not give new parameters, it provides us a large number of choice for $\psi(x)$ and $f(x)$ (see Table 1).

2. Preliminaries

Throughout this paper we shall use the following notations.

• •

$p$ is an odd prime and $q=p^{m}>1$ is a $p$-power;

• •

$\mathbb{F}_{q}$ is the finite field of $q$ elements, and $\mathbb{F}_{q}^{*}=\mathbb{F}_{q}-\{0\}$ is the multiplicative group of $\mathbb{F}_{q}$;

• •

$\alpha$ is a primitive element of $\mathbb{F}_{q}$;

• •

$\mathrm{Tr}_{q^{n}/q}(x)=\sum\limits^{n-1}_{i=0}x^{q^{i}}$ is the trace map from $\mathbb{F}_{q^{n}}$ to its subfield $\mathbb{F}_{q}$;

• •

For $t\in\mathbb{Z}$, $\langle t\rangle_{n}\in\{0,\cdots,n-1\}$ denotes the remainder of $t$ by $n$.

• •

For all sequences $(x_{t})_{t=0}^{n-1}$ indexed by a subscript $t\in\{0,\cdots,n-1\}$, we denote $x_{t}=x_{\langle t\rangle_{n}}$ for any $t\in\mathbb{Z}$.

• •

For $X$ a finite set, $\#X$ denotes the cardinality of $X$. For the map $f:X\rightarrow Y$ and $y\in Y$, $f^{-1}(y)=\{x\in X:\ f(x)=y\}$ is the set of preimages of $y$.

2.1. Strictly Optimal FHS Sets

Let $F=\{f_{0},f_{1},\ldots,f_{d^{\prime}-1}\}$ be an alphabet of $d^{\prime}$ available frequencies. Let $\mathcal{F}$ be a set of $M$ frequency-hopping sequences of the form $(x_{i})_{i=0}^{N-1}$ with $x_{i}\in F$. We also call $\mathcal{F}$ an $(N,M;d^{\prime})$-FHS set. For $X=(x_{t})_{t=0}^{N-1}$ and $Y=(y_{t})_{t=0}^{N-1}$ in $\mathcal{F}$, the partial Hamming correlation of $X$ and $Y$ over a window of length $L\in\{1,\cdots,N\}$ starting from $j\in\{0,\cdots,N-1\}$ is

[TABLE]

where $h[a,b]=1$ if $a=b$ and [math] otherwise (i.e. the Kronecker $\delta$-function). In other words,

[TABLE]

If $X=Y$ (resp. $X\neq Y$), $H_{X,X}(\tau;j\mid L)$ (resp. $H_{X,Y}(\tau;j\mid L)$) is called the partial Hamming autocorrelation (resp. cross-correlation) of $X$ (resp. $X$ and $Y$). Define

[TABLE]

In 2009, Niu et al. [23] obtained the following bound on the maximum partial Hamming correlation of FHS sets: for any $(N,M;d^{\prime})$-FHS set $\mathcal{F}$, for any window length $1\leq L\leq N$,

[TABLE]

and

[TABLE]

where $I=\left\lfloor\frac{MN}{d^{\prime}}\right\rfloor$. The partial Hamming correlation bound in  (6) is the bound proved by Eun et al. [17] for the case $M=1$, the Lempel-Greenberger bound by [2] for $L=N$ and $M=1$, and the Peng-Fan bound by [3] for $L=N$.

In 2016, inspired by the idea of Ding et al. [4], Cai et al. [18] obtained the following results: for any $(N,M;d^{\prime})$-FHS set $\mathcal{F}$,

[TABLE]

and

[TABLE]

Definition 1**.**

Let $\mathcal{F}$ be an $(N,M;d^{\prime})$-FHS set.

(1) $\mathcal{F}$ is said to be strictly optimal with respect to the partial Hamming correlation if one of the bounds in (6) or (7) is achieved for any correlation window length $1\leq L\leq N$.

(2) $\mathcal{F}$ is said to be strictly optimal with respect to family size if one of the bounds in (8) or (9) is achieved for any correlation window length $2\leq L\leq N$.

2.2. Difference-balanced functions

Definition 2**.**

A function $f(x):\mathbb{F}_{q^{m}}\rightarrow\mathbb{F}_{q}$ is called balanced if $\#f^{-1}(x)=q^{m-1}$ for each $x\in\mathbb{F}_{q}$. It is called difference-balanced if the difference function $f(\delta x)-f(x)$ is balanced for any $\delta\in\mathbb{F}_{q^{m}}\setminus\{0,1\}$.

Remark 1**.**

In the literature (see [32]), balanced and difference-balanced functions are defined over $\mathbb{F}_{q^{m}}^{*}$. However, the following is clear. If assigning $f(0)=0$ to a balanced function $f$ over $\mathbb{F}_{q^{m}}^{*}$, one gets a balanced function over $\mathbb{F}_{q^{m}}$; for a balanced function $f$ over $\mathbb{F}_{q^{m}}$, then $f-f(0)$ is a balanced function over $\mathbb{F}_{q^{m}}^{*}$. If assigning $f(0)=b$ for any $b\in\mathbb{F}_{q}$ to a difference-balanced function $f$ over $\mathbb{F}_{q^{m}}^{*}$, one get a difference-balanced function over $\mathbb{F}_{q^{m}}$; for a difference-balanced function $f$ on $\mathbb{F}_{q}$, the restriction of $f$ on $\mathbb{F}_{q^{m}}^{*}$ is a difference-balanced function over $\mathbb{F}_{q^{m}}^{*}$.

Pott-Wang [32] tells us that a difference-balanced function $f$ such that $f(0)=0$ is always balanced. Moreover, the following is the list of all known difference-balanced functions from $\mathbb{F}_{q^{m}}$ to $\mathbb{F}_{q}$ satisfying $f(0)=0$:

1. (0)

Functions which are surjective and $\mathbb{F}_{q}$-linear. 2. (1)

Functions of the form

[TABLE]

where $d$ is a positive integer prime to $q^{m}-1$. 3. (2)

Functions of Helleseth-Gong type, which was discovered in [31]. 4. (3)

Functions of Lin type

[TABLE]

where $q=3$, $m=2l+1$ and $s=2\times 3^{l}+1$. The difference balance property of functions of this type was a conjecture of Lin [25] and proved by Hu et al. [30]. 5. (4)

Functions which are composites of functions of the previous types (when the composition is legal).

Definition 3**.**

Let $d$ be an integer prime to $q-1$. A function $f(x)$ from $\mathbb{F}_{q^{n}}$ onto $\mathbb{F}_{q}$ is called a $d$-form function if

[TABLE]

for any $y\in\mathbb{F}_{q}$ and $x\in\mathbb{F}_{q^{n}}$.

By definition, it is easy to see all the known difference-balanced functions are $d$-form functions: functions of type (0) are $1$-form functions, of type (1) are $d$-form functions, and of type (2) and (3) are $1$-form functions for the case $q=p$.

Lemma 1**.**

Let $f:\mathbb{F}_{q^{2}}\rightarrow\mathbb{F}_{q}$ be a $d$-form difference-balanced function and $\delta\notin\{0,1\}$, let $f_{\delta}(x)=f(\delta x)-f(x)$. Then $f^{-1}_{\delta}(0)=\mathbb{F}_{q}\cdot a_{\delta}$ for some $a_{\delta}\in\mathbb{F}^{*}_{q^{2}}$.

Proof.

On one hand $f^{-1}_{\delta}(0)$ is of order $q$ as $f_{\delta}$ is balanced. On the other hand, $f(c)=c^{d}f(a)$ if $c\in\mathbb{F}_{q}$, hence $f_{\delta}(a)=0$ implies that $f_{\delta}(ca)=0$ for any $c\in\mathbb{F}_{q}$. ∎

Remark 2**.**

All known difference-balanced functions $f:\mathbb{F}_{q^{2}}\rightarrow\mathbb{F}_{p}$ such that $f(0)=0$ are $d$-form functions belonging to one of the following two types: (i) a surjective $\mathbb{F}_{q}$-linear map; (ii) $(x\mapsto\psi(\mathrm{Tr}_{q^{2}/q}(x^{d})))$ or $(x\mapsto\mathrm{Tr}_{q^{2}/q}(\psi(x^{d})))$ where $\psi$ is an $\mathbb{F}_{q}$-linear automorphism of $\mathbb{F}_{q^{2}}$.

3. First Construction of Optimal FHS Sets

3.1. A generic construction

From now on, let $q=p^{m}$.

Definition 4**.**

For an $(n,M;q)$-FHS set $\mathcal{U}:=\{U^{i}=(u^{i}_{t})_{t=0}^{n-1}:0\leq i\leq M-1\}$ and a function $\phi(x)$ over $\mathbb{F}_{q}$, we say that $(\mathcal{U},\phi)$ satisfies $\mathbf{A1}$ if for any given triple $(i,j,\tau)\in[0,M-1]^{2}\times[0,n-1]-\{(i,i,0)\}$,

[TABLE]

For a vector $v=(v_{0},v_{1},\ldots,v_{n^{\prime}-1})\in\mathbb{F}^{n^{\prime}}_{q}$ and a function $\psi(x)$ over $\mathbb{F}_{q}$, we say that $(v,\psi)$ satisfies $\mathbf{A2}$ if for $b\in\mathbb{F}_{q}$,

[TABLE]

Theorem 1**.**

Let $N=nn^{\prime}$ with $\gcd(n,n^{\prime})=1$ and $M$ be positive integers. Suppose $\mathcal{U}$ is an $(n,M;q)$-FHS set such that $(\mathcal{U},\phi)$ satisfies $\mathbf{A1}$. Suppose $v\in\mathbb{F}_{q}^{n^{\prime}}$ such that $(v,\psi)$ satisfies $\mathbf{A2}$. Then the FHS set $\mathcal{S}:=\{S^{i}:\ 0\leq i\leq M-1\}$ with $S^{i}=(s^{i}_{t})_{t=0}^{N-1}$ defined by

[TABLE]

is an $(N,M;q)$-FHS set and for each correlation window length $L\in\{1,\cdots,N\}$,

[TABLE]

Proof.

For $0\leq\tau\leq N-1$, $0\leq i,j\leq M-1$ and $0\leq l\leq N-1$ such that $\tau\neq 0$ if $i=j$, by Eq. (1) we have

[TABLE]

Note that $u^{i}_{t}:=u^{i}_{\langle t\rangle_{n}}$, $v_{t}:=v_{\langle t\rangle_{n^{\prime}}}$ and $s^{i}_{t}:=s^{i}_{\langle t\rangle_{N}}$ for $t\in\mathbb{Z}$ by our convention.

The triple $(i,j,\tau)$ belongs to two disjoint cases, either $i=j$ and $\langle\tau\rangle_{n}=0$ or else. In both cases (the first case is trivial and the second follows from Eq. (12))

[TABLE]

with $b=0$ if and only if $i=j$ and $\langle\tau\rangle_{n}=0$. Note that if $i=j$, then $\tau\neq 0$, so one must have $\langle\tau\rangle_{n^{\prime}}\neq 0$ or $b\neq 0$. Then by (15), one has

[TABLE]

for any integers $z$ and $l$. Write the correlation window length $L$ as $L=m_{1}n^{\prime}+m_{2}$ where $0\leq m_{2}<n^{\prime}$. From (15) and (16), if $m_{2}=0$, then

[TABLE]

if $m_{2}>0$, then

[TABLE]

In both cases, we have $H_{S^{i},S^{j}}(\tau;l\mid L)\leq\left\lceil\frac{L}{n^{\prime}}\right\rceil$. Hence

[TABLE]

which completes the proof of this theorem. ∎

Remark 3**.**

The special case that $\phi(x)=\psi(x)=x$, $(\mathcal{U},\phi)$ and $(v,\psi)$ satisfying $\mathbf{A1}$ and $\mathbf{A2}$ was used in [18] to construct optimal FHS sets.

3.2. First class of optimal FHS sets.

Lemma 2**.**

Suppose $\mathbb{F}_{p^{k}}$ is a proper subfield of $\mathbb{F}_{q}$. Let $\{a_{0}=1,a_{1},\cdots,a_{p^{m-k}-1}\}$ be a set of additive coset representatives of $\mathbb{F}_{p^{k}}$ to $\mathbb{F}_{q}$. Let $\sigma$ be a bijection from $\{0,1,\cdots,p^{k}-1\}$ to $\mathbb{F}_{p^{k}}$ such that $\sigma(0)=0$. Construct $p^{m-k}$ sequences $\mathcal{U}=\{U^{i}=(u^{i}_{t})_{t=0}^{p^{k}-1}:0\leq i\leq p^{m-k}-1\}$ by

[TABLE]

Then $\mathcal{M}(\mathcal{U},L)=0$ and $\mathcal{U}$ is an $(p^{k},p^{m-k}-1,0;p^{m})$-FHS set. Let $P(x)=x^{e}+c_{e-1}x^{e-1}+\cdots+c_{0}\in\mathbb{F}_{q}[x]$ be a polynomial prime to $x^{m}-1$, and let $\phi_{P}(x)=x^{p^{e}}+c_{e-1}x^{p^{e-1}}+\cdots+c_{0}x:\mathbb{F}_{q}\rightarrow\mathbb{F}_{q}$. Then for any $(i,j,\tau)\in[0,p^{m-k}-1]^{2}\times[0,p^{k}-1]-\{(i,i,0)\}$, for any $0\leq t\leq p^{k}-1$,

[TABLE]

Hence $(\mathcal{U},\phi_{P})$ satisfies $\mathbf{A1}$.

Proof.

This is because $\phi_{P}$ is additive and defines a bijection from $\mathbb{F}_{q}$ to itself, and $a_{i}-a_{j}+\sigma(\tau)\neq 0$ if $(i,j,\tau)\neq(i,i,0)$. ∎

Lemma 3**.**

Let $\alpha$ be a primitive root of $\mathbb{F}_{q}$ and $v=(\alpha^{t})_{t=0}^{q-2}$. For $\gcd(d,q-1)=1$, set $\psi_{d}(x)=x^{d}$, then $(v,\psi_{d})$ satisfies $\mathbf{A2}$.

Proof.

Easy to check. ∎

Theorem 2**.**

Let $(\mathcal{U},\phi_{P})$ and $(v,\psi_{d})$ be given by Lemmas 2 and 3 respectively. Then the FHS set $\mathcal{S}$ constructed from $(\mathcal{U},\phi_{P})$ and $(v,\psi_{d})$ in Theorem 3 is a $(p^{k}(p^{m}-1),p^{m-k};p^{m})$-FHS set such that for any correlation window length $1\leq L\leq p^{k}(p^{m}-1)$,

[TABLE]

Moreover, $\mathcal{S}$ is a strictly optimal FHS set with optimal partial Hamming correlation with respect to the bound in (7) and with optimal family size with respect to the bound in (8).

Proof.

By Theorem 3, we know $\mathcal{S}$ is a $(p^{k}(p^{m}-1),p^{m-k};p^{m})$-FHS set and $\mathcal{M}(\mathcal{S};L)\leq\left\lceil\frac{L}{p^{m}-1}\right\rceil$. We are left to check the equality and the optimality of the partial Hamming correlation and family size with respect to the bounds in Eq.(7) and (8).

For $(N,M,d^{\prime})=(p^{k}(p^{m}-1),p^{m-k},p^{m})$,

[TABLE]

Then Niu et al.’s bound in Eq.(7) is that for any correlation window length $1\leq L\leq p^{k}(p^{m}-1)$, for any $M$ FHS set $\mathcal{F}$ of length $N$ and alphabet size $d^{\prime}$,

[TABLE]

Hence $\mathcal{M}(\mathcal{S};L)=\left\lceil\frac{L}{p^{m}-1}\right\rceil$ and $\mathcal{S}$ has optimal partial Hamming correlation with respect to the bound in (7).

Take $L=p^{m}-1$, then $\mathcal{M}(\mathcal{S};L)=1$. The bound (8) gives

[TABLE]

Note that the actual family size of $\mathcal{S}$ is exactly $p^{m-k}$, hence $\mathcal{S}$ has an optimal family size with respect to the bound in (8). ∎

Example 1**.**

Let $p=3$, $m=2$, $d=3$, $k=1$ and $\alpha$ be a primitive element of $\mathbb{F}_{3^{2}}$ over $\mathbb{F}_{3}$ satisfying $\alpha^{2}+\alpha+2=0$. Take $a_{i}\in\{0,\alpha,2\alpha\}$ with $0\leq i\leq 2$. Suppose $\mathcal{U}=\{(0,1,2);(\alpha,\alpha+1,\alpha+2);(2\alpha,2\alpha+1,2\alpha+2)\}$ and $v=(\alpha^{t})_{t=0}^{7}$. Set

[TABLE]

and $\sigma$ is identity mapping over $\mathbb{F}_{3}$. Then the sequence set $\mathcal{S}$ in Theorem 4 consists of the following three FHSs of the length 24:

[TABLE]

By computer experiments,

[TABLE]

$\mathcal{S}$* is strictly optimal with respect to the bound in (7), and also has an optimal family size with respect to the bound in (8).*

Example 2**.**

Let $p=3$, $m=4$, $d=5$, $k=2$, and $\alpha$ be a root of a primitive polynomial $x^{4}+x+2$ over $\mathbb{F}_{3}$. Set $\beta=\alpha^{10}\in\mathbb{F}_{3^{2}}\setminus\mathbb{F}_{3}$, and

[TABLE]

One know that $\{\alpha,\alpha\beta\}$ can be viewed as a complementary basis of $\mathbb{F}_{3^{2}}$, and take $a_{i}\in\{c_{0}\alpha+c_{1}\alpha^{11}:c_{0},c_{1}\in\mathbb{F}_{3}\}$. Suppose $\sigma(a)=\beta^{a}$ satisfying $\sigma(0)=0$ for $a\in\{1,2,\ldots,8\}$. Then

[TABLE]

and the sequence set $\mathcal{S}$ in Theorem 4 consists of 9 FHSs of the length 720, for $(a_{0},a_{1},a_{2})=(0,\alpha,\alpha^{11})$, three of them are listed below:

[TABLE]

By computer experiments, it can check that $\mathcal{S}$ is strictly optimal with respect to the bound in (7), and also has an optimal family size with respect to the bound in (8).

4. Construction of Optimal FHS Sets via the trace map

Let $q=p^{m}$ and $\alpha$ be a primitive root of $\mathbb{F}_{q}^{*}$. Let $T=\frac{q-1}{p-1}$. For a nonzero vector ${\mathbf{w}}=(a_{0},a_{1},\ldots,a_{k-1})\in\mathbb{F}_{q}^{k}$, let $V_{\mathbf{w}}=\langle a_{0},\cdots,a_{k-1}\rangle_{\mathbb{F}_{p}}$ be the $\mathbb{F}_{p}$-subspace of $\mathbb{F}_{q}$ generated by $\mathbf{w}$ and let $R(\mathbf{w})=\dim_{\mathbb{F}_{p}}V_{\mathbf{w}}$. Define the map $\mathcal{T}=\mathcal{T}_{\mathbf{w}}:\mathbb{F}_{q}\rightarrow\mathbb{F}_{p}^{k}$ by

[TABLE]

Lemma 4**.**

The map $\mathcal{T}$ is an $\mathbb{F}_{p}$-linear map whose kernel $\ker(\mathcal{T})=V_{\mathbf{w}}^{\bot}$ where $V_{\mathbf{w}}^{\bot}$ is the orthogonal complementary of $V_{\mathbf{w}}$ via the nondegenerate bilinear map $\mathrm{Tr}_{q/p}$. In particular, $\dim_{\mathbb{F}_{p}}\ker(\mathcal{T})=m-R(\mathbf{w})$ and $\dim_{\mathbb{F}_{p}}\mathrm{Im}(\mathcal{T})=R(\mathbf{w})$.

Proof.

An element $x\in\ker(\mathcal{T})$ if and only if $\mathrm{Tr}_{q/p}(a_{i}x)=0$, i.e., $x\bot a_{i}$ for all $i$, in other words, $x\in V_{\mathbf{w}}^{\bot}$. ∎

We construct optimal FHS sequences based on $\mathbf{w}$ with $R(\mathbf{w})=m-1$ or $m$.

4.1. Second class of optimal FHS Sets

Theorem 3**.**

Fix $\mathbf{w}=(a_{0},\cdots,a_{k-1})\in\mathbb{F}_{q}^{k}$ such that $R(\mathbf{w})=m-1$. Let $r\mid p-1$ such that $\gcd(r,m)=1$ and $n^{\prime}=\frac{q-1}{r}$. Let $v=(\alpha^{rt})_{0\leq t\leq n^{\prime}-1}$. Let $d\in\mathbb{Z}$ such that $\gcd(d,q-1)=1$ and $\psi$ be an $\mathbb{F}_{p}$-linear automorphism of $\mathbb{F}_{q}$. Define the sequence set

[TABLE]

where

[TABLE]

Then $\mathcal{S}$ is an $(n^{\prime},r;p^{m-1})$-FHS set and for each correlation window length $1\leq L\leq n^{\prime}$,

[TABLE]

Moreover, $\mathcal{S}$ is a strictly optimal FHS set with optimal partial Hamming correlation with respect to the bound in (7) and with optimal family size with respect to the bound in (8).

Proof.

As $\gcd(d,q-1)=1$, $\alpha^{d}$ is still a primitive root of $\mathbb{F}_{q}^{*}$. Replacing $\alpha$ by $\alpha^{d}$, we may assume $d=1$.

The alphabet set $\{s^{i}_{t}:\ 0\leq i\leq r-1,\ 0\leq t\leq n^{\prime}-1\}$ of $\mathcal{S}$ is nothing but $\mathrm{Im}(\mathcal{T})$, hence it is of order $p^{m-1}$ by Lemma 4.

We are left to compute $\mathcal{M}(\mathcal{S};L)$. Note that $\ker(\mathcal{T}\circ\psi)=\psi^{-1}\ker(\mathcal{T})$ is $1$-dimensional $\mathbb{F}_{p}$-vector space, pick $0\neq a\in\ker(\mathcal{T}\circ\psi)$, then $\ker(\mathcal{T}\circ\psi)=\mathbb{F}_{p}a$. Fix $i$, $j$ and $\tau$ such that if $i=j$, then $\tau\not\equiv 0\bmod{n^{\prime}}$. The value $H_{S^{i},S^{j}}(\tau;l\mid T)$ is nothing but the number of $t\in[l,l+L-1]$ such that $s^{i}_{t}=s^{j}_{t+\tau}$, in other words, $\alpha^{i+rt}-\alpha^{j+r(t+\tau)}=\alpha^{rt}(\alpha^{i}-\alpha^{j+r\tau})\in\mathbb{F}_{p}a$. Note that $i-j-r\tau$ is not a multiple of $q-1$, hence $\alpha^{i}-\alpha^{j+r\tau}\neq 0$. Let $b=(\alpha^{i}-\alpha^{j+r\tau})^{-1}a$, then

[TABLE]

Suppose we have $\alpha^{rt_{0}}\in\mathbb{F}^{*}_{p}b$ for some $t_{0}$ (otherwise $H_{S^{i},S^{j}}(\tau;l\mid L)=0$). Then

[TABLE]

Note that $\mathbb{F}^{*}_{p}=\{\alpha^{cT}:0\leq c\leq p-2\}$ and $\gcd(r,T)=\gcd(r,m)=1$, then

[TABLE]

This means that the number of $t\in[l,l+L-1]$ such that $s^{i}_{t}=s^{j}_{t+\tau}$ is at most $\left\lceil\frac{L}{T}\right\rceil$, i.e., $H_{S^{i},S^{j}}(\tau;l\mid L)\leq\left\lceil\frac{L}{T}\right\rceil$. Hence $\mathcal{M}(\mathcal{S},L)\leq\left\lceil\frac{L}{T}\right\rceil$.

On the other hand, the bound Eq. (7) gives

[TABLE]

Thus $\mathcal{M}(\mathcal{S},L)=\left\lceil\frac{L}{T}\right\rceil$ and $\mathcal{S}$ is optimal with respect to the bound in (7).

Set $L=\frac{q-1}{r}$, then $\mathcal{M}(\mathcal{S};L)=\frac{p-1}{r}$ and $d^{\prime}\mathcal{M(S;}L)<L$. According to the bound in (8), we have

[TABLE]

Thus the side of $\mathcal{S}$ is optimal with respect to the bound in (8). This completes the proof. ∎

Remark 4**.**

The special case that $d=1$ and $\psi$ is identity mapping of $\mathbb{F}_{q}$ was used in [16] to construct optimal FHS sets.

Example 3**.**

Let $p=5$, $m=d=3$, $r=2$ and $(a_{0},a_{1})=(1,\alpha)$, where $\alpha$ is a primitive element of $F_{5^{3}}$ over $\mathbb{F}_{5}$ generated by $\alpha^{3}+\alpha+1=0$. Suppose $\psi$ is Frobenius automorphism of $\mathbb{F}_{5^{3}}$. Then the set $\mathcal{S}$ of  (19) consists of the following two FHSs:

[TABLE]

By computation,

[TABLE]

$\mathcal{S}$* is strictly optimal with respect to the bound in (7), and also has an optimal family size with respect to the bound in (8).*

4.2. Third class of optimal FHS Sets

Theorem 4**.**

Fix $\mathbf{w}=(a_{0},\cdots,a_{k-1})$ and assume $R(\mathbf{w})=m$. Let $\theta$ be a primitive element of $\mathbb{F}_{q^{2}}$ over $\mathbb{F}_{q}$, $r$ an odd factor of $q-1$, $n^{\prime}=\frac{q^{2}-1}{r}$ and $v=(\theta^{rt})_{t=0}^{n^{\prime}-1}$. Suppose $f:\mathbb{F}_{q^{2}}\rightarrow\mathbb{F}_{q}$ is a $d$-form difference-balanced function. Define a sequence set $\mathcal{S}=:\{S^{i}=(s^{i}_{t})_{0\leq t\leq n^{\prime}-1}:0\leq i\leq r-1\}$ by

[TABLE]

Then $\mathcal{S}$ is an $(n^{\prime},r;q)$-FHS set and for each correlation window length $1\leq L\leq n^{\prime}$,

[TABLE]

Moreover, $\mathcal{S}$ is a strictly optimal FHS set with optimal partial Hamming correlation with respect to the bound in (7) and with optimal family size with respect to the bound in (8).

Proof.

By Lemma 4, we know the alphabet size of $\mathcal{S}$ is $p^{m}$, and the map $\mathcal{T}$ is injective, hence

[TABLE]

Fix $i,j,\tau$ such that if $i=j$ then $\tau\not\equiv 0\bmod{n^{\prime}}$. Then $\delta=\theta^{j-i+r\tau}\notin\{0,1\}$. By Lemma 1, the set $f^{-1}_{\delta}(0)$ of solutions of $f(\delta x)-f(x)=0$ is $\mathbb{F}_{q}a$ for some $a\neq 0$. If there exists $\theta^{i+rt_{0}}\in f^{-1}_{\delta}(0)$ (otherwise $H_{S^{i},S^{j}}(\tau;l\mid L)=0$), then $\theta^{i+rt_{0}}\in\mathbb{F}_{q}^{*}a$ and

[TABLE]

Any $c\in\mathbb{F}^{*}_{q}$ is of the form $c=\theta^{(q+1)s}$, Since $(r,q+1)=1$, we have

[TABLE]

Therefore, we have $H_{S^{i},S^{j}}(\tau;l\mid L)\leq\left\lceil\frac{L}{q+1}\right\rceil$ and

[TABLE]

We now check the strictly optimality of the sequence set $\mathcal{S}$, from (7),

[TABLE]

Thus $\mathcal{M}(\mathcal{S};L)=\left\lceil\frac{L}{q+1}\right\rceil$.

Taking $L=\frac{q^{2}-1}{r}$, then $\mathcal{M}(\mathcal{S};L)=\frac{q-1}{r}$ and $d^{\prime}\mathcal{M}(\mathcal{S};L)<L$. According to the bound in (8), we have

[TABLE]

Actually, the family size of $\mathcal{S}$ is exactly $r$. ∎

Remark 5**.**

Based on the theory of Galois ring, Eun et al. [17] obtained a class of optimal individual FHS sequence, which have length $q^{2}-1$, alphabet size $q$, and maximal partial Hamming autocorrelation $\left\lceil\frac{L}{q+1}\right\rceil$ for each window length $1\leq L\leq q^{2}-1$. Compared with the construction in [17], our construction gives new parameters when $r>1$, and may be more flexible by the choice of $f(x)$.

Example 4**.**

Let $p=q=7$, $r=3$, $k=1$ and $a_{0}=1$. Assume that $\theta$ is a primitive element of $\mathbb{F}_{7^{2}}$ over $\mathbb{F}_{7}$ generated by $\theta^{2}+3\theta-1=0$. Set

[TABLE]

Then, $\mathcal{S}$ consists of the following three FHSs of length 16:

[TABLE]

By computer experiments,

[TABLE]

Thus, $\mathcal{S}$ is strictly optimal with respect to the bound in (7), and also has an optimal family size with respect to the bound in (8).

5. Conclusion

In this paper, we constructed three classes of strictly optimal FHS sets with optimal partial Hamming correlation and optimal family size.