On the Asymptotic Existence of Hadamard Matrices

Abstract

It is conjectured that Hadamard matrices exist for all orders $4t$ ($t>0$). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers $k$, there is a Hadamard matrix of order $k{2}^{[a+b{\log }_{2}k]}$, where $a$ and $b$ are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take $a=2$ and $b=0$. Since Seberry's ground-breaking result, which showed that we may take $a=0$ and $b=2$, there have been several improvements where $b$ has been by stages reduced to 3/8. In this paper, we show that for all $\epsilon >0$, the set of odd numbers $k$ for which there is a Hadamard matrix of order $k{2}^{2+[\epsilon {\log }_{2}k]}$ has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.

Keywords: Hadamard matrices, Asymptotic existence, Cocyclic Hadamard matrices, Relative difference sets, Riesel numbers, Extended Riemann hypothesis. (Received 2 August 2008, Available online 18 March 2009)