Rank and fooling set size

Abstract

Say that A is a Hadamard factorization of the identity I_n of size n if the entrywise product of A and the transpose of A is I_n. It can be easily seen that the rank of any Hadamard factorization of the identity must be at least sqrt{n}. Dietzfelbinger et al. raised the question if this bound can be achieved, and showed a boolean Hadamard factorization of the identity of rank n^{0.792}. More recently, Klauck and Wolf gave a construction of Hadamard factorizations of the identity of rank n^{0.613}. Over finite fields, Friesen and Theis resolved the question, showing for a prime p and r=p^t+1 a Hadamard factorization of the identity A of size r(r-1)+1 and rank r over F_p. Here we resolve the question for fields of zero characteristic, up to a constant factor, giving a construction of Hadamard factorizations of the identity of rank r and size (r+1)r/2. The matrices in our construction are blockwise Toeplitz, and have entries whose magnitudes are binomial coefficients.