On the asymptotic existence of Hadamard matrices

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 Seberryʹs 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 ϵ > 0 , the set of odd numbers k for which there is a Hadamard matrix of order k 2 2 + [ ϵ 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.