Linear codes and incidence structures of bent functions and their generalizations

TL;DR

This paper explores the use of linear codes and combinatorial designs to analyze and construct bent functions and their generalizations, revealing new applications and characterizations in design theory and coding theory.

Contribution

It introduces new design-theoretic characterizations of $(n,m)$-plateaued and $(n,m)$-bent functions, and applies the extended Assmus-Mattson theorem to support 2-designs from APN functions.

Findings

Linear codes of APN functions support 2-designs.

New characterization of $(n,m)$-plateaued and $(n,m)$-bent functions.

Coding and design interpretations of extendability for $(n,m)$-bent functions.

Abstract

In this paper we consider further applications of $(n,m)$-functions for the construction of 2-designs. For instance, we provide a new application of the extended Assmus-Mattson theorem, by showing that linear codes of APN functions with the classical Walsh spectrum support 2-designs. On the other hand, we use linear codes and combinatorial designs in order to study important properties of $(n,m)$-functions. In particular, we give a new design-theoretic characterization of $(n,m)$-plateaued and $(n,m)$-bent functions and provide a coding-theoretic as well as a design-theoretic interpretation of the extendability problem for $(n,m)$-bent functions.