Asymptotics of partial Hadamard counts below the cubic scale

Determine the asymptotic behavior of N_{n,4t}, the number of n×4t partial Hadamard matrices with ±1 entries and pairwise orthogonal rows, in regimes where t/n^α → ∞ with α < 3.

Background

The paper develops a Fourier-analytic approach to count partial Hadamard matrices via returns of a high-dimensional random walk. Prior results established N_{n,4t} ∼ A_{n,4t} when t/nα → ∞ for large α (α = 12 by de Launey–Levin; α = 4 by Canfield). The present work pushes this to α = 3 and identifies a leading correction when t is on the cubic scale.

The authors explicitly state that for α below 3 the asymptotics are currently unknown. They also analyze why the cubic threshold is natural: the residual (off-core) contributions can already be controlled for t ≫ n{8/3} log t, suggesting that further progress below α = 3 will require sharper control of the cubic phase on the core region of the Fourier integral.

References

Below \alpha=3 the asymptotics of N_{n,4t} are open (Figure~\ref{fig:regimes}).

— Counting partial Hadamard matrices in the cubic regime  (2603.30013 - Davis, 31 Mar 2026) in Section 1 (Introduction), paragraph following Theorem 1 and Figure 1