Fooling-sets and rank

An n×nn×n matrix MM is called a fooling-set matrix of size  nn if its diagonal entries are nonzero and Mk,ℓMℓ,k=0Mk,ℓMℓ,k=0 for every k≠ℓk≠ℓ. Dietzfelbinger, Hromkovič, and Schnitger (1996) showed that n≤(rkM)2n≤(rkM)2, regardless of over which field the rank is computed, and asked whether the exponent on rkMrkM can be improved.We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size n=(rkM+12). In nonzero characteristic, we construct an infinite family of matrices with n=(1+o(1))(rkM)2n=(1+o(1))(rkM)2.