On the Hamming Auto- and Cross-correlation Functions of a Class of Frequency Hopping Sequences of Length $ p^{n} $

In this paper, a new class of frequency hopping sequences (FHSs) of length $ p^{n} $ is constructed by using Ding-Helleseth generalized cyclotomic classes of order 2, of which the Hamming auto- and cross-correlation functions are investigated (for the Hamming cross-correlation, only the case $ p\equiv 3\pmod 4 $ is considered). It is shown that the set of the constructed FHSs is optimal with respect to the average Hamming correlation functions.

💡 Research Summary

This paper introduces a novel construction of frequency hopping sequences (FHSs) of length pⁿ, where p is an odd prime and n ≥ 2, by employing Ding‑Helleseth generalized cyclotomic classes of order two. The authors first review the fundamental concepts of Hamming correlation, including the Lempel‑Greenberg bound for single sequences, the Peng‑Fan bounds for sets of sequences, and the average Hamming correlation (AH) criterion, which serves as a more practical performance metric for large families of FHSs.

The core of the construction lies in partitioning the multiplicative group Zₚⁿ* into two basic cyclotomic classes D₀(pᵏ) and D₁(pᵏ) for each exponent 1 ≤ k ≤ n. By scaling these classes with pⁿ⁻ᵏ and arranging them alternately, the authors obtain 2ⁿ disjoint subsets C₀,…,C_{2ⁿ‑1} that together cover the whole additive group Zₚⁿ. Each subset C_i is associated with a distinct frequency symbol f_i (0 ≤ i < 2ⁿ). An FHS X_i is then defined by the support condition support_Xi(j) = C_i + j (mod 2ⁿ), which guarantees that every frequency appears exactly pⁿ times across each sequence. Consequently, the family S = {X₀,…,X_{2ⁿ‑1}} is uniformly distributed: the total number of occurrences of any symbol f in the whole set equals pⁿ, independent of f. By Theorem 1.1, uniform distribution is sufficient for AH optimality, and the authors formally prove that S meets this condition (Theorem 2.2).

To evaluate the correlation properties, the paper defines two auxiliary counting functions: Δₖ(i:τ), which measures the overlap of a class with its τ‑shift, and Δ_{l,k}(i,j:τ), which measures the overlap between classes of possibly different levels l and k after a τ‑shift. Lemma 2.1 provides the values of Δₖ(i:τ) in terms of membership of τ in Dₖ(·), while Lemma 2.2 gives exhaustive formulas for Δ_{l,k}(·) for the three regimes l < k, l = k, and l > k. These lemmas translate the combinatorial structure of the cyclotomic classes into explicit Hamming correlation values.

Theorem 3.1 presents the Hamming auto‑correlation H(i:τ) for any sequence X_i. Two distinct formulas are derived depending on whether p ≡ 1 (mod 4) or p ≡ 3 (mod 4). In the former case, τ belonging to the first‑level class D₁(0) yields H(i:τ) = ½(2pⁿ − p + 1), while τ in higher‑level classes Dₖ(·) (k ≥ 2) produces progressively smaller values, reflecting the decreasing overlap of shifted supports. The p ≡ 3 (mod 4) case is analogous but with slightly different constants, and the authors verify that all auto‑correlation values satisfy the Peng‑Fan bound.

The cross‑correlation analysis is restricted to the p ≡ 3 (mod 4) scenario, where the authors consider two distinct sequences X_i and X_j with index difference δ = (j − i) mod m (m = 2ⁿ). They introduce δ′ = ⌈δ/2⌉ and distinguish several cases: δ even, δ odd with δ′ = 1, δ odd with δ′ = n, and the generic odd case 1 < δ′ < n. Proposition 4.1 provides a comprehensive piecewise expression for H(i,i+δ:τ) in the generic odd case, enumerating the correlation value for each possible τ belonging to a specific Dₖ(·). The expressions involve simple constants such as 0, ½(p + 1), ½p(p − 1), and linear terms in p, all of which are bounded by the Peng‑Fan and Lempel‑Greenberg limits.

A concrete example with n = 3 (sequence length p³) is worked out to illustrate how the theoretical formulas translate into actual correlation tables, confirming the practicality of the construction.

In conclusion, the paper makes four principal contributions: (1) a systematic method to generate 2ⁿ FHSs of length pⁿ using Ding‑Helleseth order‑2 cyclotomy; (2) a proof that the resulting family is uniformly distributed and therefore AH‑optimal; (3) explicit closed‑form expressions for both auto‑ and cross‑correlation functions, covering all possible shifts; and (4) verification that these correlation values meet or improve upon existing theoretical bounds. The authors suggest that extending the approach to higher‑order cyclotomy or to composite moduli could yield further families of optimal FHSs for modern FH‑CDMA, radar, and sonar applications.