Difference Balanced Functions and Their Generalized Difference Sets

TL;DR

This paper characterizes difference balanced functions using generalized difference sets, proves the Gong-Song conjecture for prime q, and reveals their connection to multipliers and affine shifts, advancing understanding of their structure.

Contribution

It provides a new characterization of difference balanced functions via generalized difference sets and proves the Gong-Song conjecture for prime q.

Findings

Characterization of difference balanced functions through generalized difference sets.

Proof of the Gong-Song conjecture for prime q.

Establishment that all difference balanced functions are balanced or affine shifts of balanced functions.

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.