Difference Balanced Functions and Their Generalized Difference Sets

2013/09/30 by Alexander Pott, Qi Wang, Pott, Alexander +1
Computer Science · Engineering · Mathematics · #05B10 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Information Theory (cs.IT) #cs.IT #graph theory and CDMA systems #math.CO #math.IT #msc:05B10

paper · pdf · doi:10.48550/arxiv.1309.7842

17 pages

arxiv created 2013/09/30 · openalex publication_date 2013/09/30 · arxiv updated 2013/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Difference balanced functions from F_qⁿ^* 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.

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