Schemes, Codes and Quadratic Zero-Difference Balanced Functions

Zero-difference balanced (ZDB) functions were introduced by Ding for the constructions of optimal and perfect systems of sets and of optimal constant composition codes. In order to be used in these two areas of application, ZDB functions have to be defined on cyclic groups. In this paper, we investigate quadratic ZDB functions from the additive group of Fpn to itself of the form G(x t+1), where G is injective on the set of (p +1)-th power of Fpn . By choosing different values of p and t, such ZDB functions include certain quadratic APN and PN functions as special cases, which gives a “trans-characteristic” interpretation of these functions. We first provide a geometric characterization of such ZDB functions, and then make use of them to give a construction of a 4-class association schemes. We further determine the weight distributions of the linear codes from such ZDB functions. This includes some of the previous work on the codes generated in the same manner from APN and PN functions as special cases.