On the asymptotic existence of complex Hadamard matrices

Abstract

Let N = N(q) be the number of nonzero digits in the binary expansion of the odd integer q. A construction method is presented which produces, among other results, a block circulant complex Hadamard matrix of order 2αq, where α ≥ 2N - 1. This improves a recent result of Craigen regarding the asymptotic existence of Hadamard matrices. We also present a method that gives complex orthogonal designs of order 2α+1q from complex orthogonal designs of order 2α. We also demonstrate the existence of a block circulant complex Hadamard matrix of order 2βq, where © 1997 John Wiley & Sons, Inc. J Combin Designs 5:319–327, 1997