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arXiv · 2210.12405

Multidimensional threshold matrices and extremal matrices of order $2$

Anna A. Taranenko
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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.

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Anna A. Taranenko. 2022-10-22. Multidimensional threshold matrices and extremal matrices of order $2$. https://arxiv.org/abs/2210.12405

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