The Zero-Difference Properties of Functions and Their Applications
This work addresses a theoretical problem in combinatorial design for applications in communications, but it is incremental as it builds on existing zero-difference balanced functions.
The paper investigates whether zero-difference functions that are not balanced can have good applications similar to zero-difference balanced functions, and constructs such functions from known ones to obtain optimal difference systems of sets and frequency-hopping sequences with new parameters.
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.