Thin circulant matrices and lower bounds on the complexity of some Boolean operators

TL;DR

This paper establishes a near-optimal lower bound on the weight of Boolean circulant matrices free of all-ones submatrices, leading to new complexity bounds for Boolean systems and convolutions.

Contribution

It introduces a novel lower bound for the weight of $(k,l)$-free Boolean circulant matrices, advancing understanding of their complexity.

Findings

Derived a lower bound of arac{k+l}{k^2l^2}N^{2-rac{k+l+2}{kl}} for matrix weight.

Established new bounds on complexity measures of Boolean sums' systems.

Proved a lower bound arac{N^2}{ exta0log^6 N} for Boolean convolution monotone complexity.

Abstract

We prove a lower bound $\Omega\left(\frac{k+l}{k^2l^2}N^{2-\frac{k+l+2}{kl}}\right)$ on the maximal possible weight of a $(k,l)$-free (that is, free of all-ones $k\times l$ submatrices) Boolean circulant $N \times N$ matrix. The bound is close to the known bound for the class of all $(k,l)$-free matrices. As a consequence, we obtain new bounds for several complexity measures of Boolean sums' systems and a lower bound $\Omega(N^2\log^{-6} N)$ on the monotone complexity of the Boolean convolution of order $N$.