Counting partial Hadamard matrices in the cubic regime

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