New infinite families of Williamson Hadamard matrices

Due to the Hadamard Conjecture, Williamson matrices (WM) and Williamson type matrices (WTM) of order 4n have been primarily investigated for odd n. Several constructions for this case have been introduced, leading to finite and infinite families of WMs and WTMs. The aim of this paper is to present new families of WMs and WTMs with blocks of even order. Let q and r be prime powers congruent to 1 modulo 4. There are WMs of order 4a(q + 1) for every a ∈ {1, 11, 17, 23, 29, 33, 39, 43}. If gcd(q + 1, r + 1) = 2, then there is a WM of order 2(q + 1)(r + 1). There are WMs of order 2^b (q + 1) and WTMs with circulant blocks of order 2^c (q + 1) for every b ∈ {2, …, 7} and c ∈ {5, 6}. We prove these results and more by exploiting a recently established correspondence between perfect quaternionic sequences and relative difference sets.