On the weighing matrices of order 4n and weight 4n-2 and 2n-1

We algorithms and constructions for mathematical and computer searches which allow us to establish the existence of W( 4n, 4n 2) and 2n 1) for many orders 4n less than 4000. We compare these results with the orders for which W(4n,4n) and W(4n,2n) are known. We use new based on the theory of to obtain new T-matrices of order 43 and J M-matrices which yield W( 4n, 4n 2) for 5,7, 11,13,17,19,25,31,37,41,43,61,71,73,157. Definition 1 An orthogonal design A, of order n , and (S11 S2" .. , su), denoted o D( n; Sl, S2, .., on the commuting variables (±X1, ... , 0) is a square matrix of order n with entries ±Xk where each Xk occurs 51<: times in each row and column such that the distinct rows are pairwise orthogonal. In other words AAT = (SlX~ + ... + sux~)In where In is the identity matrix. It is known that the maximum number of variables in an orthogonal design is p(n), the Radon number, where for n 2 b, b odd, set a 4e + d, 0 :::; d < 4, then p(n) = 8e + 2d. Definition 2 A weighing matrix W = W(n, k) is a square matrix with entries 0, ±1 k non-zero entries per row and column and inner product of distinct rows zero. Hence W satisfies WW = kIn. The number k is called the weight of W. A n), for n == 0 (mod 4), 1 or 2, whose entries are ±1 only is called an Hadamard matrix. A W(n, n 1) for n == 0 (mod 4) is equivalent to an OD(n; 1, n 1) and a skew-HadamaTd matT1;x or order n. 'Written while visiting the Department of Computing Science, University of Alberta, Edmonton, T6G 2Hl, Canada. Research supported by a small ARC grant. Austn:ilasian Journal of Combinatodcs 1995), pp.157-174 There number of r01'1,p,(''t1''-'"'''' c()nc:erJmnLg VlirelJlhlng HHl<'iJ •. ' ..... "'.,.