Fooling-sets and rank in nonzero characteristic
An n × n matrix M is called a fooling-set matrix of size n, if its diagonal entries are nonzero, whereas for every k ≠ ℓ we have Mk,ℓMℓ,k = 0. Dietzfel-binger, Hromkovič, and Schnitger (1996) showed that n ≤ (rkM)2, regardless of over which field the rank is computed, and asked whether the exponent on rkM 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.
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