2.1 Notation
Let \(n\ge 2\) and \(d=\binom {n}{2}\). An \(n\times 4t\) partial Hadamard matrix has \(\pm 1\) entries with pairwise orthogonal rows. Let \(N_{n,4t}\) be the number of such matrices and \(A_{n,4t}=\binom {2n}{n}^{2t}\) the Gaussian approximation. Set \(\psi (\lambda )=\mathbb {E}[\exp (i\sum _{i{\lt}j}\lambda _{\{ i,j\} }\xi _i\xi _j)]\) for independent Rademacher signs \(\xi _i\), \(s(\lambda )=\| \lambda \| ^2=\sum _e\lambda _e^2\), and
\[ K_n:=2^{2d-n+1}(2\pi )^{-d}, \qquad G_{\mathrm{core}}(d,t):=\int _{\mathcal{D}_\mathrm {core}}e^{-2t\| \lambda \| ^2}\, d\lambda , \qquad F(d,t):=\left(\frac{\pi }{2t}\right)^{d/2}. \]