Optimal One-Coincidence Sequence Sets with a Large Alphabet and Prime Length

Abstract

The performance of a frequency-hopping spread-spectrum system is mainly dependent on the mathematical properties of its hopping sequences, which are designed to minimize interference between different users. The one-coincidence sequence frequency-hopping sequence (OC-FHS) set is one of the primary types, because it achieves the lowest possible values regarding Hamming auto- and cross-correlation. In this work, we propose an OC-FHS set of a prime length p and alphabet size $pq$ for two primes p and q using a block structure modulo $pq$. In particular, when $p=q$, our construction provides a significantly larger set size compared with a previously known OC-FHS set with the same length and the same alphabet size. Moreover, the set size is optimal with respect to the bound established by Cao, Ge, and Miao. This extended set size can be applied to FHMA systems that need to accommodate a large number of users.

1. Introduction

A frequency-hopping spread spectrum (FHSS) is a means of radio communication that quickly switches a carrier between different frequency channels in a set pattern that both the transmitter and receiver know. This method spreads the signal across a wide range of frequencies, making it difficult to intercept or jam [1,2]. An FHSS system has two main parts: a collection of available frequencies and a hopping sequence, which shows the frequency to utilize at each time interval: the hop duration. The main benefits of FHSS are as follows: (1) anti-jamming: because the signal only uses a narrow frequency for a short time, a jammer must either jam the whole frequency spectrum, which takes a lot of power, or guess the hopping pattern; (2) low probability of intercept: an eavesdropper must know the hopping sequence to follow the signal, which makes it hard to listen to someone without permission; and (3) mitigation of interference and fading: by avoiding prolonged attention on a single frequency, the system reduces multipath fading or persistent interference associated with a particular channel.

The performance of an FHSS system is critically dependent on the mathematical properties of its hopping sequence [3,4]. In a multi-user environment, such as frequency-hopping code division multiple access (FH-CDMA), sequences must be designed to minimize interference. There have been several algebraic and combinatorial designs for such frequency-hopping sequences [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. A “hit” appears when two users adopt the same frequency simultaneously, and an FHS set should be designed to minimize such occurrences. The metric to evaluate this is the Hamming correlation. Given two sequences $X=\left\{x_i\right\}$ and $Y=\left\{y_i\right\}$ of length L, their periodic Hamming correlation at a time shift $\tau$ is defined as the number of positions where their elements coincide (i.e., $x_i=y_{i+\tau }$).

• Hamming auto-correlation: This measure quantifies the correlation of an FHS with a shift of itself. For an FHS to be applied to actual systems, it must have low auto-correlation values for any nonzero shift. This is essential for a receiver to be synchronized with the incoming signal.
• Hamming cross-correlation: This represents the correlation between two FHSs utilized by two distinct users. To reduce multiple access interference (MAI), the Hamming cross-correlation values of any FHS pairs in an FHS set should be minimized for all possible time shifts.

A one-coincidence frequency-hopping sequence (OC-FHS) set is a family of non-repeating FH sequences, in which, for any two distinct sequences and any relative shift, the periodic Hamming cross-correlation never exceeds one; equivalently, because the symbols do not repeat within a period, every out-of-phase Hamming auto-correlation is zero. In other words, two users can collide at most for one hop per period regardless of their timing offset [21]. Historically, the OC criterion and the early constructions were presented in the classic survey by Shaar and Davies in 1984, which introduced several OC-FHS sets and clarified the definition that is still in use (non-repeating sequences; cross-correlation) [22]. In 2006, Cao, Ge, and Miao reframed OC-FHS sets through perfect Mendelsohn packings and provided an upper bound on the family size of OC-FHS sets [23]. Ren, Fu, Wang, and Gao used the Chinese Remainder Theorem and interleaving to extend prime length OC-FHS sets to composite lengths while preserving the maximum Hamming correlation and Lempel-Greenberger optimality – An important step toward flexible parameter sets [24]. Then, Shao and Miao utilized generalized cyclotomy to produce optimal OC-FHS sets [25]. Most recently, Chung, Ahn, and Kim proposed OC-FHS sets with large frequency set sizes [26], using the properties of the integer ring.

In this study, we propose a new construction of OC-FHS sets in the case where the alphabet size is larger than the length of the sequences using the block structure modulo $pq$ for two primes p and q with $p\le q$. Compared with the previously known OC-FHS set in [26] with the same length and the same alphabet size, our construction provides a significantly larger set size for the same length and alphabet size. Moreover, the set size is optimal with respect to the bound established by Cao, Ge, and Miao.

The remainder of this paper is organized as follows. Section 2 introduces some preliminary knowledge regarding FHS sets and OC-FHS sets, including some relevant theoretical bounds. Section 3 presents a new construction of optimal OC-FHS sets, as well as mathematical proofs for the correlation properties and optimality. Section 4 provides a discussion of the parameter comparison, system-level interpretation, and complexity. Finally, Section 5 concludes the paper.

2. Preliminaries

2.1. FHS Sets

A frequency-hopping sequence (FHS) $X={\left\{X\left(n\right)\right\}}_{n=0}^{N-1}$ of length N is defined over an alphabet $F=\{f_0,\dots ,f_{M-1}\}$ that has a one-to-one correspondence to the set of available frequencies. We assume that there are two FHSs, X and Y, of length N over an alphabet of size M. For $0\le \tau \le N-1$, the Hamming correlation $H_{X,Y}\left(\tau \right)$ is the number of coincidences (or ’hits’) between X and the $\tau$-shift of Y; that is,

$$H_{X,Y}\left(\tau \right)=\sum _{n=0}^{N-1}h[X\left(n\right),Y(n+\tau modN)],$$

where h is an indicator function such that $h[x,y]=1$ if $x=y$, and $h[x,y]=0$ otherwise. For a set of L FHSs $\mathcal{X}=\{X_0,\dots ,X_{L-1}\}$, the maximum out-of-phase Hamming auto-correlation $H_{\mathrm{a}}\left(\mathcal{X}\right)$ and the maximum Hamming cross-correlation $H_{\mathrm{c}}\left(\mathcal{X}\right)$ of $\mathcal{X}$ are defined as

$$\begin{array}{ccc}& & H_{\mathrm{a}}\left(\mathcal{X}\right)=\underset{X_i\in \mathcal{X}}{max}\underset{0<\tau \le N-1}{max}{H}_{X_i,X_i}\left(\tau \right),\hfill \\ & & H_{\mathrm{c}}\left(\mathcal{X}\right)=\underset{X_i,X_j\in \mathcal{X},i\ne j}{max}\underset{0\le \tau \le N-1}{max}{H}_{X_i,X_j}\left(\tau \right),\hfill \end{array}$$

respectively. When $max{H}_{\mathrm{a}}\left(\mathcal{X}\right),H_{\mathrm{c}}\left(\mathcal{X}\right)=\lambda$ for some nonnegative integer $\lambda$, then $\mathcal{X}$ is called an $(N,M,\lambda ;L)$ FHS set. Peng and Fan established a bound on an $(N,M,\lambda ;L)$ FHS set regarding the alphabet size, length, and the number of FHSs.

Theorem 1

(Peng–Fan Bound [4]). For an $(N,M,\lambda ;L)$-FHS set,

$$\begin{array}{c}\hfill ⌈{\displaystyle \frac{(NL-M)N}{(NL-1)M}}⌉\le \lambda .\end{array}$$

(1)

The Peng–Fan bound includes the Lembel–Greenberger bound [3] for a single FHS as a special case. Numerous algebraic and combinatorial constructions have been proposed for designing FHS sets that are optimal with respect to the Lempel–Greenberger bound or the Peng–Fan bound [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].

Recent studies have also investigated no-hit-zone and wide-gap properties in frequency-hopping sequence design (see, e.g., [17,19,20]). A no-hit-zone FHS set is designed so that the (periodic or partial) Hamming cross-correlation is zero (or tightly bounded) over a prescribed range of relative shifts, which is particularly useful when the relative timing offset is known to lie within a limited uncertainty window [19]. Wide-gap constructions, on the other hand, focus on controlling the spacing of collision events by enforcing constraints such as uniform or minimum gaps between hits, rather than requiring the strict one-coincidence condition [17,20]. In this study, we focus on OC-FHS sets, which guarantee at most one collision for every relative shift and admit a direct optimality notion via the Cao-Ge-Miao bound on the family size. Our construction is compared with some important previously known families in

Table 1. Comparison of OC-FHS sets (N: sequence length, M: alphabet size, $\lambda =1$: maximum Hamming correlation value, and L: family size).

Ref.	Structure	N	M	$\mathit{\lambda }$	L	Remarks	Bound Optimality
[23]	Difference-unit set	N	N	1	$k-1$	k: size of a difference-unit set	Optimal when N is prime
	Cyclotomy	e	$p^n$	1	${\displaystyle \frac{p^n(p^n-1)}{e}}$	p prime, and $e\mid (p-1)$	Optimal
[25]	Generalized cyclotomy	e	$v=p_1^{m_1}\dots p_k^{m_k}$	1	${\displaystyle \frac{v(v-1)}{e}}$	p prime, $n\ge 1$, and $e\mid p_i-1$ for each i	Optimal
[26]	Integer ring	p	$p^2$	1	$p^2-p$	p is prime	Not optimal
This work	Two-dimensional integer block	p	$pq$	1	$(p-1)\left(q^2+⌊{\displaystyle \frac{q}{p}}⌋p\right)$	Two primes $p\le q$	Optimal when $p=q$

.

2.2. OC-FHS Sets

An $(N,M,\lambda ;L)$ FHS set $\mathcal{X}$ is called an OC-FHS set if it achieves the optimal correlation values of $H_{\mathrm{a}}\left(\mathcal{X}\right)=0$ and $H_{\mathrm{c}}\left(\mathcal{X}\right)=1$ [21,22]. This implies that $\lambda =max\{H_{\mathrm{a}}\left(\mathcal{X}\right),H_{\mathrm{c}}\left(\mathcal{X}\right)\}=1$. The condition $H_{\mathrm{a}}\left(\mathcal{X}\right)=0$ dictates a crucial structural property: every sequence $X\in \mathcal{X}$ must be non-repeating. That is, for any X, $H_{X,X}\left(\tau \right)=0$ for all $0<\tau \le N-1$, meaning no symbol (frequency) appears more than once within one period. This property implies that the sequence length N cannot exceed the alphabet size M (i.e., $N\le M$).

An OC-FHS set is, regarding the maximum correlation value, always optimal with respect to the Peng–Fan bound given in Theorem 1. The next goal is to maximize the family size L (i.e., the number of supported users) for given parameters N and M. Cao, Ge, and Miao provided a combinatorial characterization for OC-FHS sets and established a critical upper bound on the family size for OC-FHS sets [23].

Theorem 2

(Cao–Ge–Miao Bound [23]). For an $(N,M,1;L)$ OC-FHS set, the family size L is upper bounded by

$$L\le {\displaystyle \frac{M(M-1)}{N}}.$$

(2)

An OC-FHS set $\mathcal{X}$ is called optimal if its family size L meets this bound with equality. The new construction proposed in this study provides new optimal OC-FHS sets for parameters, where the alphabet size M is significantly larger than the length N. Some recent OC-FHS sets are listed in Table 1, which are optimal or not optimal.

3. New OC-FHS Sets

3.1. Construction

Theorem 3.

Let p and q be two positive prime integers with $p\le q$ that are not necessarily distinct. Define a set $S_p$ of one-to-one functions on ${\mathbb{Z}}_p$ as

$$S_p=\left\{{\sigma }_k\right|1\le k\le p-1\},$$

(3)

where ${\sigma }_k\left(l\right)=klmodp$. For $0\le a\le p-1$, $0\le b\le ⌊{\displaystyle \frac{q}{p}}⌋-1$, and $\alpha \in S_p$, define $X_{a,b,\alpha }={\left\{X_{a,b,\alpha }\left(n\right)\right\}}_{n=0}^{p-1}$ as

$$X_{a,b,\alpha }\left(n\right)=\alpha \left(n\right)+aq+bp.$$

(4)

For $0\le c,d\le q-1$ and $\beta \in S_p$, define $Y_{c,d,\beta }={\left\{Y_{c,d,\beta }\left(n\right)\right\}}_{n=0}^{p-1}$ as

$$Y_{c,d,\beta }\left(n\right)=\beta \left(n\right)q+{\left(c\beta \left(n\right)+d\right)}_q,$$

(5)

where ${\left(x\right)}_y=(xmody)$. Construct two FHS sets $\mathcal{X}$ and $\mathcal{Y}$ as

$$\begin{array}{ccc}\hfill \mathcal{X}& =& \left\{X_{a,b,\alpha }\right|0\le a\le p-1,0\le b\le ⌊q/p⌋-1,\alpha \in S_p\},\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill \mathcal{Y}& =& \left\{Y_{c,d,\beta }\right|0\le c,d\le q-1,\beta \in S_p\}.\hfill \end{array}$$

(7)

Then, the union $\mathcal{X}\cup \mathcal{Y}$ is an $\left(p,pq,1;(p-1)\left(q^2+⌊{\displaystyle \frac{q}{p}}⌋p\right)\right)$ OC-FHS set.

Before proving the correlation properties, we briefly describe the structural roles of the two subsets $\mathcal{X}$ and $\mathcal{Y}$. Each sequence in $\mathcal{X}$ is confined to a single contiguous q-block of the alphabet, i.e., all the symbols in each sequence lie in an interval of the form $[aq,(a+1)q-1]$, and different choices of a make another disjoint block. In contrast, each sequence in $\mathcal{Y}$ selects exactly one symbol from each block of length q indexed by $\{0,1,\dots ,p-1\}$, where the permutation parameter controls which block is visited at each time index. This complementary “within one block of length q” versus “one per block of length q” design is the key idea to minimize the number of hits between two sequences.

Proofof Correlation for

$\mathcal{X}$. For every a and b, the set $\{n+aq+bp|0\le n\le p-1\}$ has p consecutive integers that are all distinct and between 0 and $pq-1$; that is, $X_{a,b,\alpha }$ is non-repeating. Thus, the Hamming auto-correlation of $X_{a,b,\alpha }$ every shift is always 0 for all nonzero shifts.

For $X_{a_1,b_1,{\alpha }_1}$ and $X_{a_2,b_2,{\alpha }_2}$, if $(a_1,b_1)\ne (a_2,b_2)$, there is no intersection between

$$\{{\alpha }_1\left(n\right)+a_{1}q+b_{1}p|0\le n\le p-1\}$$

and

$$\{{\alpha }_2\left(n\right)+a_{2}q+b_{2}p|0\le n\le p-1\},$$

which implies the Hamming cross-correlation between $X_{a_1,b_1,{\alpha }_1}$ and $X_{a_2,b_2,{\alpha }_2}$ is always 0 for every shift.

The remaining step is to verify the correlation between $X_{a,b,{\alpha }_1}$ and $X_{a,b,{\alpha }_2}$. Because the two FHSs have exactly the same set of frequencies, their Hamming cross-correlation depends on the functions ${\alpha }_1$ and ${\alpha }_2$. For ${\alpha }_1\ne {\alpha }_2$, it is well known that their Hamming correlation value is always 1 [23]. □

Proof of Correlation for

$\mathcal{Y}$. For every $c,d,$ and $\beta$, the set $\{\beta \left(n\right)q+{\left(c\beta \left(n\right)+d\right)}_q|0\le n\le p-1\}$ has exactly one element between $eq$ and $(e+1)q-1$ for $0\le e\le p-1$, which implies that $Y_{c,d,\beta }$ is a non-repeating sequence. Thus, the Hamming auto-correlation value of $Y_{c,d,\beta }$ is always 0 for all nonzero shifts.

The Hamming correlation value between $Y_{c_1,d_1,{\beta }_1}$ and $Y_{c_2,d_2,{\beta }_2}$ is less than the number of elements in the intersection of the two sets,

$$A_1\triangleq \{{\beta }_1\left(n_1\right)q+{\left(c_1{\beta }_1\left(n_1\right)+d_1\right)}_q|0\le n_1\le p-1\}$$

(8)

and

$$A_2\triangleq \{{\beta }_2\left(n_2\right)q+{\left(c_2{\beta }_2\left(n_2\right)+d_2\right)}_q|0\le n_2\le p-1\}.$$

(9)

If ${\beta }_1\left(n_1\right)q+{\left(c_1\beta \left(n_1\right)+d_1\right)}_q={\beta }_2\left(n_2\right)q+{\left(c_2\beta \left(n_2\right)+d_2\right)}_q$ for some $0\le n_1,n_2\le p-1$, then

$$\left({\beta }_1\left(n_1\right)-{\beta }_2\left(n_2\right)\right)q={\left(c_1{\beta }_1\left(n_1\right)+d_1\right)}_q-{\left(c_2{\beta }_2\left(n_2\right)+d_2\right)}_q$$

(10)

which is possible only when ${\beta }_1\left(n_1\right)={\beta }_2\left(n_2\right)$. Thus, the condition (10) can be rewritten as

$$\left(c_1-c_2\right){\beta }_1\left(n_1\right)\equiv d_2-d_{1}modq.$$

(11)

The number of $n_1$ between 0 and $p-1$, satisfying (11) is p if $(c_1,d_1)=(c_2,d_2)$, 1 if $c_1\ne c_2$, and 0 if $c_1=c_2$ and $d_1\ne d_2$. Therefore, there is at most one overlapping element if $(c_1,d_1)\ne (c_2,d_2)$. Therefore, $\left|A_1\cap A_2\right|\le 1$ if $(c_1,d_1)\ne (c_2,d_2)$, which implies that the Hamming correlation value between $Y_{c_1,d_1,{\beta }_1}$ and $Y_{c_2,d_2,{\beta }_2}$ is less than or equal to 1 for $(c_1,d_1)\ne (c_2,d_2)$. Finally, the Hamming correlation between $Y_{c,d,{\beta }_1}$ and $Y_{c,d,{\beta }_2}$ is always 1, which is equal to the Hamming correlation between the two functions ${\beta }_1$ and ${\beta }_2$. □

Proof of Correlation between

$\mathcal{X}$ and $\mathcal{Y}$. Let $t=\lfloor q/p\rfloor$. We first describe the q-block structure of the values.

(i)

Values of $X_{a,b,\alpha }$ lie in a single q-block. For $0\le n\le p-1$, note that

$$X_{a,b,\alpha }\left(n\right)=\alpha \left(n\right)+aq+bp.$$

Since $\alpha$ is a permutation of $\{0,1,\dots ,p-1\}$, we have $0\le \alpha \left(n\right)\le p-1$. Moreover, $0\le b\le t-1$ implies $0\le bp\le (t-1)p\le q-p$. Hence,

$$aq\le X_{a,b,\alpha }\left(n\right)\le aq+(q-p)+(p-1)=aq+q-1=(a+1)q-1.$$

Therefore, all values of $X_{a,b,\alpha }$ belong to the single q-block

$$[aq,(a+1)q-1].$$

(ii)

$Y_{c,d,\beta }$ contains exactly one element from each q-block. For $0\le n\le p-1$, we have

$$Y_{c,d,\beta }\left(n\right)=\beta \left(n\right)q+{(c\beta \left(n\right)+d)}_q.$$

Since $0\le {(c\beta \left(n\right)+d)}_q\le q-1$, each value $Y_{c,d,\beta }\left(n\right)$ lies in the block

$$\left[\beta \right(n)q,(\beta \left(n\right)+1)q-1].$$

Because $\beta$ is a permutation of $\{0,1,\dots ,p-1\}$, for each $e\in \{0,1,\dots ,p-1\}$, there is a unique index $m_e$ such that $\beta \left(m_e\right)=e$. Consequently, $Y_{c,d,\beta }$ contains exactly one element in each block $[eq,(e+1)q-1]$.

(iii)

There is at most one coincidence for any shift. Fix an arbitrary shift $0\le \tau \le p-1$. Suppose there exist indices $n_1,n_2$ such that

$$X_{a,b,\alpha }\left(n_i\right)=Y_{c,d,\beta }(n_i+\tau modp),i=1,2.$$

By (i), the left-hand side belongs to the block $[aq,(a+1)q-1]$; hence,

$$Y_{c,d,\beta }(n_i+\tau modp)\in [aq,(a+1)q-1],i=1,2.$$

By (ii), we have

$$\beta (n_i+\tau modp)=a,i=1,2.$$

Since $\beta$ is a permutation, there is a unique $m_a$ with $\beta \left(m_a\right)=a$. Thus, $n_i+\tau \equiv m_a(modp)$, which implies $n_1\equiv n_2(modp)$; hence, $n_1=n_2$ in $\{0,1,\dots ,p-1\}$. Therefore, for the fixed shift $\tau$, there is at most one index n satisfying $X_{a,b,\alpha }\left(n\right)=Y_{c,d,\beta }(n+\tau )$, and thus, $H_{X_{a,b,\alpha },Y_{c,d,\beta }}\left(\tau \right)\le 1$.

□

As a concrete illustration,

Table 2. Example of periodic Hamming cross-correlation between $X\in \mathcal{X}$ and $Y\in \mathcal{Y}$ for $p=5$.

X	Y	${\mathit{H}}_{\mathit{X},\mathit{Y}}\left(0\right)$	${\mathit{H}}_{\mathit{X},\mathit{Y}}\left(1\right)$	${\mathit{H}}_{\mathit{X},\mathit{Y}}\left(2\right)$	${\mathit{H}}_{\mathit{X},\mathit{Y}}\left(3\right)$	${\mathit{H}}_{\mathit{X},\mathit{Y}}\left(4\right)$
$(5,6,7,8,9)$	$(0,5,10,15,20)$	0	1	0	0	0
$(5,6,7,8,9)$	$(0,10,20,5,15)$	0	0	0	1	0
$(5,6,7,8,9)$	$(0,15,5,20,10)$	0	0	1	0	0
$(5,6,7,8,9)$	$(0,20,15,10,5)$	0	0	0	0	1

lists periodic Hamming cross-correlation values for some combinations of sequences from $\mathcal{X}$ and $\mathcal{Y}$ when $p=q=5$.

3.2. Examples

Consider the smallest-number case with $p<q$, namely $(p,q)=(3,5)$. Then, the length is $N=p=3$, and the alphabet size is $M=pq=15$. Moreover, $S_3=\{{\sigma }_1,{\sigma }_2\}$, where ${\sigma }_k\left(n\right)=knmod3$. For brevity, we write $X_{a,b,k}:=X_{a,b,{\sigma }_k}$ and $Y_{c,d,k}:=Y_{c,d,{\sigma }_k}$. Since $\lfloor q/p\rfloor =\lfloor 5/3\rfloor =1$, the parameter b is fixed to $b=0$. Hence, $\left|\mathcal{X}\right|=p(p-1)\lfloor q/p\rfloor =6$, $\left|\mathcal{Y}\right|=q^2(p-1)=50$, and the total family size is $L=56$. The full OC-FHS set is illustrated in

Table 3. Construction example for $(p,q)=(3,5)$ (for brevity, $X_{a,b,k}$ and $Y_{c,d,k}$ denote $X_{a,b,{\sigma }_k}$ and $Y_{c,d,{\sigma }_k}$, respectively).

	Index	Sequence	Index	Sequence	Index	Sequence
$\mathcal{X}$	$X_{0,0,1}$	$(0,1,2)$	$X_{0,0,2}$	$(0,2,1)$
	$X_{1,0,1}$	$(5,6,7)$	$X_{1,0,2}$	$(5,7,6)$
	$X_{2,0,1}$	$(10,11,12)$	$X_{2,0,2}$	$(10,12,11)$
$\mathcal{Y}$	$\beta ={\sigma }_1$ (i.e., $k=1$)
	$Y_{0,0,1}$	$(0,5,10)$	$Y_{0,1,1}$	$(1,6,11)$	$Y_{0,2,1}$	$(2,7,12)$
	$Y_{0,3,1}$	$(3,8,13)$	$Y_{0,4,1}$	$(4,9,14)$	$Y_{1,0,1}$	$(0,6,12)$
	$Y_{1,1,1}$	$(1,7,13)$	$Y_{1,2,1}$	$(2,8,14)$	$Y_{1,3,1}$	$(3,9,10)$
	$Y_{1,4,1}$	$(4,5,11)$	$Y_{2,0,1}$	$(0,7,14)$	$Y_{2,1,1}$	$(1,8,10)$
	$Y_{2,2,1}$	$(2,9,11)$	$Y_{2,3,1}$	$(3,5,12)$	$Y_{2,4,1}$	$(4,6,13)$
	$Y_{3,0,1}$	$(0,8,11)$	$Y_{3,1,1}$	$(1,9,12)$	$Y_{3,2,1}$	$(2,5,13)$
	$Y_{3,3,1}$	$(3,6,14)$	$Y_{3,4,1}$	$(4,7,10)$	$Y_{4,0,1}$	$(0,9,13)$
	$Y_{4,1,1}$	$(1,5,14)$	$Y_{4,2,1}$	$(2,6,10)$	$Y_{4,3,1}$	$(3,7,11)$
	$Y_{4,4,1}$	$(4,8,12)$
	$\beta ={\sigma }_2$ (i.e., $k=2$)
	$Y_{0,0,2}$	$(0,10,5)$	$Y_{0,1,2}$	$(1,11,6)$	$Y_{0,2,2}$	$(2,12,7)$
	$Y_{0,3,2}$	$(3,13,8)$	$Y_{0,4,2}$	$(4,14,9)$	$Y_{1,0,2}$	$(0,12,6)$
	$Y_{1,1,2}$	$(1,13,7)$	$Y_{1,2,2}$	$(2,14,8)$	$Y_{1,3,2}$	$(3,10,9)$
	$Y_{1,4,2}$	$(4,11,5)$	$Y_{2,0,2}$	$(0,14,7)$	$Y_{2,1,2}$	$(1,10,8)$
	$Y_{2,2,2}$	$(2,11,9)$	$Y_{2,3,2}$	$(3,12,5)$	$Y_{2,4,2}$	$(4,13,6)$
	$Y_{3,0,2}$	$(0,11,8)$	$Y_{3,1,2}$	$(1,12,9)$	$Y_{3,2,2}$	$(2,13,5)$
	$Y_{3,3,2}$	$(3,14,6)$	$Y_{3,4,2}$	$(4,10,7)$	$Y_{4,0,2}$	$(0,13,9)$
	$Y_{4,1,2}$	$(1,14,5)$	$Y_{4,2,2}$	$(2,10,6)$	$Y_{4,3,2}$	$(3,11,7)$
	$Y_{4,4,2}$	$(4,12,8)$

. Furthermore, the case where $p=q=5$ is presented in

Table A1. The full set of the $(5,25,1;120)$ OC-FHS set in Theorem 3.

Subsets	${\mathit{\sigma }}_1$	${\mathit{\sigma }}_2$	${\mathit{\sigma }}_3$	${\mathit{\sigma }}_4$
$\mathcal{X}$	$(0,1,2,3,4)$	$(0,2,4,1,3)$	$(0,3,1,4,2)$	$(0,4,3,2,1)$
	$(5,6,7,8,9)$	$(5,7,9,6,8)$	$(5,8,6,9,7)$	$(5,9,8,7,6)$
	$(10,11,12,13,14)$	$(10,12,14,11,13)$	$(10,13,11,14,12)$	$(10,14,13,12,11)$
	$(15,16,17,18,19)$	$(15,17,19,16,18)$	$(15,18,16,19,17)$	$(15,19,18,17,16)$
	$(20,21,22,23,24)$	$(20,22,24,21,23)$	$(20,23,21,24,22)$	$(20,24,23,22,21)$
$\mathcal{Y}$	$(0,5,10,15,20)$	$(0,10,20,5,15)$	$(0,15,5,20,10)$	$(0,20,15,10,5)$
	$(1,6,11,16,21)$	$(1,11,21,6,16)$	$(1,16,6,21,11)$	$(1,21,16,11,6)$
	$(2,7,12,17,22)$	$(2,12,22,7,17)$	$(2,17,7,22,12)$	$(2,22,17,12,7)$
	$(3,8,13,18,23)$	$(3,13,23,8,18)$	$(3,18,8,23,13)$	$(3,23,18,13,8)$
	$(4,9,14,19,24)$	$(4,14,24,9,19)$	$(4,19,9,24,14)$	$(4,24,19,14,9)$
	$(0,6,12,18,24)$	$(0,12,24,6,18)$	$(0,18,6,24,12)$	$(0,24,18,12,6)$
	$(1,7,13,19,20)$	$(1,13,20,7,19)$	$(1,19,7,20,13)$	$(1,20,19,13,7)$
	$(2,8,14,15,21)$	$(2,14,21,8,15)$	$(2,15,8,21,14)$	$(2,21,15,14,8)$
	$(3,9,10,16,22)$	$(3,10,22,9,16)$	$(3,16,9,22,10)$	$(3,22,16,10,9)$
	$(4,5,11,17,23)$	$(4,11,23,5,17)$	$(4,17,5,23,11)$	$(4,23,17,11,5)$
	$(0,7,14,16,23)$	$(0,14,23,7,16)$	$(0,16,7,23,14)$	$(0,23,16,14,7)$
	$(1,8,10,17,24)$	$(1,10,24,8,17)$	$(1,17,8,24,10)$	$(1,24,17,10,8)$
	$(2,9,11,18,20)$	$(2,11,20,9,18)$	$(2,18,9,20,11)$	$(2,20,18,11,9)$
	$(3,5,12,19,21)$	$(3,12,21,5,19)$	$(3,19,5,21,12)$	$(3,21,19,12,5)$
	$(4,6,13,15,22)$	$(4,13,22,6,15)$	$(4,15,6,22,13)$	$(4,22,15,13,6)$
	$(0,8,11,19,22)$	$(0,11,22,8,19)$	$(0,19,8,22,11)$	$(0,22,19,11,8)$
	$(1,9,12,15,23)$	$(1,12,23,9,15)$	$(1,15,9,23,12)$	$(1,23,15,12,9)$
	$(2,5,13,16,24)$	$(2,13,24,5,16)$	$(2,16,5,24,13)$	$(2,24,16,13,5)$
	$(3,6,14,17,20)$	$(3,14,20,6,17)$	$(3,17,6,20,14)$	$(3,20,17,14,6)$
	$(4,7,10,18,21)$	$(4,10,21,7,18)$	$(4,18,7,21,10)$	$(4,21,18,10,7)$
	$(0,9,13,17,21)$	$(0,13,21,9,17)$	$(0,17,9,21,13)$	$(0,21,17,13,9)$
	$(1,5,14,18,22)$	$(1,14,22,5,18)$	$(1,18,5,22,14)$	$(1,22,18,14,5)$
	$(2,6,10,19,23)$	$(2,10,23,6,19)$	$(2,19,6,23,10)$	$(2,23,19,10,6)$
	$(3,7,11,15,24)$	$(3,11,24,7,15)$	$(3,15,7,24,11)$	$(3,24,15,11,7)$
	$(4,8,12,16,20)$	$(4,12,20,8,16)$	$(4,16,8,20,12)$	$(4,20,16,12,8)$

Note: The sequences in blue comprise the set presented in Table A2 of [26].

of Appendix A, including a comparison with the construction in [26] for the same parameter. Generation of numbers and analysis of correlation properties were conducted using Python (version 3.12).

3.3. Optimality

By plugging the parameters of $\mathcal{X}\cup \mathcal{Y}$ into (2), we obtain

$$(p-1)q^2+p(p-1)⌊{\displaystyle \frac{q}{p}}⌋\le p{q}^2-q.$$

(12)

In particular, when $p=q$ in (12), we have

$$(p-1)p^2+p(p-1)=p^3-p.$$

(13)

Corollary 1.

If $p=q$, the set $\mathcal{X}\cup \mathcal{Y}$ in Theorem 3 is an $(p,p^2,1;p^3-p)$ OC-FHS set, optimal with respect to the Cao–Ge–Miao bound [23].

Lemma 1.

Let p and q be primes with $p\le q$. Consider the parameters $(N,M)=(p,pq)$, and let

$$U(p,q)={\displaystyle \frac{M(M-1)}{N}}=p{q}^2-q$$

and

$$L(p,q)=(p-1)\left(q^2+p⌊{\displaystyle \frac{q}{p}}⌋\right)$$

be the family size of the proposed OC-FHS set in Theorem 3. Writing $t=\lfloor q/p\rfloor$, the difference satisfies

$$U(p,q)-L(p,q)=q^2-q-t(p^2-p)=q^2-q-t{p}^2+tp.$$

Moreover,

$$q(q-p)\le U(p,q)-L(p,q)\le q^2-q-p^2+p.$$

In particular, $U(p,q)=L(p,q)$ if and only if $p=q$. Finally, for fixed p and $q\to \infty$,

$${\displaystyle \frac{L(p,q)}{U(p,q)}}⟶{\displaystyle \frac{p-1}{p}}.$$

Proof. 

Let $t=\lfloor q/p\rfloor$. By direct substitution,

$$\begin{array}{cc}\hfill U(p,q)-L(p,q)& =(p{q}^2-q)-(p-1)\left(q^2+pt\right)\hfill \\ \hfill & =p{q}^2-q-(p-1)q^2-(p-1)pt\hfill \\ \hfill & =q^2-q-t(p^2-p).\hfill \end{array}$$

Since $p\le q$, we have $t\ge 1$, and $t=\lfloor q/p\rfloor \le q/p$. Because the expression $q^2-q-t(p^2-p)$ is decreasing in t, it follows that

$$q^2-q-{\displaystyle \frac{q}{p}}(p^2-p)\le U(p,q)-L(p,q)\le q^2-q-(p^2-p).$$

Because ${\displaystyle \frac{1}{p}}(p^2-p)=p-1$, we obtain

$$q^2-q-q(p-1)=q(q-p)\le U(p,q)-L(p,q)\le q^2-q-p^2+p.$$

When $p=q$, the equality holds. Finally, dividing $L(p,q)$ by $U(p,q)$ and using $t\sim q/p$ as $q\to \infty$ yields ${\displaystyle \frac{L(p,q)}{U(p,q)}}\to {\displaystyle \frac{p-1}{p}}$. □

4. Discussion

4.1. Comparison of Parameters

In [26], three constructions for OC-FHS sets were presented, which are based on integer rings. However, they do not meet the Cao–Ge–Miao bound [23]. Our construction gives an optimal OC-FHS set when the length is a prime p, and the alphabet size is $p^2$. In particular, when $p=5$, the family size is expanded by a factor of 6 (from 20 to 120) compared with the work in [26]. In Table A1 of Appendix A, the optimal OC-FHS set of length 5 and alphabet size 25 from our construction is presented. The $(5,25,1;20)$ OC-FHS set in [26] is a subset of the OC-FHS set from Theorem 3. This expansion is not a trivial generalization obtained merely by applying permutations to the construction in [26].

4.2. System-Level Interpretation in FHMA

As a simple system-level illustration under admission control, we model the assignment of hopping sequences as a loss system with L identical resources (one sequence per active user/session, no reuse) [27]. Under the classical Erlang-B model with offered traffic E (Poisson arrivals and exponential holding times), the blocking probability is

$$P_b(E,L)={\displaystyle \frac{\frac{E^L}{L!}}{{\sum }_{k=0}^L\frac{E^k}{k!}}}.$$

(14)

For a target grade of service $P_b=0.01$, solving (14) yields $E\approx 12.03$ Erlangs for $L=20$ (the $(5,25)$ construction in [26]), whereas, it yields $E\approx 102.96$ Erlangs for $L=120$ (our $(5,25)$ optimal case). This comparison does not model physical-layer throughput or BER; rather, it quantifies how enlarging the available sequence pool can increase the admissible offered traffic under a standard blocking model.

4.3. Computational Complexity and Implementation Aspects

All sequences in the proposed OC-FHS family have length $N=p$ and are generated using simple modular arithmetic. For the subset $\mathcal{X}$, each symbol has the form $X_{a,b,\alpha }\left(n\right)=\alpha \left(n\right)+aq+bp$, where $\alpha$ is a permutation of $\{0,1,\dots ,p-1\}$ (e.g., $\alpha \left(n\right)=knmodp$). Hence, generating one symbol requires one multiplication modulo p (to compute $\alpha \left(n\right)$) and a constant number of integer additions. Thus, generating a length-p sequence costs $O\left(p\right)$ operations. For the subset $\mathcal{Y}$, each symbol has the form $Y_{c,d,\beta }\left(n\right)=\beta \left(n\right)q+{(c\beta \left(n\right)+d)}_q$; so, each symbol requires one multiplication modulo p (for $\beta \left(n\right)$), one multiplication and one reduction modulo q, and a constant number of additions. Therefore, generation of any length-p sequence in $\mathcal{Y}$ requires $O\left(p\right)$ modular operations. If we want to compute the entire codebook offline, the total family size is

$$L=(p-1)\left(q^2+p⌊\frac{q}{p}⌋\right);$$

therefore, the explicit enumeration costs $O\left(pL\right)$ arithmetic operations and stores $pL$ integers in $[0,pq-1]$ (for $p=q$, this is $O\left(p^4\right)$ time and $O\left(p^4\right)$ symbols). However, in practical FHMA/FH-CDMA implementation, full codebook storage is not necessary. Each user (or a controller) only needs to store a small tuple such as $(a,b,k)$ for $X_{a,b,{\sigma }_k}$ or $(c,d,k)$ for $Y_{c,d,{\sigma }_k}$, together with $(p,q)$. The hop at time n is then computed directly from the closed-form expression, which reduces the memory to $O\left(1\right)$ per user. Moreover, the sequence generation across different parameter tuples is parallel, making the offline pre-computation straightforward.

5. Conclusions

This paper presented a new algebraic construction of OC-FHS sets based on a block structure modulo $pq$. When $p=q$, the resulting OC-FHS family achieves the Cao–Ge–Miao bound with equality. For instance, when the period is 5, and the alphabet size is 25, the proposed construction yields 120 distinct sequences, whereas a recent OC-FHS set in [26] for the same parameters provided 20 sequences. This improvement implies a significant increase in the user capacity. In addition, we provided a system-level interpretation of the parameters, as well as an analysis for implementation complexity. Future work includes performance analysis under implementation of practical scenarios and generalization of the construction to the non-prime length and alphabet size.