Rank and fooling set size

Say that A is a Hadamard factorization of the identity In of size n if A ◦AT = In, where ◦ denotes the Hadamard or entrywise product, and AT denotes the transpose of A. As n = rk(In) = rk(A ◦ AT ) ≤ rk(A)2, it is clear that the rank of any Hadamard factorization of the identity must be at least √ n. Dietzfelbinger et al. [DHS96] raised the question if this bound can be achieved, and showed a boolean Hadamard factorization of the identity with rk(A) ≤ n0.792.... More recently, Klauck and Wolf [KW13] gave a construction of Hadamard factorizations of the identity of rank n0.613.... Over finite fields, Friesen and Theis [FT12] resolved the question, showing for a prime p and r = pt + 1 a Hadamard factorization of the identity A of size r(r − 1) + 1 and rkp(A) = r, where rkp(·) indicates the rank over Fp. 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 2 ) . The matrices in our construction are blockwise Toeplitz, and have entries whose magnitudes are binomial coefficients.