The Zero-Difference Properties of Functions and Their Applications

Abstract

A function $f$ from an Abelian group $(A,+)$ to an Abelian group $(B,+)$ is $(n,m,S)$ zero-difference (ZD), if $S=\{\hat{I}{»}_{\hat{I}}\pm \mid \hat{I}\pm \in A\setminus \{0\}\}$ where $n=|A|$, $m=|f(A)|$ and $\hat{I}{»}_{\hat{I}}\pm =|\{x\in A\mid f(x+\hat{I}\pm )=f(x)\}|$. A function is called zero-difference balanced (ZDB) if $S=\{\hat{I}»\}$ where $\hat{I}»$ 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.

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