Orthogonal Designs IV: Existence Questions

In [5] Raghavarao showed that if n = 2 (mod 4) and A is a {O, 1, -1} matrix satisfying AAt = (n 1) In. then n 1 = a2 b2 for a, b integers. In [4] van Lint and Seidel giving a proof modeled on a proof of the Witt cancellation theorem, proved more generally that if n is as above and A is a rational matrix satisfying AAt = kIn then k = q12 + q22 (q1, q2 E Q, the rational numbers). Consequently, if k is an integer then k = a2 + b2 for two integers a and b. In [1] we showed that if, in addition, A = -At then k = S2. Disciplines Physical Sciences and Mathematics Publication Details Anthony V. Geramita and Jennifer Seberry Wallis, Orthogonal designs IV: Existence questions, Journal of Combinatorial Theory, Ser. A., 19, (1975), 66-83. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/964 Reprinted from JOURr\AL OF COl\IBI~.-\TORT \1. THEORY All Rights Reserved by Academic Press, :\ew York and London Yo1. 19, :\0, \, July 1975 Printed in Belgium Orthogonal Designs IV: Existence Questions ANTHONY V, GERAMITAI Department of Mathematics, Queen's Unirersity, Kingston, Ontario, Canada