Counting partial Hadamard matrices in the cubic regime

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

Figures (4)

• Figure 1: A $3\times 8$ partial Hadamard matrix (left) and the running pairwise-product sums (right). The matrix is partial Hadamard exactly when all three sums return to zero at $k=8$.
• Figure 2: Each marker indicates the smallest $\alpha$ for which the corresponding work establishes $N_{n,4t}\sim A_{n,4t}$ as $t/n^\alpha\to\infty$. Corollary \ref{['cor:uniform']} extends this to $\alpha=3$. Theorem \ref{['thm:main-intro']} further shows that $N_{n,4t}/A_{n,4t}$ has a nonvanishing correction when $t/n^3$ converges to a constant.
• Figure 3: Composite overview. (a) The torus $\mathbb{T}^3$ decomposes into primary boxes $\mathcal{B}_\delta$ (blue), even residual cells (orange), and odd cells (hatched). (b) A single even cell $\mathcal{B}_{\pi/4}$; the blue $\mathcal{B}_\delta$ is expanded in (c). (c) Inside $\mathcal{B}_\delta$: the core $\mathcal{D}_{\mathrm{core}}$ (red), annulus (yellow), and corners (blue).
• Figure 4: Partner counting when $\mathcal{O}(M)$ is a triangle on $\{i,j,k\}$. The three blue edges have odd exponent (fixed by $\mathcal{O}$); the two orange edge-slots carry the remaining multiplicity. Left: the shared odd support. Middle: the doubled edge lands on a triangle edge ($3$ choices). Right: the doubled edge goes to a new vertex $\ell$ ($O(n)$ choices).

Theorems & Definitions (48)

• Theorem 1.1
• Lemma 1.2: Cosine-product bound DL10
• Lemma 1.3: Lattice structure
• Proposition 1.4: Primary-secondary decomposition DL10
• Lemma 1.5: Residual decomposition DL10
• proof
• Lemma 1.6: Odd-cell bound
• proof
• Lemma 1.7: Positivity of $\operatorname{Re}\psi$ DL10
• proof