Fooling-sets and rank

An n i? n matrix M is called a fooling-set matrix of size n if its diagonal entries are nonzero and M k , ? M ? , k = 0 for every k ? ? . Dietzfelbinger, Hromkovic, and Schnitger (1996) showed that n ? ( 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. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size n = ( rk M + 1 2 ) . In nonzero characteristic, we construct an infinite family of matrices with n = ( 1 + o ( 1 ) ) ( rk M ) 2 .

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