Sets of Zero-Difference Balanced Functions and Their Applications

TL;DR

This paper characterizes zero-difference balanced functions, provides a generic construction method for generating many new classes, and explores their applications in coding and sequence design.

Contribution

It introduces a new generic construction of ZDB functions and extends it to sets of related ZDB functions, enhancing their applicability.

Findings

New classes of ZDB functions are generated.

A set of ZDB functions with uniform relations is constructed.

Applications in coding theory and sequence design are demonstrated.

Abstract

Zero-difference balanced (ZDB) functions can be employed in many applications, e.g., optimal constant composition codes, optimal and perfect difference systems of sets, optimal frequency hopping sequences, etc. In this paper, two results are summarized to characterize ZDB functions, among which a lower bound is used to achieve optimality in applications and determine the size of preimage sets of ZDB functions. As the main contribution, a generic construction of ZDB functions is presented, and many new classes of ZDB functions can be generated. This construction is then extended to construct a set of ZDB functions, in which any two ZDB functions are related uniformly. Furthermore, some applications of such sets of ZDB functions are also introduced.