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on Asymptotic counting exponent for partial Hadamard matrices ($C_{23c}$)

The statement

For integers $n \ge 2$ and $t \ge 1$ , an $n \times t$ partial Hadamard matrix is a matrix with entries in $\{\pm 1\}$ whose rows are pairwise orthogonal. Let $N_{n,t}$ denote the number of such matrices. For every fixed $n$ one has

N_{n,4t} = [1+o(1)]\,A_{n,4t} \qquad\text{as } t \to \infty,

where

A_{n,4t} := 2^{4nt+(n-1)^2}(8\pi t)^{-n(n-1)/4}.

[Davis2026-def-partial] [Davis2026-fixed-n-A]

Historically, de Launey–Levin proved the fixed- $n$ asymptotic $N_{n,4t} \sim A_{n,4t}$ ; Canfield later obtained the uniform regime $t/n^4 \to \infty$ ; and the current best result improves this to $t/n^3 \to \infty$ and shows that $N_{n,4t}/A_{n,4t}$ does not converge to $1$ when $t=\Theta n^3$ with $\Theta$ large fixed. Thus the remaining open problem is to determine the correct asymptotics of $N_{n,4t}$ below the cubic scale. [Davis2026-fixed-n-A] [Can2011] [Davis2026-prior-exponents] [Davis2026-change-cubic] [Davis2026-open-below3]

We define $C_{23c}$ to be the smallest admissible exponent $\alpha \ge 1$ for which the count $N_{n,4t}$ admits a uniform asymptotic formula throughout the regime $t/n^\alpha \to \infty$ . The restriction $\alpha \ge 1$ is forced by the linear-algebra obstruction $4t \ge n$ . The current best-established range is

1 \le C_{23c} \le 3.

[DL2010-linear-obstruction] [Davis2026-open-below3] [Davis2026-abstract-cubic]

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