Bent Vectorial Functions, Codes and Designs

TL;DR

This paper explores the use of bent vectorial functions to construct binary linear codes that form 2-designs, providing new coding-theoretic insights and expanding the applications of bent functions beyond traditional limits.

Contribution

It introduces a novel construction of binary linear codes from bent vectorial functions that do not meet the Assmus-Mattson theorem but still generate 2-designs.

Findings

Constructed a two-parameter family of codes from bent vectorial functions.

Demonstrated codes form 2-designs despite not satisfying Assmus-Mattson conditions.

Provided a new coding-theoretic characterization of bent vectorial functions.

Abstract

Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group $(\gf(2^{2m}), +)$, have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold $2$-designs. A new coding-theoretic characterization of bent vectorial functions is presented.