Multidimensional convolution matrices and perfect colorings of subspace hypergraphs applied for bent functions and related designs

TL;DR

This paper introduces new methods using multidimensional convolution matrices and perfect colorings of subspace hypergraphs to analyze combinatorial designs related to bent functions, providing a unified framework for their study.

Contribution

It establishes a novel correspondence between convolution eigenfunctions and perfect colorings, applying this to characterize bent functions and related combinatorial structures.

Findings

Eigenfunctions of convolution matrices represent bent and plateaued Boolean functions.

Perfect colorings of subspace hypergraphs correspond to two-valued eigenfunctions.

New eigenvalues of convolution matrices over finite fields are identified.

Abstract

The main aim of the present paper is to introduce new methods for the study of combinatorial designs related to bent functions. They are based on interpretations of convolution on finite abelian groups as multiplication by a multidimensional matrix and designs as perfect colorings of subspace hypergraphs of $\mathbb{F}_2^n$. We establish a correspondence between eigenfunctions of convolution matrices and perfect colorings of subspace hypergraphs, show that perfect colorings of subspace hypergraphs admit a characterization in terms of convolution and that two-valued eigenfunctions of subspace hypergraphs correspond to perfect colorings. As applications, we represent partial difference sets, bent and plateaued Boolean functions, spreads, and strong bent partitions of $\mathbb{F}_2^n$ as eigenfunctions of convolution matrices and as perfect colorings of subspace hypergraphs. We also find some eigenvalues of convolution matrices over $\mathbb{F}_2^n$ and $\mathbb{F}_3^n$.