Difference balanced functions and their generalized difference sets

Difference balanced functions from F∗ q 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 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.