Fooling-sets and rank in nonzero characteristic (extended abstract)

TL;DR

This paper proves that for matrices over fields with nonzero characteristic, the known rank bound for fooling-set matrices is asymptotically tight, using constructions based on linear recurring sequences.

Contribution

It establishes the tightness of the rank bound for fooling-set matrices in nonzero characteristic fields through explicit constructions.

Findings

The bound n ≤ (rk M)^2 is asymptotically tight in nonzero characteristic fields.

Constructs a family of matrices achieving the bound using linear recurring sequences.

Answers an open question about the exponent in the rank bound for fooling-set matrices.

Abstract

An n\times n matrix M is called a fooling-set matrix of size n, if its diagonal entries are nonzero, whereas for every k\ne \ell we have M_{k,\ell} M_{\ell,k} = 0. Dietzfelbinger, Hromkovi\v{c}, and Schnitger (1996) showed that n \le (\rk M)^2, regardless of over which field the rank is computed, and asked whether the exponent on \rk M can be improved. We settle this question for nonzero characteristic by constructing a family of matrices for which the bound is asymptotically tight. The construction uses linear recurring sequences.