Multidimensional threshold matrices and extremal matrices of order $2$

TL;DR

This paper characterizes extremal multidimensional (0,1)-matrices related to polydiagonals, showing they are threshold matrices, and explores their properties, enumeration, and specific structures for order 2.

Contribution

It proves that extremal matrices of order 2 are exactly selfdual threshold Boolean functions and analyzes their distribution and enumeration.

Findings

Extremal matrices of order 2 are selfdual threshold Boolean functions.

Different threshold matrices have distinct hyperplane distributions.

The paper provides asymptotics for the number of extremal matrices of order 2.

Abstract

The paper is devoted to multidimensional $(0,1)$-matrices extremal with respect to containing a polydiagonal (a fractional generalization of a diagonal). Every extremal matrix is a threshold matrix, i.e., an entry belongs to its support whenever a weighted sum of incident hyperplanes exceeds a given threshold. Firstly, we prove that nonequivalent threshold matrices have different distributions of ones in hyperplanes. Next, we establish that extremal matrices of order $2$ are exactly selfdual threshold Boolean functions. Using this fact, we find the asymptotics of the number of extremal matrices of order $2$ and provide counterexamples to several conjectures on extremal matrices. Finally, we describe extremal matrices of order $2$ with a small diversity of hyperplanes.