Williamson Hadamard matrices and Gauss sums

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...

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...

An angular momentum approach to quadratic Fourier transform , Hadamard matrices , Gauss sums , mutually unbiased bases , unitary group and Pauli group

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and analysed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl p...

New Hadamard Matrices of Order 4p obtained from Jacobi Sums

Let p ≡ 7 mod 16 be a prime. Then there are integers a, b, c, d with a ≡ 15 mod 16, b ≡ 0 mod 4, p = a + 2(b + c + d), and 2ab = c − 2cd − d. We show that there is a regular Hadamard matrix of order 4p provided that p = a± 2b or p = a+ δ1b+4δ2c+4δ1δ2d with δi = ±1. ∗This research was done during a visit of the first two authors at the University of Augsburg

Jacket matrices constructed from Hadamard matrices and generalized Hadamard matrices