On the Asymptotic Existence of Hadamard Matrices

TL;DR

This paper demonstrates that for any small positive epsilon, a positive density of odd numbers exist for which Hadamard matrices of a certain asymptotic order can be constructed, advancing the understanding of their existence.

Contribution

It shows that for all epsilon>0, a positive density of odd numbers have associated Hadamard matrices of order k2^{2+[epsilon log_2 k]}, using a number-theoretic approach.

Findings

Positive density of odd numbers with Hadamard matrices of specified order.

Extension of asymptotic existence results for Hadamard matrices.

Application of Erdos and Odlyzko's number-theoretic methods.

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 $k2^{[a+b\log_2k]}$, 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 $k2^{2+[\epsilon\log_2k]}$ 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.