Asymptotic Enumeration of Difference Matrices over Cyclic Groups

TL;DR

This paper establishes a connection between random walks on Euclidean lattices and difference matrices over cyclic groups, using Fourier analysis to derive their asymptotic enumeration as columns grow.

Contribution

It introduces a novel approach linking random walk analysis with the enumeration of difference matrices over cyclic groups.

Findings

Derived asymptotic formulas for the number of difference matrices

Connected random walk return probabilities with difference matrix enumeration

Applied Fourier analysis to estimate combinatorial quantities

Abstract

We identify a relationship between a certain family of random walks on Euclidean lattices and difference matrices over cyclic groups. We then use the techniques of Fourier analysis to estimate the return probabilities of these random walks, which in turn yields the asymptotic number of difference matrices over cyclic groups as the number of columns increases.