Density of Free Modules over Finite Chain Rings

TL;DR

This paper analyzes the density and asymptotic behavior of free modules over finite chain rings, providing bounds and implications for coding theory, especially regarding random codes and the Gilbert-Varshamov bound.

Contribution

It introduces new density calculations for free modules over finite chain rings and explores their asymptotics, linking these results to coding theory performance.

Findings

Density of free modules can be bounded by Andrews-Gordon identities.

Asymptotic behavior of modules generated by random matrices is characterized.

Random codes over finite chain rings achieve the Gilbert-Varshamov bound with high probability.

Abstract

In this paper we focus on modules over a finite chain ring $\mathcal{R}$ of size $q^s$. We compute the density of free modules of $\mathcal{R}^n$, where we separately treat the asymptotics in $n,q$ and $s$. In particular, we focus on two cases: one where we fix the length of the module and one where we fix the rank of the module. In both cases, the density results can be bounded by the Andrews-Gordon identities. We also study the asymptotic behaviour of modules generated by random matrices over $\mathcal{R}$. Since linear codes over $\mathcal{R}$ are submodules of $\mathcal{R}^n$ we get direct implications for coding theory. For example, we show that random codes achieve the Gilbert-Varshamov bound with high probability.