Construction of Williamson type matrices

Recent advances in the construction of Hadamard matrices have depeaded on the existence of Baumert-Hall arrays and four (1, ?1) matrices A B C Dof order m which are of Williamson type, that is they pair-wise satisfy

i) MN^T = NM^T , ∈ {A B C D} and

ii) AA^T + BB^T + CC^T + DD^T = 4mI_m .

It is shown that Williamson type matrices exist for the orders m = s(4 ? 1)m = s(4s + 3) for s∈ {1, 3, 5, …, 25} and also for m = 93. This gives Williamson matrices for several new orders including 33, 95,189.

These results mean there are Hadamard matrices of order

i) 4s(4s ?1)t, 20s(4s ? 1)t,s ∈ {1, 3, 5, …, 25};

ii) 4s(4:s + 3)t, 20s(4s + 3)t s ∈ {1, 3, 5, …, 25};

iii) 4.93t, 20.93t

for

t ∈ {1, 3, 5, … , 61} ∪ {1 + 2 ^a 10 ^b 26 ^c a b c nonnegative integers}, which are new infinite families.

Also, it is shown by considering eight-Williamson-type matrices, that there exist Hadamard matrices of order 4(p + 1)(2p + l)r and 4(p + l)(2p + 5)r when p ≡ 1 (mod 4) is a prime power, 8ris the order of a Plotkin array, and, in the second case 2p + 6 is the order of a symmetric Hadamard matrix. These classes are new.