An infinite class of supplementary difference sets and Williamson matrices (Q1180560)

With skilful calculations the authors obtain the following results. Theorem 1: Let \(q\) be a prime power, \(v=q^ 2\), \(k=q(q-1)/2\) and \(\lambda=4k-v\). Then there exists a \(4\)-\(\{v;k,k,k,k;\lambda\}\) supplementary set. Theorem 2 (proof using Theorem 1): Let \(p\) be an odd prime and \(S=\{2k+1; 0\leq k\leq 16\}\cup\{37,59,61,67\}\cup\{2^ i10^ j26^ k+1; i,j,k\geq 0\}\). Further let \(n=tp^{2r}\) or \(tp^{4r}\) according as \(p\equiv 1\pmod 4\) or \(p\equiv 3\pmod 4\), \(t\in S\) and \(n=p^ r\) if \(p=2^ i10^ j26^ k+1\equiv 1\pmod 4\). Then there exists an Hadamard matrix of order \(4n\).