On the Density of the Set of Known Hadamard Orders

TL;DR

This paper improves the lower bound on the density of known Hadamard matrix orders, showing that their count grows faster than previously established, using advanced number theory techniques.

Contribution

It demonstrates that considering products of Paley matrix orders significantly increases the known density of Hadamard orders, surpassing earlier bounds.

Findings

Established a new lower bound involving exponential growth with iterated logarithms.

Showed that existing construction methods do not improve the bound further.

Introduced the use of multiplicative monoids in analyzing Hadamard matrix orders.

Abstract

Let $S(x)$ be the number of $n \leq x$ for which a Hadamard matrix of order $n$ exists. Hadamard's conjecture states that $S(x)$ is about $x/4$. From Paley's constructions of Hadamard matrices, we have that \[ S(x) = \Omega(x/\log x). \] In a recent paper, the first author suggested that counting the products of orders of Paley matrices would result in a greater density. In this paper we use results of Kevin Ford to show that it does: \begin{equation}\label{eq:abs} S(x) \geq x/\log x \exp((C+o(1))(\log \log \log x)^2)\,, \nonumber \end{equation} where $C=0.8178...$. This bound is surprisingly hard to improve upon. We show that taking into account all the other major known construction methods for Hadamard matrices does not shift the bound. Our arguments use the notion of a (multiplicative) monoid of natural numbers. We prove some initial results concerning these objects. Our techniques may be useful when assessing the status of other existence questions in design theory.