Optimal FHSs and DSSs via near zero-difference balanced functions

Zero-difference balanced (ZDB) functions were introduced by Ding in connection with constructions of optimal constant composition codes and optimal and perfect difference systems of sets. Based on such functions, people have constructed optimal constant weight codes and optimal frequency-hopping sequences. In order to obtain more optimal cryptographic objects, the zero-difference balanced (ZDB) function is generalized to the near zero-difference balanced (N-ZDB) function in the present paper, whose characterizations are partially given. Furthermore, we prove that near zero-difference balanced (N-ZDB) functions are equivalent to partitioned almost difference families (PADFs) in design theory. As the main contribution of this paper, three classes of the N-ZDB functions are proposed by means of the partition of Zn, where n is an odd positive integer. Employing these N-ZDB functions, we obtain at the same time optimal frequency-hopping sequences and optimal difference systems of sets with flexible parameters.