On skew Hadamard matrices

Recently I have proved that for every odd integer q there exists integers t and s (dependent on q) so that there is an Hadamard matrix of order 2tq and a symmetric Hadamard matrix with constant diagonal order 2s q2. We conjecture that "for every odd integer q there exists an integer t (dependent on q) so that there is a skew-Hadamard matrix of order 2tq”. This paper makes progress toward proving this conjecture. In particular we prove the result when q = 5 (mod 8) = s2 + 4r2 is a prime power and all orthogonal designs of type (l, a, b, c, c+│r│), where 1+a+b+2c+│r│ = 2t, exist in order 2t.