Counting partial Hadamard matrices in the cubic regime
This presentation explores a breakthrough in asymptotic enumeration of partial Hadamard matrices, revealing precisely how many such objects exist when the number of columns scales with the cube of the number of rows. The work dramatically sharpens prior results by identifying the exact threshold where Gaussian behavior emerges and explicitly computing the first-order correction term tied to triangular substructures. Through sophisticated Fourier analysis and random walk techniques, the authors solve a long-standing quantitative gap in combinatorial design theory.Script
How many ways can you arrange plus-ones and minus-ones into a matrix where every pair of rows is perfectly orthogonal? For decades, mathematicians could answer this only when the matrix had vastly more columns than rows. This paper cracks the cubic threshold, revealing the precise count when columns scale with the cube of rows, a regime previously out of reach.
A partial Hadamard matrix is deceptively simple: rows of plus-ones and minus-ones that, taken two at a time, always dot to zero. The full Hadamard conjecture asks whether square versions exist for all dimensions divisible by 4, a question open for over a century. But counting how many such matrices exist for a given size has remained even more mysterious.
This work solves the counting problem in a regime no prior method could touch.
Earlier methods required the number of columns to dwarf the number of rows by a factor of n to the 4th or even n to the 12th before the asymptotics became sharp. The new approach isolates the cubic threshold, deriving an explicit formula with a correction term proportional to one over t, and demonstrating that all higher-order deviations vanish as t over n cubed grows.
The proof recasts the counting problem as estimating the probability that a high-dimensional random walk returns to the origin after t steps. By decomposing the integration domain into a Gaussian core, negligible tails, and strongly contracting regions, the authors isolate the leading correction, which emerges from cubic phase oscillations near the critical points. Controlling this phase in one integrated step is the technical breakthrough.
Knowing precisely when the count transitions to Gaussian asymptotics has immediate consequences for random construction algorithms and probabilistic existence arguments in design theory. The explicit first-order correction, tied to the number of triangles in the complete graph, reveals the combinatorial structure governing deviations. The analytic techniques extend naturally to related enumeration problems in coding theory and extremal combinatorics.
From a century-old conjecture to a sharp asymptotic formula, this work shows how modern Fourier analysis can illuminate the deep structure of combinatorial designs. Visit EmergentMind.com to explore more research and create your own videos.
