On the asymptotic existence of complex Williamson Hadamard matrices

It is shown that for each odd integer q, there is a complex WilliamsonHadamard matrix of order 2(q)+1 ·2n (q)+1 . q. In a recent paper Craigen, Holzmann and Kharaghani [1] showed that for every odd integer q) there is an integer N( q) which does not exceed twice the number of nonzero digits in the binary expansion of q, such that the existence of an Orthogonal Design (OD) of order 2N (q)-1 implies the existence of a Complex Orthogonal Design (COD) of the same number of variables and of order 2N (q)q. Although ODs of order 2m for small values of m are known, not much is known when m is 7 or more. We first give a method of constructing some crucial ODs of order 2 ) for m 2:: 7. Then we use these ODs and present a simple method of extending a classical method of Williamson [3) to any class of 2 circulant ±l-matrices, leading to an asymptotic existence theorem for complex Williamson matrices. A (Complex) Orthogonal Design of order n and type (81, S2, ... , Sk), Si positive integers, denoted (C)OD(njSl,s2,'" ,Sk), is a matrix X of order n, with entries in {0,CX1,cX2, .. . ,cXk}, c E {±1} (c E {±l,±i}), satisfying XX* = 2::7=1 (six;)In. A (complex) Hadamard matrix is a special (C)OD with Xi = 1, for all i and no zero entries. A set {Al, A 2 , ••• , Am} of (0, ±1, ±i)-matrices of order n is called msupplementary of weight w if 2::~1 AiAi = wIn. An m-supplementary set of circulant matrices of weight nm is called a set of m-complex Williamson matrices if Ai Ai for all i. A pair of matrices X, Y is called amicable (antiamicable) if XY* = YX* (XY* = -YX*). For integer n = 2 q, q odd, write c = 4a + b, 0::; b < 4. pC n) = 8a + 2b is called the Radon number of n. It is easy to see that p(22n+l_lq) = 2n+2 , for any odd integer q. Australasian Journal of Combinatorics 1O( 1994), pp.225-230 Our main reference is [2] and we refer the reader to this reference for terminology not defined here. . We begin with a well known result. Theorem 1 For every positive integer n, there is an OD(n; 1,1, ... ,1) in p(n)variables. Equivalently, there are pen) (0, ±l)-matrices, P1 , P2, ... , pp(n) , of order n such that: (i) Pi * Pj = 0, i i= j (ii) PiPl = I (iii) PiP] = Pj Pl , i i= j. PROOF. See page 2 of [2]. I Following Craigen, ,(0, ±l)-matrices satisfying (ii) above are called signed permutations. For matrices A, • A2 • let L2(A, • A,) (~:~:). Inductively. for k > 1 and matrices All A 2, ... , A2k, let L2k(AI' A2, ... , A2k) = L2(L2k-l(A1 , A2, ... , A2k-l), L2k-l (A2k-l+1,"" A2k)). For example, We call such a matrix an L 2k-matrix constructed from 2k matrices AI, A2, . .. , A2k. Obviously, different ordering of Ai's give different L 2k-matrices. Lemma 2 Let {PI, P2, ... , P2k}, k a positive integer, be a set of mutually antiamicable signed permutations of order n. Let H be an Hadamard matrix of order n. Then any L2k-matrix constructed from 2k matrices XIPIH, X2P2H, .. . , X2kP2kH, is an OD(2kn; n, n, ... , n) in 2k-variables. PROOF. We use induction on k. For k 1, note that (XiPiH)(XiPiHY = nx;In, so XiPiH is an OD(nin) for all i. PI,P2 are antiamicable, so are XIPIH,X2P2H. Hence L2(XIPIH,X2P2H) is an OD(2njn,n). Assume that X = L2t(X1P1 H, X2P2H, ... , X2lP2lH) and Y = L2l(X2L+1P2l+1H, ... , X2l+1 P2lt1 H) are OD(2ln; n, n, ... ,n). It follows now from the assumption on the Pi's that X and Yare antiamicable ODs. So L2ltl(X IPIH, ... , X2l+1P2ltlH) is an OD(2l+1 . ). 2l+1 . bl n,n,n, ... ,n In -varIa es. I