Difference balanced functions and their generalized difference sets

Abstract

Difference balanced functions from F * q n to F q are closely related to combinatorial designs and naturally define p-ary sequences with the ideal two-level autocorrelation. In the literature, all existing such functions are associated with the d-homogeneous property, and it was conjectured by Gong and Song that difference balanced functions must be d-homogeneous. First we characterize difference balanced functions by generalized difference sets with respect to two exceptional subgroups. We then derive several necessary and sufficient conditions for d-homogeneous difference balanced functions. In particular, we reveal an unexpected equivalence between the d-homogeneous property and multipliers of generalized difference sets. By determining these multipliers, we prove the Gong-Song conjecture for q prime. Furthermore, we show that every difference balanced function must be balanced or an affine shift of a balanced function.

Three New Families of Zero-Difference Balanced Functions With Applications

“…Zero-difference balanced functions were first introduced by Ding in constructing optimal constant composition codes [2] and optimal and perfect difference systems of sets [3]. In the literature, perfect nonlinear functions [8], [11], [15], [19], [20] and difference balanced functions [16], [21] are special types of ZDB functions. ZDB functions unify different subjects in combinatorics, algebra and finite geometry, and they have found applications not only in these three areas but also in communications, coding theory and cryptography.…”

9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures

“…We remark that p-ary sequences with ideal two-level autocorrelation {−1, v} are equivalent to relative difference sets with Singer parameters, and are characterized by the d-homogeneous property [86,87]. A natural question to ask is whether there exist binary sequences with two-level autocorrelation in the other three categories for which v ≡ 3 (mod 4).…”

“…We remark that p-ary sequences with ideal two-level autocorrelation {−1, v} are equivalent to relative difference sets with Singer parameters, and are characterized by the d-homogeneous property [86,87].…”

On Generalized Zero-Difference Balanced Functions

“…It is well known that perfect nonlinear functions [7,8,12,16,17] and difference balanced functions [13,18] are special types of ZDB functions. ZDB functions unify different subjects in combinatorics, algebra and finite geometry, and they have been found applications not only in these three areas but also in communications, coding theory and cryptography.…”