Some infinite classes of Williamson matrices and weighing matrices

Williamson type matrices A, B, C 1 D will be called nice if ABT + C DT = 0, perfect if ABT + CDT = ACT + BDT 0, special if ABT + CDT = ACT + BDT = ADT + BCT = o. Type 1 (1 , -1 )-matrices A, B, C, D of order n will be called tight Williamson-like matrices if AAT + BBT + CCT + DDT 4nIn and ABT + BAT + CDT + DCT o. Write N = 32T . piTl ••• p!Tn , where Pj 3(mod 4), Pj > 3, j = 1, ... ,n and r, rl , ... 1 rn are non-negative integers. In this paper we prove: (i) if there exist special Williamson type matrices of order n then there exist two disjoint amicable W(2n, n), whose sum and difference are (1 ,-1 )-matrices, and four disjoint and amicable W( 4n, n), whose sum is a (1 , -1 )-matrix; (ii) there exists an Hadamard matrix of order 4mn, where m is the order of tight Williamson-like matrices and n is the order of nice Williamson type matrices. Definition 1 Williamson type matrices A, B, C) D will be called nice if ABT + CDT = 0, perfect if ABT + CDT = ACT + BDT = 0, special if ABT + CDT = ACT + BDT ADT + BCT = 0 (see Definition 4, [2]). Definition 2 Type 1 (1 , -1 )-matrices A, B, C, D of order n will be called tight Williamson-like matrices if AAT + B BT + C CT + DDT = 4nIn and ABT + BAT + CDT + DCT = 0 (see Definition 5, [2]). Notation 1 Write N = 32r . pirl ... p!rn, where Pj 3(mod 4), Pj > 3, j = 1, ... ,n and r, rI, ... ,rn are non-negative integers. Australasian Journal of Combinatorics §( 1992), pp.107-110 Theorem 1 If there exist special Williamson type matrices of order n then there exist two disjoint amicable W(2n, n), whose sum and difference are (1 , -1}-matrices. Proof. Let AI, A2, A3, A4 be the Williamson type matrices of order n. Set Q = ~ [ Al A2 A3 A4] 2 A3 A4 Al A2 . Then P and Q are the required two W(2n, n). o Remark. W(2n, n), n odd, exist only if n is the sum of two squares (see Corollary 2.11 [1]). Coronary 1 There exist two disjoint and amicable W(2N, N), whose sum and difference are (1,-1}-matrices. Proof. N. From Theorem 5 [3] there exist special Williamson type matrices of order o Theorem 2 If there exist special Williamson type matrices of order n then there exist four disjoint and amicable W ( 4n, n), whose sum is a (1 , -1 }-matrix. Proof Set E = [~ n, F = [~ ~], G = [~ ~], H [~~], where P, Q were given in the proof of Theorem 1. Then E, F, G, H are the required weighing matrices. 0 Coronary 2 There exist four disjoint and amicable W(4N, N), whose sum is a (1 J -1}-matrix.