Counting partial Hadamard matrices in the cubic regime

Abstract

We give a precise asymptotic formula for the number of $n\times 4t$ partial Hadamard matrices in the regimes $t/n^3\to \infty$ and $t/n^3\to \hat{I}˜$ for sufficiently large fixed $\hat{I}˜$. This strengthens earlier results of de~Launey and Levin, who obtained the asymptotic for $t/n^{12}\to \infty$, and of Canfield, who extended this to $t/n^4\to \infty$.