A Comment on the Hadamard Conjecture

Fix n. Let r(n) denote the largest number r for which there is an r×n (1, −1)-matrix H satisfying the matrix equation HH⊤=nIr. The Hadamard conjecture states that for n divisible by 4 we have r(n)=n. Let ε>0. In this paper, we show that the Extended Riemann Hypothesis and recent results on the asymptotic existence of Hadamard matrices imply that for n sufficiently large r(n)>(12−ε) n.