Optimal Constructions of Low-Hit Zone Frequency-Hopping Sequence Set Based on m-Sequence

Abstract

Quasi-Synchronous Frequency hopping (FH) Multiple Access (QS-FHMA) systems feature high communication efficiency, strong flexibility, and low operational costs, and they have been widely used in various FH communication scenarios such as satellite communication, military communication, and radio measurement. The low-hit zone (LHZ) FH sequences set (LHZ FHS set) plays a critical role in QS-FHMA systems, enabling user access with permissible time-delay offsets while maintaining superior performance. In this paper, three new methods to construct LHZ FHS sets based on m-sequences are proposed. The newly constructed sequence sets achieve optimality with respect to the Peng–Fan bound. Compared with existing LHZ FHS sets constructed from m-sequences, these new sequence sets offer more flexible parameters. Furthermore, due to the simple structure of m-sequences and their extensive adoption in engineering applications, the proposed new sequence sets possess significant practical value for engineering implementation.

1. Introduction

1.1. Background

Frequency hopping (FH) is a spread-spectrum technology that achieves spectral broadening through carrier frequency hopping [1,2,3,4]. In an FH system, the transmitting end employs pseudorandom sequences to control the central frequency of the data-modulated carrier wave, causing it to randomly hop over frequency sets according to a specific pattern. Correspondingly, the receiving end follows the same pattern for synchronization, demodulation, and signal recovery. The pseudorandom sequence is termed the FH sequence (FHS). When multiple users share the same spectrum resources, during certain time intervals, different users may simultaneously occupy the same frequency channel, and interference occurs due to the signals from different users overlapping. This type of interference is what is referred to as multiple-access interference (MAI). Studies have shown that MAI is the primary factor affecting the bit error rate (BER) performance of FH systems [5]. The severity of MAI between users is determined by the number of frequency collisions between their used FHSs.

As one type of FH system, Quasi-Synchronous FH Multiple Access (QS-FHMA) systems seamlessly integrate FH technology with multiple-access (MA) technology. In these systems, multiple users communicate over different frequencies according to predetermined FHSs (FH patterns) to achieve spectrum sharing and interference avoidance. QS-FHMA systems exhibit relatively relaxed requirements for time synchronization, allowing for a certain degree of time delay or frequency offset. For QS-FHMA systems, the concept of low-hit zone (LHZ) FHS set was introduced [6,7]. The asynchronicity of QS-FHMA systems is constrained near zero delay, and the MAI between users is determined by the Hamming correlation values of different FHSs within the synchronization error range. An LHZ FHS set possesses a delay zone Z near zero delay within which both the Hamming autocorrelation sidelobes of each sequence and the Hamming cross-correlation values between distinct sequences remain low, where Z is termed “LHZ”. For an LHZ FHS set, when the delay does not exceed Z, the Hamming correlation values are small and the number of sequences is large. QS-FHMA systems, supported by well-designed LHZ FHS sets, are integral to a variety of modern applications. For instance, in satellite communication, due to the long signal transmission distances and complex environments, interference issues are particularly prominent; the LHZ FHS sets support MA communication, optimize spectrum utilization, and significantly enhance the anti-interference capability and security of satellite communication. In military communications, the LHZ FHS sets are employed to prevent interception and jamming, thereby ensuring the confidentiality and integrity of transmitted information [8,9]. Additionally, radar systems utilize the LHZ FHS sets to evade detection and countermeasures, enhancing the reliability of signals in complex operational environments.

Current research on LHZ FHS sets focuses primarily on two aspects. (1) The first is exploring theoretical limits on the relationships between different parameters of LHZ FHS sets. Studies have shown that parameters such as the Hamming correlation values, sequence counts, frequency slot numbers, and sequence lengths of LHZ FHS sets are subject to certain theoretical constraints. These mathematical relationships are often referred to as “theoretical bounds” [10,11,12], serving as an important method for evaluating LHZ FHS set performance and guiding their design. (2) The second aspect is constructing LHZ FHS sets that achieve these theoretical bounds. An LHZ FHS set is termed “optimal” if its parameters satisfy the equality of the corresponding theoretical bound. The goal of LHZ FHS set design is to construct optimal LHZ FHS sets. In recent years, domestic and international scholars have actively engaged in research regarding the construction of LHZ FHS sets. Through prolonged exploration, several optimal or near-optimal LHZ FHS sets have been successfully developed (see [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28], for example). To sum up, there are several optimal construction methods: constructions based on m-sequences [13,14,15], constructions based on interleaving techniques [16,17,18,19], constructions based on Cartesian products [15,20], and constructions based on cyclotomic theory [21,22].

1.2. Motivation

The m-sequence features a relatively simple structure and has been extensively studied, making its engineering implementation particularly straightforward and widely adopted in engineering practice. Consequently, the generation mechanism of FHS sets based on m-sequences is simple and direct with strong technical adaptability, low engineering costs, and highly flexible scalability. These characteristics facilitate achieving a favorable balance between the performance of FHS and the complexity of system implementation. These motivate us to design optimal LHZ FHS sets from m-sequences and their d-decimated sequences to enhance the overall communication quality and performance of QS-FHMA systems as well as improve flexibility in engineering applications.

1.3. Contribution

Let the length of LHZ be $L_z$ and the length of FHS be N. The known constructions of LHZ FHS sets have limitations, such as Z divides N. Furthermore, the constructions in the literature [14,15] based on q-ary m-sequences with degree n and their d-decimated sequences have the limitations that it does not get the desired results for $q=2$ and $q^n-1$ is Mersenne prime. In this paper, we resolve the aforementioned problem by utilizing shift operation on m-sequences and design three classes of LHZ FHS sets with optimal maximum periodic Hamming correlation (MPHC) based on m-sequences, which share the same efficiency, computational complexities, and cost as the constructions based on m-sequences [13,14,15]. The contributions of this paper can be summarized as follows:

a. All the new constructions do not need to meet the requirement of $L_z$ divides N. That is, compared with the known constructions, the new ones have more flexible parameters and can generate more LHZ FHS sets with different parameters.

b. The first two methods among the new construction approaches derive from any q-ary m-sequences with degree n, even though $q=2$ and $q^n-1$ is Mersenne prime, and include some previously known parameters as special cases. The last one is based on the d-decimated sequences of any q-ary m-sequences with degree n, $q\ne 2$, which have parameters that are not covered in the literature.

1.4. Organization

The remainder of the paper is organized as follows: Section 2 summarizes some terminologies, notations related to FHS sets and the concept of q-ary m-sequences and their d-decimated sequences. Section 3 presents three new constructions of LHZ FHS sets with an optimal MPHC. Lastly, the conclusions of this paper are outlined in Section 4.

2. Preliminaries

For simplicity, in the remainder of this paper, we use $(N,M,q,L_z,h)$-LHZ FHS set to denote an LHZ FHS set of size M and length N over a frequency slot set of size q whose LHZ is $L_z$, and the MPHC is h within the LHZ. Additionally, the notations and definitions presented in

Table 1. Notations and definitions employed in the remainder of this paper.

Notations & Definitions	Description
${\left[x\right]}_y$	the maximum quotient when x is divided by y.
${〈x〉}_y$	the least positive integer of x modulo y.
$L^{\left(\zeta \right)}$	the operator for cyclic left shift by $\zeta$ positions.
$N(\mathbb{S},g)$	the number of g that occurs in any sequence $\mathbb{S}$.
${\mathrm{𝔽}}_q$, ${\mathrm{𝔽}}_{q^n}$	“the finite field” and “the extension field of a finite field”.
$T{r}_q^{q^n}\left(x\right)$	the trace function from ${\mathrm{𝔽}}_{q^n}$ to ${\mathrm{𝔽}}_q$.
${\mathrm{𝔽}}_{q^n}/\left\{0\right\}$	the set obtained by removing the element 0 from ${\mathrm{𝔽}}_{q^n}$.
$⌈x⌉$	the greatest positive integer not exceeding x.
$\lfloor x\rfloor$	the smallest positive integer that is not less than x.
⊕	the addition modulo q over ${\mathrm{𝔽}}_q$.
$x\nmid y$	x does not divide y.
$x\mid y$	x divides y.

will also be utilized.

2.1. Hamming Correlation

Assume that 𝔽 is a frequency set and $\mathbf{S}$ is an FHS set containing M FHSs with length N over 𝔽. For any two FHSs $\mathbf{x}=(x_0,x_1,\dots ,x_{N-1})$ and $\mathbf{y}=(y_0,y_1,\dots ,y_{N-1})$ in $\mathbf{S}$, at the time delay $\tau$, $1\le \tau <N$ if $\mathbf{x}=\mathbf{y}$ and $0\le \tau <N$ if $\mathbf{x}\ne \mathbf{y}$, the periodic Hamming correlation $H(\mathbf{x},\mathbf{y},\tau )$ of $\mathbf{x}$ and $\mathbf{y}$ is defined as follows.

$$H(\mathbf{x},\mathbf{y},\tau )=\sum _{t=0}^{N-1}h[x_t,y_{t+\tau \mathrm{mod}N}]$$

(1)

where $h[x_t,y_{t+\tau \mathrm{mod}N}]=1$ if $x_t=y_{t+\tau \mathrm{mod}N}$ and 0 otherwise. If $\mathbf{x}=\mathbf{y}$, then it is called the periodic Hamming autocorrelation. Otherwise, it is called the periodic Hamming cross-correlation.

Furthermore, the maximum periodic Hamming autocorrelation (MPHAC) $H_a\left(\mathbf{S}\right)$, the maximum periodic Hamming cross-correlation (MPHCC) $H_c\left(\mathbf{S}\right)$ and the MPHC $H\left(\mathbf{S}\right)$ are, respectively, defined by

$$\begin{array}{c}{H}_a\left(\mathbf{S}\right)=max\left\{H(\mathbf{x},\mathbf{x},\tau )\right|1\le \tau \le N-1,\forall \mathbf{x}\in \mathbf{S}\},\hfill \\ H_c\left(\mathbf{S}\right)=max\left\{H(\mathbf{x},\mathbf{y},\tau )\right|0\le \tau \le N-1,\forall \mathbf{x},\mathbf{y}\in \mathbf{S},\mathbf{x}\ne \mathbf{y}\},\hfill \\ H\left(\mathbf{S}\right)=max\{H_a\left(\mathbf{S}\right),H_c\left(\mathbf{S}\right)\}.\hfill \end{array}$$

(2)

2.2. LHZ FHS Set and Theoretical Bound

Let integers $L_a>0$, $L_c>0$. Then, the LHZ $L_z$ of $\mathbf{S}$ is defined as follows.

$$\begin{array}{cc}\hfill L_z& =min\{\underset{1\le \tau \le T^{\prime }}{max}\left\{T^{\prime }\right|H(\mathbf{x},\mathbf{x},\tau )\le L_a,\forall \mathbf{x}\in \mathbf{S}\},\hfill \\ \hfill & \underset{0\le \tau \le T^{\prime }}{max}\left\{T^{\prime }\right|H(\mathbf{x},\mathbf{y},\tau )\le L_c,\forall \mathbf{x},\mathbf{y}\in \mathbf{S},\mathbf{x}\ne \mathbf{y}\left\}\right\}.\hfill \end{array}$$

(3)

$\mathbf{S}$ is called an LHZ FHS set if $L_z>1$. In 2006, Peng et al. [10] obtained the lower bound on the MPHC of the LHZ FHS set.

Lemma 1

(Peng–Fan bound). For any set $\mathbf{S}$ of M FHSs with length N over F of size q, the low-hit zone Z, and the MPHC $H\left(\mathbf{S}\right)$, one has

$$H\left(\mathbf{S}\right)\ge ⌈{\displaystyle \frac{(MZ+M-q)N}{(MZ+M-1)q}}⌉.$$

(4)

Definition 1.

Let $\mathbf{S}$ be any LHZ FHS set. For the MPHC, $\mathbf{S}$ is said to be optimal if its parameters let the equal sign in (4) be true.

2.3. q-Ary m-Sequences and Its Decimated Sequence

Using the trace function $T{r}_q^{q^n}\left(x\right)$ from ${\mathrm{𝔽}}_{q^n}$ to ${\mathrm{𝔽}}_q$, any q-ary m-sequence of degree n over ${\mathrm{𝔽}}_q$, such as $\mathbf{e}$, can be presented as

$$\mathbf{e}=(T{r}_q^{q^n}\left(\beta \right),T{r}_q^{q^n}\left(\beta \alpha \right),\dots ,T{r}_q^{q^n}\left(\beta {\alpha }^{q^n-2}\right))$$

(5)

where $\beta \in {\mathrm{𝔽}}_{q^n}/\left\{0\right\}$, and $\alpha$ is a primitive element of ${\mathrm{𝔽}}_{q^n}$.

Any q-ary m-sequences have many good properties, and scholars such as Gong et al. [29]. have conducted in-depth research on them. Among these properties, the ideal k-tuple distribution property and shift-and-add property will be utilized in the designs of LHZ FHS sets in this paper.

Lemma 2

(Ideal k-tuple distribution property). For any q-ary m-sequence of degree n such as $\mathbf{e}$, for any k with $1\le k\le n$, the zero k-tuple $(0,0,...,0)$ occurs in $\mathbf{e}$ exactly $q^{n-k}-1$ times, and each nonzero k-tuple $({\lambda }_0,{\lambda }_1,\dots ,{\lambda }_{k-1})\in {\mathrm{𝔽}}_q^k$ occurs exactly $q^{n-k}$ times.

Lemma 3

(Shift-and-add property). Let $\mathbf{e}$ be any q-ary m-sequence of degree n. Then, for each pair $(i,j)$ with $0\le i,j\le q^n-1$, either there exists some k with $0\le g\le q^n-2$ such that

$$L^{\left(i\right)}\left(\mathbf{e}\right)+L^{\left(j\right)}\left(\mathbf{e}\right)=L^{\left(g\right)}\left(\mathbf{e}\right)$$

or $L^{\left(i\right)}\left(\mathbf{e}\right)+L^{\left(j\right)}\left(\mathbf{e}\right)$ is the zero sequence.

Assume that q is a prime power and $q\ne 2$, d is any positive integer satisfying $d\left|\right(q-1)$, and j is any integer with $0\le j\le d-1$. If the elements of a sequence $\mathbf{b}=(b_j\left(0\right),b_j\left(1\right),\dots ,b_j({\displaystyle \frac{q^n-1}{d}}-1))$ can be defined by

$$b_j\left(i\right)=e(j+id)=T{r}_q^{q^n}\left(\beta {\alpha }^{j+id}\right),$$

(6)

we call $\mathbf{b}$ a d-decimated sequence of $\mathbf{e}$. Note that there are d different d-decimated sequences of length ${\displaystyle \frac{q^n-1}{d}}$.

For the d-decimated sequence of any q-ary m-sequence with degree n, under the condition that $gcd(d,n)=1$, Zhou et al. [30] gave the run property stated in the following lemma.

Lemma 4.

For any d-decimated sequence of q-ary m-sequence with degree n, q is a prime power, $q\ne 2$, $gcd(d,n)=1$, for any k with $1\le k\le n$, and the run of zero with length k occurs exactly ${\displaystyle \frac{q^{n-k}-1}{d}}$ times.

3. LHZ FHS Sets with Optimal Maximum Periodic Hamming Correlation

In this section, based on a q-ary m-sequence of degree n and its d-decimated sequence, we construct three classes of LHZ FHS sets. Furthermore, we discuss and verify their optimality through analysis and simulation experiments.

Construction A: design based on any m-sequence.

Step 1: Let q be any prime power, and select any q-ary m-sequence of degree n, which is denoted as $\mathbf{e}$.

Step 2: Select any integer $L_z$ with $2\le L_z\le ⌊{\displaystyle \frac{q^n-1}{2}}⌋-1$ and then design the sequence set $\mathbf{C}=\{c_0,c_1,...,c_{⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋-1}\}$, $c_i=(c_i\left(0\right),c_i\left(1\right),\dots ,c_i(q^n-2)),0\le i\le ⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋-1$, where $c_i\left(t\right)$ is defined by

$$c_i\left(t\right)=e(t+(i+1)(L_z+1))$$

(7)

for $0\le t\le q^n-2$.

Step 3: Construct the desired FHS set ${\mathbf{S}}_{\mathbf{a}}=\{s_{a_0},s_{a_1},...,$$s_{a_{q^k⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋-1}}\}$, $s_{a_j}=(s_{a_j}\left(0\right),s_{a_j}\left(1\right),\dots ,s_{a_j}(q^n-2))$, $s_{a_j}\left(t\right)$ of which is given as

$$\begin{array}{cc}\hfill s_{a_j}\left(t\right)& =(c_{j^{\prime }}\left(t\right)\oplus {\mu }_0)+q(c_{j^{\prime }}(t+1)\oplus {\mu }_1)+\dots +\hfill \\ \hfill & q^{k-2}(c_{j^{\prime }}(t+k-2)\oplus {\mu }_{k-2})+q^{k-1}(c_{j^{\prime }}(t+k-1)\oplus {\mu }_{k-1})\hfill \end{array}$$

(8)

where $1\le k\le n$, $0\le j\le q^k⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋-1$, $j^{\prime }={\left[j\right]}_{q^k}$, $\mu ={〈j〉}_{q^k}$, $\mu ={\mu }_0+q{\mu }_1+\dots +q^{k-2}{\mu }_{k-2}+q^{k-1}{\mu }_{k-1}$, ${\mu }_{\delta }\in {𝔽}_q$, $\delta =0,1,..,k-1$.

Theorem 1.

According to the Peng–Fan bound (4), ${\mathbf{S}}_{\mathbf{a}}$ is a $(q^n-1,q^k⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋,q^k,L_z,q^{n-k})$-LHZ FHS set with optimal MPHC.

Proof. 

Let ${\mathbf{s}}_{a_{\rho }},{\mathbf{s}}_{a_{\gamma }}$ be any two FHSs in ${\mathbf{S}}_{\mathbf{a}}$. $v^{\prime }={\left[\rho \right]}_{q^k}$, $l^{\prime }={\left[\gamma \right]}_{q^k}$, $v={〈\rho 〉}_{q^k}=v_0+q{v}_1+\dots +q^{k-2}{v}_{k-2}+q^{k-1}{v}_{k-1}$, $l={〈\gamma 〉}_{q^k}=l_0+q{l}_1+\dots +q^{k-2}{l}_{k-2}+q^{k-1}{l}_{k-1}$, $v_{\eta }\in {𝔽}_q$, $l_{\eta }\in {𝔽}_q$, $\eta =0,1,..,k-1$. The periodic Hamming correlation $H({\mathbf{s}}_{a_{\rho }},{\mathbf{s}}_{a_{\gamma }},\tau )$ of ${\mathbf{s}}_{a_{\rho }}$ and ${\mathbf{s}}_{a_{\gamma }}$ at the time delay $\tau$, $1\le \tau <q^n-1$ if $\rho =\gamma$ and $0\le \tau <q^n-1$ if $\rho \ne \gamma$, equals the number of solutions to the equation ${\mathbf{s}}_{a_{\rho }}-{\mathbf{s}}_{a_{\gamma }}=0$ for $t=0,1,\dots ,q^n-2$; that is,

$$\begin{array}{c}(c_{v^{\prime }}(t+\tau )\oplus v_0)+q(c_{v^{\prime }}(t+\tau +1)\oplus v_1)+\dots +q^{k-1}(c_{v^{\prime }}(t+\tau +k-1)\oplus v_{k-1})\hfill \\ -((c_{l^{\prime }}\left(t\right)\oplus l_0)+q(c_{l^{\prime }}(t+1)\oplus l_1)+\dots +q^{k-1}(c_{l^{\prime }}(t+k-1)\oplus l_{k-1}))=0\hfill \end{array}$$

(9)

for $t=0,1,\dots ,q^n-2$. Since $1,q,q^2\dots ,q^{k-1}$ is linear independence, we have the following system of equations:

$$\left\{\begin{array}{c}\left(c_{v^{\prime }}(t+\tau )\oplus v_0\right)-\left(c_{l^{\prime }}\left(t\right)\oplus l_0\right)=0\hfill \\ \left(c_{v^{\prime }}(t+\tau +1)\oplus v_1\right)-\left(c_{l^{\prime }}(t+1)\oplus l_1\right)=0\hfill \\ \dots \hfill \\ \left(c_{v^{\prime }}(t+\tau +k-1)\oplus v_{k-1}\right)-\left(c_{l^{\prime }}(t+k-1)\oplus l_{k-1}\right)=0\hfill \end{array}\right.$$

(10)

for $t=0,1,\dots ,q^n-2$.

Since ⊕ is the addition modulo q over the finite field $F_q$, one can have

$$\left\{\begin{array}{c}\left(c_{v^{\prime }}(t+\tau )-c_{l^{\prime }}\left(t\right)\right)-\left(l_0-v_0\right)=0\hfill \\ \left(c_{v^{\prime }}(t+\tau +1)-c_{l^{\prime }}(t+1)\right)-\left(l_1-v_1\right)=0\hfill \\ \dots \hfill \\ \left(c_{v^{\prime }}(t+\tau +k-1)-c_{l^{\prime }}(t+k-1)\right)-\left(l_{k-1}-v_{k-1}\right)=0\hfill \end{array}\right.$$

(11)

for $t=0,1,\dots ,q^n-2$.

Thus, $H({\mathbf{s}}_{a_{\rho }},{\mathbf{s}}_{a_{\gamma }},\tau )$ can be expressed as

$$\begin{array}{cc}\hfill & H({\mathbf{s}}_{a_{\rho }},{\mathbf{s}}_{a_{\gamma }},\tau )=\sum _{t=0}^{q^n-2}h\left[\right(c_{v^{\prime }}(t+\tau )-c_{l^{\prime }}\left(t\right),c_{v^{\prime }}(t+\tau +1)-c_{l^{\prime }}(t+1),\dots ,\hfill \\ \hfill & c_{v^{\prime }}(t+\tau +k-1)-c_{l^{\prime }}(t+k-1)),(l_0-v_0,l_1-v_1,...,l_{k-1}-v_{k-1})].\hfill \end{array}$$

(12)

According to (7), when $t=0,1,\dots ,p^n-2$, $c_{v^{\prime }}(t+\tau )$ goes through each element in sequence $L^{(\tau +(v^{\prime }+1)(L_z+1))}\left(\mathbf{e}\right)$. Similarly, $c_{l^{\prime }}\left(t\right)$ goes through each element in sequence $L^{\left((l^{\prime }+1)(L_z+1)\right)}\left(\mathbf{e}\right)$. So $c_{v^{\prime }}(t+\tau )-c_{l^{\prime }}\left(t\right)$ goes through each element in sequence $L^{(\tau +(v^{\prime }+1)(L_z+1))}\left(\mathbf{e}\right)-L^{\left((l^{\prime }+1)(L_z+1)\right)}\left(\mathbf{e}\right)$, and it does the same in the sequence

$${\mathbf{S}}_{\theta }=L^{(\tau +(v^{\prime }-l^{\prime })(L_z+1))}\left(\mathbf{e}\right)-\mathbf{e}.$$

(13)

$(c_{v^{\prime }}(t+\tau )-c_{l^{\prime }}\left(t\right),c_{v^{\prime }}(t+\tau +1)-c_{l^{\prime }}(t+1),\dots ,c_{v^{\prime }}(t+\tau +k-1)-c_{l^{\prime }}(t+k-1))$ is the consecutive k-tuples in ${\mathbf{S}}_{\theta }$. Thus, (12) is equivalent to (14).

$$\begin{array}{cc}\hfill H({\mathbf{s}}_{a_{\rho }},{\mathbf{s}}_{a_{\gamma }},\tau ),\tau )& =\sum _{t=0}^{q^n-2}h[({\mathbf{S}}_{\theta }\left(t\right),{\mathbf{S}}_{\theta }(t+1),...,{\mathbf{S}}_{\theta }(t+k-1)),\hfill \\ \hfill & (l_0-v_0,l_1-v_1,\dots ,l_{k-1}-v_{k-1})]\hfill \\ \hfill & =N({\mathbf{S}}_{\theta },(l_0-v_0,l_1-v_1,...,l_{k-1}-v_{k-1})).\hfill \end{array}$$

(14)

We then analyze the MPHAC, MPHCC and MPHC of $\mathbf{S}$, and the following cases exist here.

Case 1: $\rho =\gamma$. Obviously, $(l_0-v_0,l_1-v_1,...,l_{k-1}-v_{k-1})=(0,0,...,0)$.

In this case, according to Lemma 3 and (13), ${\mathbf{S}}_{\theta }=L^{\left(\tau \right)}\left(\mathbf{e}\right)-\mathbf{e}$ is still a q-ary m-sequence of degree n for any $\tau$. Consequently, according to Lemma 2, $H({\mathbf{s}}_{a_{\rho }},{\mathbf{s}}_{a_{\gamma }},\tau )=q^{n-k}-1$ for any $\rho =\gamma$. The MPHAC can be obtained as follows.

$$H_a\left({\mathbf{S}}_{\mathbf{a}}\right)=q^{n-k}-1.$$

(15)

Case 2: $\rho \ne \gamma$.

Case 2. 1: $v^{\prime }=l^{\prime }$.

$(l_0-v_0,l_1-v_1,...,l_{k-1}-v_{k-1})\ne (0,0,...,0)$ because of $\rho \ne \gamma$. Based on Lemma 3, ${\mathbf{S}}_{\theta }=L^{\left(\tau \right)}\left(\mathbf{e}\right)-\mathbf{e}$ is still a q-ary m-sequence of degree n for any $\tau$. Thus, it follows that $H({\mathbf{s}}_{a_{\rho }},{\mathbf{s}}_{a_{\gamma }},\tau )=q^{n-k}$ from Lemma 2.

Case 2. 2: $v^{\prime }\ne l^{\prime }$.

${\mathbf{S}}_{\theta }$ changes to a sequence whose all items are zero when $\tau =(l^{\prime }-v^{\prime })(L_z+1)$. Consequently, from (14), it can be deduced that

$$\begin{array}{c}\hfill H({\mathbf{s}}_{a_{\rho }},{\mathbf{s}}_{a_{\gamma }},\tau )=\left\{\begin{array}{cc}{q}^n-1\hfill & \begin{array}{c}\mathrm{if}\left(l_0-v_0,...,l_{k-1}-v_{k-1}\right)=\left(0,...,0\right),\hfill \end{array}\\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(16)

${\mathbf{S}}_{\theta }$ is still a q-ary m-sequence of degree n for any $\tau$ when $\tau \ne (l^{\prime }-v^{\prime })(L_z+1)$ from Lemma 3. It then follows, based on (14), that

$$\begin{array}{c}\hfill H({\mathbf{s}}_{a_{\rho }},{\mathbf{s}}_{a_{\gamma }},\tau )=\left\{\begin{array}{cc}{q}^{n-k}-1\hfill & \begin{array}{c}\mathrm{if}\left(l_0-v_0,...,l_{k-1}-v_{k-1}\right)=\left(0,...,0\right),\hfill \end{array}\\ q^{n-k}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(17)

Based on the above discussions, for $\tau \le L_z$, the MPHCC of $\mathbf{S}$ is

$$\begin{array}{c}\hfill H_c\left({\mathbf{S}}_{\mathbf{a}}\right)=q^{n-k}.\end{array}$$

(18)

Therefore, ${\mathbf{S}}_{\mathbf{a}}$ is an LHZ FHS set, the MPHC of which is equal to

$$\begin{array}{c}\hfill H\left({\mathbf{S}}_{\mathbf{a}}\right)=q^{n-k}.\end{array}$$

(19)

for $\tau \le L_z$.

Let ${\lambda }_{right}$ denote the right side of Peng–Fan bound (4), input all the parameters of ${\mathbf{S}}_{\mathbf{a}}$, and ${\lambda }_{right}$ can be given by

$$\begin{array}{cc}\hfill {\lambda }_{right}& =⌈{\displaystyle \frac{q^k⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-q^k}{q^k⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1}}·{\displaystyle \frac{q^n-1}{q^k}}⌉\hfill \\ \hfill & =⌈{\displaystyle \frac{q^n-1}{q^k}}-{\displaystyle \frac{q^k-1}{q^k⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1}}·{\displaystyle \frac{q^n-1}{q^k}}⌉\hfill \\ \hfill & =⌈q^{n-k}-{\displaystyle \frac{(q^k⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1)+(q^k-1)(q^n-1)}{(q^k⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1)q^k}}⌉.\hfill \end{array}$$

(20)

It is obvious that ${\displaystyle \frac{(q^k⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1)+(q^k-1)(q^n-1)}{(q^k⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1)q^k}}>0$. Furthermore, let $q^n-1=g(L_z+1)+R$, $0\le R\le L_z\le ⌊{\displaystyle \frac{q^n-1}{2}}⌋-1\le {\displaystyle \frac{q^n-1}{2}}-1$, and we have

$$\begin{array}{cc}\hfill & (q^k⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋(L_z+1)-1)q^k-(q^k⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋(L_z+1)-1)-(q^k-1)(q^n-1)\hfill \\ \hfill & =(q^k-1)(q^k⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋(L_z+1)-q^n)\hfill \\ \hfill & =(q^k-1)(q^k(q^n-1-R)-q^n)\hfill \\ \hfill & >(q^k-1)(q^k{\displaystyle \frac{q^n+1}{2}}-q^n)\hfill \\ \hfill & >0.\hfill \end{array}$$

(21)

Thus, $0<{\displaystyle \frac{(q^k⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1)+(q^k-1)(q^n-1)}{(q^k⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1)q^k}}<1$. One can have

$${\lambda }_{right}=q^{n-k}.$$

(22)

The equal sign in Peng–Fan bound (4) is true. Thus, ${\mathbf{S}}_{\mathbf{a}}$ is an optimal LHZ FHS set.

The proof is then completed. □

Remark 1.

When $L_z={\displaystyle \frac{q^n-1}{q-1}}$-1, ${\mathbf{S}}_{\mathbf{a}}$ is a $(q^n-1,q^k(q-1),q^k,{\displaystyle \frac{q^n-1}{q-1}}-1,q^{n-k})$-LHZ FHS set which is exactly the sequence set constructed by Han et al. [14]. And when $T(L_z+1)=q^n-1$, ${\mathbf{S}}_{\mathbf{a}}$ is a $(q^n-1,q^{k}T,q^k,L_z,q^{n-k})$-LHZ FHS set which is exactly the sequence set constructed by Zhou et al. [15]. However, their constructions have minor issues. For instance, we do not get the desired results in the construction of Han et al. when $q=2$ and in the design of Zhou et al. when $q^n-1$ is Mersenne prime. It is obvious that ${\mathbf{S}}_{\mathbf{a}}$ in this paper can be designed by any m-sequence including a binary m-sequence under the condition of $(L_z+1)\nmid (q^n-1)$. That is, the designs of Han et al. and Zhou et al. are all the special cases of construction A.

We then illustrate the construction A by the following example, which is designed by a binary m-sequence.

Example 1.

Let $q=2$, $n=9$. A binary m-sequence $\mathbf{e}$ with degree 9, which is generated by $f\left(x\right)=x^7+x^6+x^5+x^4+1$, is listed below.

$\mathbf{e}$ = (1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, …).

Let $L_z=49$. After performing a shift operation on $\mathbf{e}$, we obtain $\mathbf{C}=\{c_0,c_1,...,c_9\}$ as follows.

$\mathbf{C}=$ {$c_0$ = (0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, …),

                  …

$c_9$ = (1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, …).}.

Let $k=4$. Then, according to step 3 of construction A, ${\mathbf{S}}_{\mathbf{a}}=\{s_{a_0},s_{a_1},...,s_{a_{159}}\}$ is obtained as follows.

${\mathbf{S}}_{\mathbf{a}}=$ {$s_{a_0}$ = (0, 0, 0, 0, 0, 1, 3, 7, 15, 14, 12, 8, 0, 1, 2, 4, 8, 1, 3, 7, 15, 14, 13, 10, 4, 9, 3, 6, 12, 9, 2, 4, 9, 2, 4, 8, 0, 1, 2, 5, 11, 7, 15, 14, 12, 8, 1, 3, 6, 12, 9, 3, 7, 15, 14, …),

                  …

$s_{a_{159}}$ = (3, 7, 15, 14, 12, 8, 1, 2, 5, 11, 6, 13, 11, 7, 14, 12, 9, 2, 5, 10, 4, 9, 2, 4, 8, 0, 0, 1, 2, 4, 9, 3, 6, 12, 9, 3, 7, 14, 13, 10, 4, 8, 1, 3, 7, 15, 15, 14, 13, 11, 7, 15, 14, 12, 8, …)}.

Computer experiments show that the MPHAC of ${\mathbf{S}}_{\mathbf{a}}$ is equal to 31 for $1\le \tau \le 510$, and the MPHCC distribution is identical to the MPHC distribution, as illustrated in Figure 1. Thus, the MPHC of ${\mathbf{S}}_{\mathbf{a}}$ equals 32 for $\tau \le 49$. It is easy to check that ${\mathbf{S}}_{\mathbf{a}}$ is an optimal $(511,160,16,49,32)$-LHZ FHS set according to Peng–Fan bound (4).

Compared with the construction of Zhou et al. [15], different values are taken for $L_Z$, and we can obtain more LHZ FHS sets with optimal MPHC by Example 1, which are listed in

Table 2. LHZ FHS sets with optimal MPHC constructed by Example 1 and existing ones.

Ref.	Parameters	Parameters
this paper	(511, 32, 16, 170–254, 32)	(511, 48, 16, 127–169, 32)
	(511, 64, 16, 102–126, 32)	(511, 80, 16, 85–101, 32)
	(511, 96, 16, 73–84, 32)	(511, 112, 16, 63–72, 32)
	(511, 128, 16, 56–62, 32)	(511, 144, 16, 51–57, 32)
	…	…
	(511, 2032, 16, 3, 32)	(511, 2720, 16, 2, 32)
[15]	(511, 1168, 16, 6, 32)	(511, 112, 16, 72, 32)

. Furthermore, under the same conditions as Example 1, the construction of Han et al. [14] fails to yield the desired results.

Construction B: design based on any m-sequence.

First, let q be any prime power, and construct a q-ary m-sequence with degree n, such as $\mathbf{e}$.

Second, select any integer $L_z$ with $2\le L_z\le ⌊{\displaystyle \frac{q^n-1}{2}}⌋-1$, define a sequence set $\mathbf{a}=(a_0,a_1,...,a_{⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋-1})$, $a_i=(a_i\left(0\right),a_i\left(1\right),\dots ,a_i(q^n-2))$, and $a_i\left(t\right)$ is given as

$$a_i\left(t\right)=e(t+(i+1)(L_z+1)).$$

(23)

Last, define the desired FHS set ${\mathbf{S}}_{\mathbf{b}}=\{s_{b_0},s_{b_1},\dots ,s_{b_{⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋-1}}\}$ where $s_{b_w}=(s_{b_w}\left(0\right),s_{b_w}\left(1\right),$$...,s_{b_w}(q^n-2)),0\le w\le ⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋-1$, $s_{b_w}\left(t\right)$ is defined by

$$s_{b_w}\left(t\right)=a_w\left(t\right)+q{a}_w(t+1)+\dots +q^{k-1}{a}_w(t+k-1),$$

(24)

where $0\le t\le q^n-2$ and $1\le k\le n$.

Theorem 2.

According to Peng–Fan bound (4), ${\mathbf{S}}_{\mathbf{b}}$ is a $(q^n-1,⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋,q^k,L_z,q^{n-k}-1)$-LHZ FHS set with an optimal MPHC.

Proof. 

Let ${\mathbf{s}}_{{\mathbf{b}}_{\mathbf{w}}},{\mathbf{s}}_{{\mathbf{b}}_{\mathbf{l}}}$ be any two FHSs in ${\mathbf{S}}_{\mathbf{b}}$. According to the proof of Theorem 1, the periodic Hamming correlation $H({\mathbf{s}}_{{\mathbf{b}}_{\mathbf{w}}},{\mathbf{s}}_{{\mathbf{b}}_{\mathbf{l}}},\tau )$ of ${\mathbf{s}}_{{\mathbf{b}}_{\mathbf{w}}}$ and ${\mathbf{s}}_{{\mathbf{b}}_{\mathbf{l}}}$ at the time delay $\tau$ can be calculated as follows.

$$\begin{array}{cc}\hfill H({\mathbf{s}}_{{\mathbf{b}}_{\mathbf{w}}},{\mathbf{s}}_{{\mathbf{b}}_{\mathbf{l}}},\tau )& =\sum _{t=0}^{q^n-1}h\left[\right(e(t+(w+1)(L_z+1)+\tau )-e(t+(l+1)(L_z+1)),\hfill \\ \hfill & e(t+(w+1)(L_z+1)+\tau +1)-e(t+(l+1)(L_z+1)+1),\hfill \\ \hfill & e(t+(w+1)(L_z+1)+\tau +k-1)\hfill \\ \hfill & -e(t+(l+1)(L_z+1)+k-1)),(0,0,...,0)]\hfill \end{array}$$

(25)

where $1\le \tau <q^n-1$ if $w=l$ and $0\le \tau <q^n-1$ if $w\ne l$.

It is easy to check that $H\left({\mathbf{S}}_{\mathbf{b}}\right)=q^{n-k}-1$ for $\tau \le L_z$.

Then, we input all the parameters of ${\mathbf{S}}_{\mathbf{b}}$ on the right side of Zhou–Peng bound (4), the value of which is as follows.

$$\begin{array}{cc}\hfill {\lambda }_{right}& =⌈{\displaystyle \frac{⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-q^k}{⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1}}·{\displaystyle \frac{q^n-1}{q^k}}⌉\hfill \\ \hfill & =⌈{\displaystyle \frac{q^n-1}{q^k}}-{\displaystyle \frac{q^k-1}{⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1}}·{\displaystyle \frac{q^n-1}{q^k}}⌉\hfill \\ \hfill & =⌈q^{n-k}-1-{\displaystyle \frac{(q^k-1)(q^n-⌊\frac{q^n-1}{L_z+1}⌋(L_z+1))}{\left(⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1\right)q^k}}⌉.\hfill \end{array}$$

(26)

It can be easily verified that $0<{\displaystyle \frac{(q^k-1)(q^n-⌊\frac{q^n-1}{L_z+1}⌋(L_z+1))}{\left(⌊\frac{q^n-1}{L_z+1}⌋(L_z+1)-1\right)q^k}}<1$ which leads to

$${\lambda }_{right}=q^{n-k}-1.$$

(27)

The results show that the theorem holds. □

Construction C: design based on d-decimated sequence of q-ary m-sequence, $q\ne 2$.

Step 1: Let q be any prime power with $q\ne 2$, and construct a d-decimated sequence set $\mathbf{a}=\{a_0,a_1,...,a_{d-1}\}$ of any q-ary m-sequence with degree n, such as

$$\begin{array}{c}\hfill a_i=(T{r}_q^{q^n}\left(\beta {\alpha }^i\right),T{r}_q^{q^n}\left(\beta {\alpha }^{i+d}\right),\dots ,T{r}_q^{q^n}(\beta {\alpha }^{i+({\displaystyle \frac{q^n-1}{d}}-1)d}))\end{array}$$

(28)

where $0\le i\le d-1$, $d\mid (q-1)$ and $gcd(d,n)=1$.

Step 2: Select any integer $L_z$ with $2\le L_z\le ⌊{\displaystyle \frac{q^n-1}{2d}}⌋-1$ and define a sequence set $\mathbf{b}=\{b_0,b_1,$$...,b_{⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋d-1}\}$, $b_j=(b_j\left(0\right),b_j\left(1\right),\dots ,b_j({\displaystyle \frac{q^n-1}{d}}-1))$, where $b_j\left(t\right)$ is defined by

$$b_j\left(t\right)=a_{{\left[j\right]}_{⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋}}(t+({〈j〉}_{⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋}+1)(L_z+1))$$

(29)

where $0\le j\le ⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋d-1$.

Step 3: Design the FHS set ${\mathbf{S}}_{\mathbf{c}}=\{s_{c_0},s_{c_1},\dots ,s_{c_{⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋d-1}}\}$, $s_{c_h}=(s_{c_h}\left(0\right),s_{c_h}\left(1\right),$$...,s_{c_h}({\displaystyle \frac{q^n-1}{d}}-1))$, where $s_{c_h}\left(t\right)$ is defined by

$$s_{c_h}\left(t\right)=b_h\left(t\right)+q{b}_h(t+1)+...+q^{k-1}{b}_h(t+k-1)$$

(30)

where $0\le h\le ⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋d-1$, $1\le k\le n$.

Theorem 3.

According to Peng–Fan bound (4), ${\mathbf{S}}_{\mathbf{c}}$ is an optimal LHZ FHS set with parameters $({\displaystyle \frac{q^n-1}{d}},⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋d,q^k,$$L_z,{\displaystyle \frac{q^{n-k}-1}{d}})$-LHZ FHS set under the MPHC.

Proof. 

Let ${\alpha }^{\prime }={\left[\alpha \right]}_{⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋}$, ${\alpha }_0={〈\alpha 〉}_{⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋}$, ${\beta }^{\prime }={\left[\beta \right]}_{⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋}$, ${\beta }_0={〈\beta 〉}_{⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋}$. For any two FHSs ${\mathbf{s}}_{c_{\alpha }},{\mathbf{s}}_{c_{\beta }}$ in ${\mathbf{S}}_{\mathbf{c}}$, the periodic Hamming correlation $H({\mathbf{s}}_{c_{\alpha }},{\mathbf{s}}_{c_{\beta }},\tau )$ of ${\mathbf{s}}_{c_{\alpha }}$ and ${\mathbf{s}}_{c_{\beta }}$ at the time delay $\tau$ , $1\le \tau <{\displaystyle \frac{q^n-1}{d}}-1$ if $\alpha =\beta$ and $0\le \tau <{\displaystyle \frac{q^n-1}{d}}-1$ if $\alpha \ne \beta$, is equal to the number of solutions to the equation ${\mathbf{s}}_{c_{\alpha }}-{\mathbf{s}}_{c_{\beta }}=0$ for $t=0,1,\dots ,{\displaystyle \frac{q^n-1}{d}}-1$, that is,

$$\begin{array}{c}{b}_{\alpha }(t+\tau )+q{b}_{\alpha }(t+\tau +1)+\dots +q^{k-1}{b}_{\alpha }(t+\tau +k-1)\hfill \\ -(b_{\beta }\left(t\right)+q{b}_{\beta }(t+1)+\dots +q^{k-1}{b}_{\beta }(t+k-1)=0\hfill \end{array}$$

(31)

for $t=0,1,\dots ,{\displaystyle \frac{q^n-1}{d}}-1$.

It is obvious that $1,q,q^2,\dots ,q^{k-1}$ are linearly independent. Accordingly, we have

$$\left\{\begin{array}{c}{b}_{\alpha }(t+\tau )-b_{\beta }\left(t\right)=0\hfill \\ b_{\alpha }(t+\tau +1)-b_{\beta }(t+1)=0\hfill \\ \dots \hfill \\ b_{\alpha }(t+\tau +k-1)-b_{\beta }(t+k-1)=0\hfill \end{array}\right.$$

(32)

for $t=0,1,\dots ,{\displaystyle \frac{q^n-1}{d}}-1$.

Furthermore, based on the expression (29), we can obtain

$$\left\{\begin{array}{c}{a}_{{\alpha }^{\prime }}(t+({\alpha }_0+1)(L_z+1)+\tau )-a_{{\beta }^{\prime }}(t+({\beta }_0+1)(L_z+1))=0\hfill \\ a_{{\alpha }^{\prime }}(t+({\alpha }_0+1)(L_z+1)+\tau +1)-a_{{\beta }^{\prime }}(t+({\beta }_0+1)(L_z+1)+1)=0\hfill \\ ...\hfill \\ a_{{\alpha }^{\prime }}(t+({\alpha }_0+1)(L_z+1)+\tau +k-1)-a_{{\beta }^{\prime }}(t+({\beta }_0+1)(L_z+1)+k-1)=0\hfill \end{array}\right.$$

(33)

for $t=0,1,\dots ,{\displaystyle \frac{q^n-1}{d}}-1$.

Thus, $H({\mathbf{s}}_{a_{\rho }},{\mathbf{s}}_{a_{\gamma }},\tau )$ can be expressed as

$$\begin{array}{cc}\hfill H({\mathbf{s}}_{c_{\alpha }},{\mathbf{s}}_{c_{\beta }},\tau )& =\sum _{t=0}^{(q^n-1)/d-1}h\left[\right(a_{{\alpha }^{\prime }}(t+({\alpha }_0+1)(L_z+1)+\tau )-a_{{\beta }^{\prime }}(t+({\beta }_0+1)(L_z+1)),\hfill \\ \hfill & a_{{\alpha }^{\prime }}(t+({\alpha }_0+1)(L_z+1)+\tau +1)-a_{{\beta }^{\prime }}(t+({\beta }_0+1)(L_z+1)+1),\hfill \\ \hfill & \dots ,a_{{\alpha }^{\prime }}(t+({\alpha }_0+1)(L_z+1)+\tau +k-1)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -a_{{\beta }^{\prime }}(t+({\beta }_0+1)(L_z+1)+k-1)),(0,0,...,0)]\hfill \end{array}$$

(34)

where $1\le \tau \le {\displaystyle \frac{q^n-1}{d}}-1$ if $\alpha =\beta$ and $0\le \tau \le {\displaystyle \frac{q^n-1}{d}}-1$ if $\alpha \ne \beta$.

We then analyze $H({\mathbf{s}}_{c_{\alpha }},{\mathbf{s}}_{c_{\beta }},\tau )$ from the perspective of the shift-and-add property of m-sequences. According to the definition of $\mathbf{a}$ (28), it is easy to verify that $L^{(\tau +({\alpha }_0+1)(L_z+1))}\left(a_{{\alpha }^{\prime }}\right)$ is a d-decimated sequence of $L^{({\alpha }^{\prime }+(\tau +({\alpha }_0+1)(L_z+1))d)}\left(e\right)$ and $L^{\left(({\beta }_0+1)(L_z+1)\right)}\left(a_{{\beta }^{\prime }}\right)$ is a d-decimated sequence of $L^{({\beta }^{\prime }+({\beta }_0+1)(L_z+1)d)}\left(e\right)$. Thus,

$${\mathbf{S}}_{\#}=L^{(\tau +({\alpha }_0+1)(L_z+1))}\left(a_{{\alpha }^{\prime }}\right)-L^{\left(({\beta }_0+1)(L_z+1)\right)}\left(a_{{\beta }^{\prime }}\right)$$

(35)

is a d-decimated sequence of

$$L^{({\alpha }^{\prime }+(\tau +({\alpha }_0+1)(L_z+1))d)}\left(e\right)-L^{({\beta }^{\prime }+({\beta }_0+1)(L_z+1)d)}\left(e\right)=L^{({\alpha }^{\prime }-{\beta }^{\prime }+(\tau +({\alpha }_0-{\beta }_0)(L_z+1))d)}\left(e\right)-e.$$

(36)

Therefore, $(a_{{\alpha }^{\prime }}(t+({\alpha }_0+1)(L_z+1)+\tau )-a_{{\beta }^{\prime }}(t+({\beta }_0+1)(L_z+1)),a_{{\alpha }^{\prime }}(t+({\alpha }_0+1)(L_z+1)+\tau +1)$ $-a_{{\beta }^{\prime }}(t+({\beta }_0+1)(L_z+1)+1),\dots ,a_{{\alpha }^{\prime }}(t+({\alpha }_0+1)(L_z+1)+\tau +k-1)-a_{{\beta }^{\prime }}(t+({\beta }_0+1)$ $(L_z+1)+k-1\left)\right)$ is the consecutive k-tuples in ${\mathbf{S}}_{\#}$. Hence, (34) can be rewritten as (37).

$$\begin{array}{cc}\hfill H({\mathbf{s}}_{c_{\alpha }},{\mathbf{s}}_{c_{\beta }},\tau )& =\sum _{t=0}^{(q^n-1)/d-1}h[({\mathbf{S}}_{\#}\left(t\right),{\mathbf{S}}_{\#}(t+1),...,{\mathbf{S}}_{\#}(t+k-1)),(0,0,...,0)]\hfill \\ \hfill & =N({\mathbf{S}}_{\vartheta },(0,0,...,0))\hfill \end{array}$$

(37)

The following cases exist here.

Case 1: $\alpha =\beta$.

According to (36) and Lemma 3, ${\mathbf{S}}_{\#}$ is a d-decimated sequence of $L^{\left(\tau d\right)}\left(e\right)-e$ which is still a q-ary m-sequence of degree n for any $\tau \ne 0$. It follows that $H({\mathbf{s}}_{c_{\alpha }},{\mathbf{s}}_{c_{\beta }},\tau )={\displaystyle \frac{q^{n-k}-1}{d}}$ from Lemma 4. Thus, the MPHAC of ${\mathbf{S}}_{\mathbf{c}}$ is

$$H_a\left({\mathbf{S}}_{\mathbf{c}}\right)={\displaystyle \frac{q^{n-k}-1}{d}}.$$

(38)

Case 2: $\alpha \ne \beta$.

Case 2. 1: ${\alpha }^{\prime }={\beta }^{\prime }$.

${\alpha }_0\ne {\beta }_0$ because of $\alpha \ne \beta$ and ${\alpha }^{\prime }={\beta }^{\prime }$. Therefore, ${\mathbf{S}}_{\#}$ is still a d-decimated sequence of $L^{\left((\tau +({\alpha }_0-{\beta }_0)(L_z+1))d\right)}\left(e\right)-e$.

When $\tau =({\beta }_0-{\alpha }_0)(L_z+1)$, ${\mathbf{S}}_{\#}$ is a sequence in which all elements are zero. Thus, $H({\mathbf{s}}_{c_{\alpha }},{\mathbf{s}}_{c_{\beta }},\tau )={\displaystyle \frac{q^n-1}{d}}$.

When $\tau \ne ({\beta }_0-{\alpha }_0)(L_z+1)$, ${\mathbf{S}}_{\#}$ is still a q-ary m-sequence of degree n for any $\tau$ from Lemma 3. In this case, $H({\mathbf{s}}_{c_{\alpha }},{\mathbf{s}}_{c_{\beta }},\tau )={\displaystyle \frac{q^{n-k}-1}{d}}$.

Case 2. 2: ${\alpha }^{\prime }\ne {\beta }^{\prime }$. ${\mathbf{S}}_{\#}$ is a d-decimated sequence of $L^{({\alpha }^{\prime }-{\beta }^{\prime }+(\tau +({\alpha }_0-{\beta }_0)(L_z+1))d)}\left(e\right)-e$ which is still a q-ary m-sequence of degree n for any $\tau$ according to Lemma 3. From Lemma 4, we have $H({\mathbf{s}}_{c_{\alpha }},{\mathbf{s}}_{c_{\beta }},\tau )={\displaystyle \frac{q^{n-k}-1}{d}}$.

Thus, the MPHCC of ${\mathbf{S}}_{\mathbf{c}}$ is equal to

$$H_c\left({\mathbf{S}}_{\mathbf{c}}\right)={\displaystyle \frac{q^{n-k}-1}{d}}$$

(39)

for $\tau \le L_z$.

To sum up, ${\mathbf{S}}_{\mathbf{c}}$ is an LHZ FHS set, and the MPHC is

$$H\left({\mathbf{S}}_{\mathbf{c}}\right)={\displaystyle \frac{q^{n-k}-1}{d}}$$

(40)

for $\tau \le L_z$.

Then, input all the parameters of ${\mathbf{S}}_{\mathbf{c}}$ into Peng–Fan bound (4), and we have

$$\begin{array}{cc}\hfill {\lambda }_{right}& =⌈{\displaystyle \frac{⌊\frac{q^n-1}{d(L_z+1)}⌋d(L_z+1)-q^k}{⌊\frac{q^n-1}{d(L_z+1)}⌋d(L_z+1)-1}}·{\displaystyle \frac{q^n-1}{d{q}^k}}⌉\hfill \\ \hfill & =⌈{\displaystyle \frac{q^n-1}{d{q}^k}}-{\displaystyle \frac{q^k-1}{⌊\frac{q^n-1}{d(L_z+1)}⌋d(L_z+1)-1}}·{\displaystyle \frac{q^n-1}{d{q}^k}}⌉\hfill \\ \hfill & =⌈{\displaystyle \frac{q^{n-k}-1}{d}}-{\displaystyle \frac{(q^k-1)(q^n-⌊\frac{q^n-1}{d(L_z+1)}⌋d(L_z+1))}{\left(⌊\frac{q^n-1}{d(L_z+1)}⌋d(L_z+1)-1\right)d{q}^k}}⌉.\hfill \end{array}$$

(41)

It is easy to check that ${\displaystyle \frac{(q^k-1)(q^n-⌊\frac{q^n-1}{d(L_z+1)}⌋d(L_z+1))}{\left(⌊\frac{q^n-1}{d(L_z+1)}⌋d(L_z+1)-1\right)d{q}^k}}>0$. And let $q^n-1=⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋d(L_z+1)+R$, $0\le R\le L_z\le {\displaystyle \frac{q^n-1}{2d}}-1$. We have

$$\begin{array}{cc}\hfill & q^n-⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋d(L_z+1)-(⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋d(L_z+1)-1)d\hfill \\ \hfill & =q^n+d-(d+1)⌊{\displaystyle \frac{q^n-1}{d(L_z+1)}}⌋d(L_z+1)\hfill \\ \hfill & =q^n+d-(d+1)(q^n-1-R)\hfill \\ \hfill & \le q^n+d-(d+1){\displaystyle \frac{(2d-1)q^n+1}{2d}}\hfill \\ \hfill & \le 0,\hfill \end{array}$$

(42)

Therefore, ${\displaystyle \frac{(q^k-1)(q^n-⌊\frac{q^n-1}{d(L_z+1)}⌋d(L_z+1))}{\left(⌊\frac{q^n-1}{d(L_z+1)}⌋d(L_z+1)-1\right)d{q}^k}}<1$. One can have

$${\lambda }_{right}={\displaystyle \frac{q^{n-k}-1}{d}}.$$

(43)

The equal sign in Peng–Fan bound (4) is true, which indicates ${\mathbf{S}}_{\mathbf{c}}$ is an optimal LHZ FHS set. □

The parameter d determines the sampling width and the number of FHSs in ${\mathbf{S}}_{\mathbf{c}}$. When $d\mid (q-1)$, it ensures that each element in the q-ary m-sequence with degree n appears exactly once in the d-decimated sequence. Furthermore, if $gcd(d,n)\ne 1$, Lemma 4 does not hold and in such cases the Hamming correlation of ${\mathbf{S}}_{\mathbf{c}}$ cannot achieve optimality.

Now, we use the following example to illustrate the construction C.

Example 2.

Let $q=7$, $k=1$, $n=3$, $d=2$. α is a primitive element of $F_{7^3}$ which is generated by ${\alpha }^3+{\alpha }^2+\alpha +2=0$. We first obtain a set of d-decimated sequence $\mathbf{a}=\{a_0,a_1\}$ derived from 7-ary m-sequence with degree 3 as follows.

$a_0$ = (0, 1, 0, 3, 1, 1, 0, 0, 4, 3, 2, 2, 2, 5, 2, 0, 5, 3, 5, 3, 3, 5, 2, 4, 1, 5, 0, 5, 1, 0, …),

$a_1$ = (0, 6, 6, 5, 2, 2, 3, 4, 6, 4, 2, 6, 2, 3, 6, 4, 5, 3, 0, 1, 4, 6, 3, 3, 2, 5, 6, 3, 0, 5, …).

Take $L_z=28$. Then, according to step 2 and step 3 of construction C, ${\mathbf{S}}_{\mathbf{c}}=\{s_{c_0},s_{c_1},...,s_{a_9}\}$ is obtained as follows.

$s_{c_0}$ = (0, 1, 0, 3, 1, 1, 0, 0, 4, 3, 2, 2, 2, 5, 2, 0, 5, 3, 5, 3, 3, 5, 2, 4, 1, 5, 0, 5, 1, …),

$s_{c_1}$ = (0, 2, 2, 4, 3, 3, 1, 6, 2, 6, 3, 2, 3, 1, 2, 6, 4, 1, 0, 5, 6, 2, 1, 1, 3, 4, 2, 1, 0, …),

$s_{c_2}$ = (4, 0, 5, 4, 4, 0, 0, 2, 5, 1, 1, 1, 6, 1, 0, 6, 5, 6, 5, 5, 6, 1, 2, 4, 6, 0, 6, 4, 0, …),

                  …

$s_{c_8}$ = (0, 4, 6, 6, 0, 0, 3, 4, 5, 5, 5, 2, 5, 0, 2, 4, 2, 4, 4, 2, 5, 3, 6, 2, 0, 2, 6, 0, 5, …),

$s_{c_9}$ = (5, 3, 4, 4, 6, 1, 5, 1, 4, 5, 4, 6, 5, 1, 3, 6, 0, 2, 1, 5, 6, 6, 4, 3, 5, 6, 0, 3, 0, …).

Simulation experiments demonstrate that the MPHAC and the MPHCC of ${\mathbf{S}}_{\mathbf{c}}$ equal 24 for $\tau \le 28$, and the MPHCC and the MPHC of which have the same distribution for $0\le \tau \le 170$. As illustrated in Figure 2, the MPHC of ${\mathbf{S}}_{\mathbf{c}}$ is equal to 24 for $\tau \le 28$. One can check that ${\mathbf{S}}_{\mathbf{c}}$ is an optimal $(171,10,7,28,24)$ LHZ FHS set according to Peng–Fan bound (4).

Compared with the constructions of Zhou et al. [15] and Han et al. [14], different values are taken for $L_z$, and we can obtain more LHZ FHS sets with optimal MPHC by Example 2, which are listed in

Table 3. LHZ FHS sets with optimal MPHC constructed by Example 2 and existing ones.

Ref.	Parameters	Parameters
this paper	(171, 4, 7, 57–84, 24)	(171, 6, 7, 42–56, 24)
	(171, 8, 7, 34–41, 24)	(171, 10, 7, 28–33, 24)
	(171, 12, 7, 24–27, 24)	(171, 14, 7, 21–23, 24)
	(171, 16, 7, 19 or 20, 24)	(171, 18, 7, 17 or 18, 24)
	…	…
	(171, 84, 7, 3, 24)	(171, 114, 7, 2, 24)
[15]	(171, 3, 7, 56, 24)	(171, 57, 7, 2, 24)
	(171, 9, 7, 18, 24)	(171, 19, 7, 8, 24)
[14]	(171, 6, 7, 56, 24)

.

4. Conclusions

In this correspondence, we have constructed three classes of LHZ FHS set. The first two are designed by any q-ary m-sequence of degree n, including binary m-sequences, and the last one is designed by the d-decimated sequence of q-ary m-sequence of degree n, $q\ne 2$. The results show that the first construction includes some known constructions as special instances, and all new classes of LHZ FHS set exhibit optimal maximum periodic Hamming correlation properties, good randomness properties and complexity, etc. Furthermore, new LHZ FHS sets have flexible parameters. A comprehensive comparison of our results with similar constructions reported in the existing literature is presented in

Table 4. Comparisons between some known optimal LHZ FHS sets and new ones.

Ref.	Parameters	Constraints
[13]	$(q(q^n-1),q|I|,q,Z,q^n)$	$q^n-1=M(Z+1)$, $M\ne 1$
	$(q(q^n-1),q^n-1,q,q-1,q^n)$	-
	$(s(q^n-1),M,q,Z,s(q^{n-1}-1))$	$q^n-1=M(Z+1)$,$gcd(s,q^n-1)=1$, $s<M$
[14]	$(q^n-1,q^k(q-1),q^k,{\displaystyle \frac{q^n-1}{q-1}}-1,q^{n-k})$	$1\le k\le n$
	$({\displaystyle \frac{q^n-1}{t}},q-1,q^k,{\displaystyle \frac{q^n-1}{q-1}}-1,{\displaystyle \frac{q^{n-k}-1}{t}})$	$1\le k\le n,t\left|\right(q-1),gcd(t,n)=1$
[15]	$(q^n-1,q^{k}T,q^k,L_Z,q^{n-k})$	$q^n-1=T(L_Z+1),1\le k\le n$
	$({\displaystyle \frac{q^n-1}{t}},T,q^k,L_Z,{\displaystyle \frac{q^{n-k}-1}{t}})$	$q^n-1=T(L_Z+1),1\le k\le n$
[16]	$(T{\displaystyle \frac{q^n-1}{l}},Ml,q,w-1,T{\displaystyle \frac{q^{n-1}-1}{l}})$	$l\left|\right(q^n-1)$, ${\displaystyle \frac{q^n-1}{l}}=Mw$,$gcd(l,n)=1$,
		$T=\lambda w+1$,$\lambda \ge 1$, $T<{\displaystyle \frac{lq(q^n-2)}{q-1}}$
	$(T{\displaystyle \frac{r(q^n-1)}{l}},M\lfloor {\displaystyle \frac{l}{r}}\rfloor ,q,w-1,T{\displaystyle \frac{r(q^{n-1}-1)}{l}})$	$l\left|\right(q^n-1)$,
		$2\le r\le l$,${\displaystyle \frac{r(q^n-1)}{l}}=Mw$,$gcd(l,n)=1$
		$T=\lambda w+1$,$\lambda \ge 1$,$T<{\displaystyle \frac{lq(q^n-2)}{q-1}}$
	$(T(q^n-1),M{q}^k,q^k,w-1,T(q^{n-k}-1))$	$1\le k\le n,q^n-1=Mw$,
		$T=\lambda w+1$,$\lambda \ge 1$,$T<{\displaystyle \frac{q^k(q^n-2)}{q^k-1}}$
[17]	$(Tn,Ml,Q,w-1,T{H}_m)$	$T\ge 2,Mw=n,T=\lambda w+1$
[20]	$(p^2(q_1-1),p{q}_1,p{q}_1,min\{p^2-1,q_1-2\},p)$	$gcd(p^2,q_1-1)=1$,$2p\le q_1-1$
	$(p^2(q_1-1)(q_2-1),p{q}_1{q}_2,p{q}_1{q}_2,min\{p^2-1,q_1-2,q_2-2\},p)$	$gcd(p,q_1-1,q_2-1)=1,3p<min\{q_1-2,q_2-2\}$,
	$({\displaystyle \frac{(q_1-1)\dots (q_k-1)}{d}},q_1\dots q_k,q_1\dots q_k,q_1-1,1)$	$2<q_1<\dots <q_k$,$d=gcd(q_1,q_2,\dots ,q_k)$
	$(k_1{k}_2(q_1-1)(q_2-1),{\displaystyle \frac{(q_1-1)(q_2-1)}{k_1{k}_2}},$	$k_1|q_1-1,k_2|q_2-1,gcd(k_1(q_1-1),k_2(q_2-1))=1$
	$q_1{q}_2,min\{q_1-1,q_2-1\},k_1{k}_2)$	$k_1(q_1-1)<k_2(q_2-1)$ and $q_1>k_1{k}_2^2+2k_1{k}_2$
		or $k_1(q_1-1)>k_2(q_2-1)$ and $q_2>k_2{k}_1^2+2k_1{k}_2$
[12]	$((q_1-1)(q_2^n-1),q_1,q_1{q}_2,q_2^n-2,q_2^{n-1}-1)$	$q_1>q_2^n$,$gcd(q_1-1,q_2^n-1)=1$
	$(p^2(p^2-1),p,p^2,p^2-2,p(p-1))$	$gcd(p^2,p^2-1)=1$
[21]	$(l(p^n-1),e,e+1,p^n-2,lf)$	$p^n-1=ef,lf{e}^2<(f{e}^2-1)(e+1-lf)$
	$(l(p^n-1),e,e+1,{\displaystyle \frac{p^n-1-m}{m}},lf)$	$p^n-1=ef,lf{e}^{2}m<(f{e}^2-m)(e+1-lf)$
[22]	$(lp,\theta ,\theta +1,p-1,l\beta )$	$p-1=\theta \beta$,$l\theta p<(\theta p-1)(l+\theta +1-l\beta )$
	$(2p,\theta ,\theta ,p-1,2(\beta +1\left)\right)$	$p-1=\theta \beta$, $\beta$ is even
Theorem 1	$(q^n-1,q^k⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋,q^k,L_z,q^{n-k})$	$2\le L_z\le \lfloor {\displaystyle \frac{q^n-1}{2}}\rfloor -1$
Theorem 2	$(q^n-1,⌊{\displaystyle \frac{q^n-1}{L_z+1}}⌋,q^{n-k},L_z,q^{n-k}-1)$	$2\le L_z\le \lfloor {\displaystyle \frac{q^n-1}{2}}\rfloor -1$
Theorem 3	$({\displaystyle \frac{q^{\prime n}-1}{d}},⌊{\displaystyle \frac{q^{\prime n}-1}{d(L_z+1)}}⌋d,q^{\prime n-k},$$L_z,{\displaystyle \frac{q^{\prime n-k}-1}{d}})$	$2\le L_z\le \lfloor {\displaystyle \frac{q^{\prime n}-1}{2d}}\rfloor -1$, $q^{\prime }$ is odd prime or odd prime power

q is prime or prime power. $p,q_1,q_2,...,q_k$ are prime numbers. $Q,k,k_1,n,m,w,t,r,l,M,T,Z,\theta ,\beta$ are positive integers.

. Constructing FHS sets based on new sampling patterns of q-ary m-sequences of degree n is worthy of further investigation.