Density of free modules over finite chain rings

Abstract

In this paper we focus on modules over a finite chain ring R of size qs. We compute the density of free modules of Rn, 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 R. Since linear codes over R are submodules of Rn we get direct implications for coding theory. For example, we show that random codes achieve the Gilbert-Varshamov bound with high probability.

Topics & Concepts