Counting partial Hadamard matrices in the cubic regime

TL;DR

This paper derives precise asymptotic formulas for counting partial Hadamard matrices in specific regimes, extending previous asymptotic results to broader parameter ranges.

Contribution

It provides a more accurate asymptotic count of partial Hadamard matrices for larger regimes, improving upon earlier asymptotic formulas.

Findings

Established asymptotic formulas for $t/n^3 o ext{large}$ and $t/n^3 o ext{finite}$ regimes.

Extended previous results from $t/n^{12} o ext{large}$ and $t/n^4 o ext{large}$ regimes.

Strengthened understanding of the enumeration of partial Hadamard matrices.

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 $t/n^3\to\Theta$ for sufficiently large fixed $\Theta$. 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 $t/n^4\to\infty$.