Product of Four Hadamard Matrices

We prove that if there exist Hadamard matrices of order 4m, 4n, 4p, and 4q then there exists an Hadamard matrix of order 16mnpq. This improves and extends the known result of Agayan that there exists a Hadamard matrix of order 8mn if there exist Hadamard matrices of order 4m and 4n. Disciplines Physical Sciences and Mathematics Publication Details R. Craigen, Jennifer Seberry and Xian-Mo Zhang, Product of four Hadamard matrices, Journal of Combinatorial Theory (Ser A), 59, (1992), 318-320. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/1063 Reprinted from JOURNAL OF COMBINATORIAL THEORY, Series A All Rights Reserved by Academic Press, New York and London Note Vol. 59, No.2, March 1992 Printed in Belgium Product of Four Hadamard Matrices R CRAIGEN Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3Gl JENNIFER SEBERRY AND XIAN-Mo ZHANG Department of Computer Science, University of Wollongong, Wollongong, NSW, 2500, Australia Communicated by V. Pless Received September 24, 1990 We prove that if there exist Hadamard matrices of order 4m, 4n, 4p, and 4q then there exists an Hadamard matrix of order 16mnpq. This improves and extends the known result of Agayan that there exists a Hadamard matrix of order 8mn if there exist Hadamard matrices of order 4m and 4n. © 1992 Academic Press, Inc. A weighing matrix [3] of order n with weight k, denoted W= Wen, k), is a (0, ± 1) matrix satisfying WW = kInA Wen, n) is an Hadamard matrix. Two matrices X and Yare said to be amicable if Xyt = Yxt. They are disjoint if X (\ Y = ° (here, n denotes the Hadamard, or entry-wise, product. of matrices ). LEMMA 1. If there exist Hadamard matrices of order 4m and 4n then there exist two (± 1) matrices, Sand R of order 4mn, satisfying (i) SST + RRT = 8mnI4mn , (ii) SR = RS T = 0. Proof We write 318 0097-3165/92 $3.00 Copyright i[; 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.