函数的零差分性质及其应用 (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 $S=\{λ_α\mid α\in A\setminus\{0\}\}$ where $n=|A|$, $m=|f(A)|$ and $λ_α=|\{x \in A \mid f(x+α)=f(x)\}|$. A function is called zero-difference balanced (ZDB) if $S=\{λ\}$ where $λ$ 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.

翻译：若从阿贝尔群$(A,+)$到阿贝尔群$(B,+)$的函数$f$满足$S=\{λ_α\mid α\in A\setminus\{0\}\}$，其中$n=|A|$，$m=|f(A)|$且$λ_α=|\{x \in A \mid f(x+α)=f(x)\}|$，则称$f$为$(n, m, S)$零差分（ZD）函数。若$S=\{λ\}$（$λ$为常数），则称该函数为零差分平衡（ZDB）函数。ZDB函数具有诸多优良应用。然而需要指出的是，许多已知的零差分平衡函数已通过划分差族（PDF）的语言给出。本文研究了零差分“非平衡”函数是否仍能像ZDB函数那样具有良好应用的问题。通过采用变更点技术，我们从已知的ZDB函数构造出具有优良应用的零差分函数，进而获得了具有新参数的最优差集系统（DSS）与最优跳频序列（FHS）。此外，本文给出了这些对象达到最优性的充分必要条件。