Applications of n-to-1 Mappings in Difference Families and Optimal Constant Weight Codes

Abstract

This paper investigates novel applications of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-to-1 mappings over finite fields to the construction of difference families, constant weight codes, and constant composition codes, which play crucial roles in optical code-division multiple access, frequency-hopping systems, and error-correcting codes, respectively. First, we propose two general constructions for difference families: one based on the preimage sets and the other on the image sets of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-to-1 mappings. For the construction using image sets, we establish necessary and sufficient conditions that guarantee the resulting collection forms a difference family over either the additive or multiplicative group of the finite field. By selecting suitable <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-to-1 mappings, we obtain several infinite families of difference families with new parameters, and simultaneously recover a number of previously known families. In particular, we also construct <inline-formula> <tex-math notation="LaTeX">$(v,k,k-1)$ </tex-math></inline-formula> disjoint difference families, which are related to an open problem posed by Chang and Ding concerning the existence of additional families of this type. Furthermore, we demonstrate an application of the obtained difference families to the construction of Gaussian integer periodic complementary sequences of degree two. Second, we give two systematic constructions of constant weight codes and constant composition codes from <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-to-1 mappings. We establish some sufficient conditions for these constructions to yield constant weight codes or constant composition codes. Using concrete <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-to-1 mappings, we obtain several infinite families of optimal constant weight codes and several infinite families of optimal constant composition codes.