Counting results for thin Butson 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 $M=2,3,\dots$ 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.

14 pages