Counting results for thin Butson matrices

TL;DR

This paper studies the enumeration of partial Butson matrices with small numbers of rows and large numbers of columns, extending previous counting methods to a new 'thin' regime.

Contribution

It introduces new counting techniques for partial Butson matrices in the thin regime, expanding understanding of their enumeration for small row counts.

Findings

Derived counting formulas for small M and large N

Extended existing methods to the thin matrix regime

Provided asymptotic estimates for the number of such matrices

Abstract

A partial Butson matrix is a matrix $H\in M_{M\times N}(\mathbb Z_q)$ having its rows pairwise orthogonal, where $\mathbb Z_q\subset\mathbb C^\times$ is the group of $q$-th roots of unity. We investigate here the counting problem for these matrices in the "thin" regime, where $M=2,3,\ldots$ is small, and where $N\to\infty$ (subject to the condition $N\in p\mathbb N$ when $q=p^k>2$). The proofs are inspired from the de Launey-Levin and Richmond-Shallit counting results.