Fooling-sets and rank in nonzero characteristic (extended abstract)
arXiv:1305.2468
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, Hromkovič, 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.
This is an extended abstract; the full paper is arXiv:1208.2920