Publication Details

Jennifer Seberry Wallis, Williamson matrices of even order, Combinatorial Mathematics: Proceedings of the Second Australian Conference, (D.A. Holton, (Ed.)), Lecture Notes in Mathematics, 403, Springer–Verlag, Berlin–Heidelberg–New York, (1974), 132-142.


Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and Williamson-type matrices. These latter are four (1,-1) matrices A,B,C,D, of order m, which pairwise satisfy

(i) MNT = NMT, M,N E (A,B,C,D),

and (ii) AAT +BBT +CCT +DDT = 4mIm, where I is the identity matrix.

Currently Williamson matrices are known to exist for all orders less than 100 except: 35,39,47,53,59,65,67,70,71,73,76,77,83,89,94.

This paper gives two constructions for Williamson matrices of even order, 2n. This is most significant when no Williamson matrices of order n are known. In particular we give matrices for the new orders 2.39,2.203,2.303,2.333.2.689,2.915. 2.1603.