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

Vladimir N. Potapov, Anna A. Taranenko

TL;DR

The work develops a multidimensional convolution framework that pairs eigenfunctions of convolution matrices with perfect colorings of subspace hypergraphs to study combinatorial designs tied to bent and plateaued Boolean functions. By connecting $A_G^{(t)}$ with hypergraph adjacency and leveraging Fourier analysis, the authors derive explicit eigenvalues for $\mathbb{F}_2^n$ and $\mathbb{F}_3^n$, and establish tight correspondences between partial difference sets, bent/plateaued functions, spreads, and strong bent partitions with perfect colorings. These correspondences yield constructive and spectral criteria for the existence and structure of designs, enabling a unified algebraic-hypergraph view of design theory in the Boolean and ternary settings. The framework promises new tools for identifying designs via eigenfunction analysis and for interpreting classical objects (PDS, spreads, bent partitions) as colorings of highly regular subspace hypergraphs. The work thereby bridges spectral hypergraph theory with combinatorial designs and Boolean function theory, offering both theoretical insights and avenues for new constructions.

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 $\mathbb{F}_3^n$.

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

TL;DR

The work develops a multidimensional convolution framework that pairs eigenfunctions of convolution matrices with perfect colorings of subspace hypergraphs to study combinatorial designs tied to bent and plateaued Boolean functions. By connecting with hypergraph adjacency and leveraging Fourier analysis, the authors derive explicit eigenvalues for and , and establish tight correspondences between partial difference sets, bent/plateaued functions, spreads, and strong bent partitions with perfect colorings. These correspondences yield constructive and spectral criteria for the existence and structure of designs, enabling a unified algebraic-hypergraph view of design theory in the Boolean and ternary settings. The framework promises new tools for identifying designs via eigenfunction analysis and for interpreting classical objects (PDS, spreads, bent partitions) as colorings of highly regular subspace hypergraphs. The work thereby bridges spectral hypergraph theory with combinatorial designs and Boolean function theory, offering both theoretical insights and avenues for new constructions.

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 . 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 as eigenfunctions of convolution matrices and as perfect colorings of subspace hypergraphs. We also find some eigenvalues of convolution matrices over and .
Paper Structure (9 sections, 25 theorems, 65 equations)

This paper contains 9 sections, 25 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Eigenvalues of convolution matrices of $\mathbb{F}_2^n$ and $\mathbb{F}_3^n$
  4. Relations between convolution matrices, subspace hypergraphs, and perfect colorings
  5. Designs are perfect colorings and eigenfunctions of subspace hypergraphs
  6. Partial difference sets
  7. Bent and plateaued functions
  8. Spreads
  9. Bent partitions

Key Result

Theorem 1

my.hyperperfcolor Let $\mathcal{G}$ be a $d$-uniform hypergraph with $d$-dimensional adjacency matrix $M$. Then a coloring $F$ of $\mathcal{G}$ into $k$ colors is perfect if and only if there exists the $d$-dimensional parameter matrix $S$ of order $k$ such that $M \circ F = F \circ S$. Moreover, th where every color $j$ appears in the multiset $\{i_2, \ldots, i_d \}$ exactly $d_j$ times, and ${d-

Theorems & Definitions (42)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 1
  • proof
  • ...and 32 more