Signed groups, sequences, and the asymptotic existence of Hadamard matrices

Abstract

We use the newly developed theory of signed groups and some known sequences with zero autocorrelation to derive new results on the asymptotic existence of Hadamard matrices. New values of t are obtained such that, for any odd number p, there exists an Hadamard matrix of order 2 t p. These include: t = 2 N, where N is the number of nonzero digits in the binary expansion of p, and t = 4⌈ 1 6 log 2( (p − 1) 2 ⌉ + 2 . Both numbers improve on all previous general results, but neither uses the full power of our method. We also discuss some of the implications of our method in terms of signed group Hadamard matrices and signed group weighing matrices: There exists a circulant signed group Hadamard matrix of every even order n, using a suitable signed group. This result stands in striking contrast to the known results for Hadamard matrices and complex Hadamard matrices, and the circulant Hadamard matrix conjecture. Signed group weighing matrices of even order n always exist, with any specified weight w ⩽ n.