Difference balanced functions and their generalized difference sets

Difference balanced functions from Fqn⁎ to Fq 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 have the homogeneous property, and it was conjectured by Gong and Song that difference balanced functions must be d-homogeneous for some d with gcd⁡(d,q−1)=1. Here we characterize difference balanced functions by difference sets with respect to two exceptional subgroups. We then reveal an unexpected equivalence between the homogeneous property and certain multipliers of the corresponding difference sets. By determining these multipliers, we prove the Gong-Song conjecture for q prime.