Hadamard Matrices of the Williamson Type

Unified Particle Swarm Optimization for Hadamard matrices of Williamson type

In this work we apply the recently proposed Unified Particle Swarm Optimization (UPSO) method to the search for Hadamard matrices of the Williamson type. The objective functions that arise from the classical Williamson construction, are ideally suited for UPSO algorithms. This is the first time that swarm intelligence methods are applied to this problem. Mathematics Subject Classification (2000...

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 impl...

A new construction for Williamson-type matrices

It is shown that if q is a prime power then there are Williamson-type matrices of order (i) 1/2q2(q + 1) when q ≡ 1 (mod 4), (ii)1/4q2(q + 1) when q ≡ 3 (mod 4) and there are Williamson-type matrices of order l/4(q + 1). This gives Williamson-type matrices for the new orders 363, 1183, 1805, 2601, 3174, 5103. The construction can be combined with known results on orthogonal designs to give an H...

Semi Williamson Type Matrices and the W (2n; N) Conjecture

Four (1,-1, 0)-matrices of order m, X = (x ij), Y = (y ij), Z = (z ij), U = (u ij) satisfying will be called semi Williamson type matrices of order m. In this paper we prove that if there exist Williamson type matrices of order n 1 ;: :: ;n k then there exist semi Williamson type matrices of order N = Q k j=1 n rj j , where r j are non-negative integers. As an application, we obtain a W(4N;2N)....

Jacket matrices constructed from Hadamard matrices and generalized Hadamard matrices