Papers › Fooling sets and rank
Fooling sets and rank
Mirjam Friesen, Aya Hamed, Troy Lee, Dirk Oliver Theis
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An n×n matrix M is called a \textit{fooling-set matrix of size n} if its diagonal entries are nonzero and M_(k,ℓ) M_(ℓ,k) = 0 for every kℓ. Dietzfelbinger, Hromkovi{\v{c}}, and Schnitger (1996) showed that n ≤( M)², regardless of over which field the rank is computed, and asked whether the exponent on M can be improved. We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size n = M+12. In nonzero characteristic, we construct an infinite family of matrices with n= (1+o(1))( M)².
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