The Zero-Difference Properties of Functions and Their Applications

A function $f$ from an Abelian group $(A,+)$ to an Abelian group $(B,+)$ is $(n, m, S)$ zero-difference (ZD), if $S={\lambda_\alpha \mid \alpha \in A\setminus{0}}$ where $n=|A|$, $m=|f(A)|$ and $\lambda_\alpha=|{x \in A \mid f(x+\alpha)=f(x)}|$. A function is called zero-difference balanced (ZDB) if $S={\lambda}$ where $\lambda$ is a constant number. ZDB functions have many good applications. However it is point out that many known zero-difference balanced functions are already given in the language of partitioned difference family (PDF). The problem that whether zero-difference ``not balanced" functions still have good applications as ZDB functions, is investigated in this paper. By using the change point technic, zero-difference functions with good applications are constructed from known ZDB functions. Then optimal difference systems of sets (DSS) and optimal frequency-hopping sequences (FHS) are obtained with new parameters. Furthermore the sufficient and necessary conditions of these objects being optimal, are given.

💡 Research Summary

The paper introduces the concept of zero‑difference (ZD) functions as a natural generalization of zero‑difference balanced (ZDB) functions. For a mapping f from an Abelian group (A,+) to another Abelian group (B,+), the zero‑difference value for a non‑zero shift α∈A is defined as λ_α = |{x∈A | f(x+α)=f(x)}|. The multiset S = {λ_α | α≠0} characterises an (n,m,S) ZD function, where n=|A| and m=|f(A)|. When S contains a single element λ, the function is called ZDB. The authors point out that many recent ZDB constructions are merely restatements of known partitioned difference families (PDFs), and they raise the question whether functions that are not perfectly balanced (i.e., S contains more than one value) can still be useful for applications that traditionally rely on ZDB functions.

The paper first establishes several elementary properties of ZD functions. Lemma 1 shows the fundamental identity Σ_b r_b(r_b−1)=Σ_α λ_α, linking the pre‑image sizes r_b=|f^{-1}(b)| with the zero‑difference counts. Lemma 2 provides a lower bound on the average λ, expressed in terms of n, m, and the division n = km + ε (0≤ε<m). Lemma 3 bounds each r_b within n±√Δ/m, where Δ is a quadratic expression in n, m, λ, and the minimum µ of S. These results generalize known bounds for ZDB and near‑ZDB functions.

The core construction technique, called the “change‑point technique,” starts from a well‑known class of ZDB functions built from a finite ring R and a multiplicative subgroup G⊂R^×. For each x∈R, the original ZDB function f_G(x) = h_G(g_G(x)) maps the coset rG containing x to a distinct integer via a bijection h_G. The condition (G−1){0}⊂R^× guarantees that f_G is an (n, n−1/k+1, k−1) ZDB function, where k=|G| and n=|R|.

To obtain a ZD function, the authors modify f_G only at the point x=0, defining f₀_G(0)=f_G(1) and leaving all other values unchanged. Theorem 1 proves that f₀_G is an (n, n−1/k, S) ZD function, where S consists of either {k}, {k−1,k}, or {k−1,k+1} depending on whether −1 belongs to G and on the size of k. Lemma 10 characterises the inclusion of −1 in G: it occurs iff k is even or the characteristic of R equals 2. This subtle change adds at most one extra solution to the equation f₀_G(x+α)=f₀_G(x) for each shift α, thereby keeping the zero‑difference values tightly clustered around k.

Having constructed ZD functions, the authors investigate when these functions yield optimal combinatorial objects. They show that if λ attains the lower bound of Lemma 2 and the pre‑image sizes are as balanced as possible (i.e., each r_b equals either ⌊n/m⌋ or ⌈n/m⌉), then the associated partition‑type difference packing (PDP) leads to optimal difference systems of sets (DSS). The parameters of the resulting DSS satisfy the known optimality condition λ = (n−ε)(n+ε−m)/(m(n−1)). Similarly, by interpreting the blocks of the PDP as frequency‑hopping sequences (FHS) on the cyclic group Z_n, the constructed FHS achieve the Welch bound on maximum Hamming correlation, and their linear complexity can be analysed because the underlying function is defined via explicit algebraic operations.

The paper proceeds to instantiate these general results with concrete families. Using rings Z_n and products of finite fields, the authors generate ZD functions with a wide range of parameters (different n, k, and whether −1∈G). For each family they explicitly compute S, λ, and the distribution of pre‑image sizes, then derive the corresponding constant‑weight codes (CWC), DSS, and FHS. In every case the codes meet the Singleton‑type bound for CWCs, the optimality criteria for DSS, and the correlation bound for FHS, thereby extending the catalogue of known optimal constructions.

In the concluding section the authors emphasize that ZD functions bridge the gap between the highly restrictive ZDB functions and the more flexible but previously under‑exploited “non‑balanced” functions. The change‑point technique demonstrates that a minimal perturbation of a known ZDB function preserves most of its desirable algebraic structure while relaxing the strict balance condition, which in turn yields new optimal combinatorial designs. The paper suggests future work on extending the framework to non‑abelian groups, exploring other small perturbations, and investigating cryptographic properties such as differential uniformity and resistance to algebraic attacks.