A comment on the Hadamard conjecture (Q5940309)

Let \(r(n)\) be the largest integer such that there exists an \(r \times r\) matrix with entries \(1\) and \(-1\) satisfying \(HH^T=nI_{r(n)}\). The authors prove that, under the extended Riemann hypothesis, for any \(\varepsilon>0\) one has \(r(n)>(\frac{1}{2}-\varepsilon)n\) for all suffiently large \(n\). By applications of Hadamard matrices this applies to error-correcting codes, sets of mutually orthogonal \(F\)-squares, etc.