More Optimal Difference Systems of Sets and Frequency-Hopping Sequences From Zero-Difference Functions

Abstract

Difference systems of sets (DSS) and frequency-hopping sequences (FHS) are two objects with many applications in wireless communication. Zero-difference balanced function (ZDBF) and near zero-difference balanced functions (NZDBF) are two types of functions which can be used to obtain optimal DSSs and FHSs. In order to obtain more optimal DSSs and FHSs, zero-difference function (ZDF) as a generalization of ZDBF and NZDBF was recently proposed. In this paper, four classes of ZDFs with good applications are given from some known ZDBFs. It is noticed that these ZDFs are neither ZDBFs nor NZDBFs. As a result, more optimal DSSs and FHSs with new flexible parameters are obtained. INDEX TERMSDifference system of sets, frequency-hopping sequence, zero-difference balanced function, zero-difference function. I. INTRODUCTION Difference systems of sets (DSS) are related with comma-free codes [21], [29], authentication codes and secrete sharing schemes [15], [27]. Frequency-hopping sequences are used to reduce the interferes between the wireless devices in CDMA communication [6], [11], [12], [14]. Definition 1 [8]: Let (A, +) and (B, +) be two finite Abelian groups. A function from A to B is an (n, m, λ) zero-difference balanced function (ZDBF), if there exists a constant number λ such that for any nonzero element a ∈ A, |{x ∈ A | f (x + a) − f (x) = 0}| = λ, where n = |A| and m = |f (A)|. Ding first proposed the concept of ZDBF in 2008 [8], [8]. Since optimal DSSs and FHSs can be obtained from ZDBFs, many researchers have been working on constructing ZDBFs (see [3]-[5], [8], [9], [13], [22], [31], [33], [35], [37]-[39] and the references therein). It is worth mentioning that in The associate editor coordinating the review of this article and approving it for publication was Zilong Liu.

A class of optimal frequency hopping sequences based on the decimated m-sequences

“…Example: Let q = 61, m = 2 and e = 15. Let θ be the primitive element of F 61 2 with θ 2 + 6θ + 2 = 0 and g = 1, then the FHS X g defined by ( 3) is (2,31,24,37,29,4,49,11,46,24,55,38,24,21,21,54,40,36,5,3,38,47,2,33,25,46,53,39,16,6,10,0,12,6,54,22,27,46,31,28,39,47,52,58,6,36,13,7,13,21,20,23,5,24,18,…”