On Counting Subring-Subcodes of Free Linear Codes Over Finite Principal Ideal Rings

TL;DR

This paper investigates the enumeration of free linear codes over finite principal ideal rings and their Galois extensions, correcting previous inaccuracies in the finite field case.

Contribution

It provides a precise count of subring-subcodes of free linear codes over finite principal ideal rings, addressing and correcting earlier errors in the finite field scenario.

Findings

Derived formulas for counting free S-linear codes with given ranks

Corrected previous incorrect results in the finite field case

Extended enumeration techniques to finite principal ideal rings

Abstract

Let $R$ be a finite principal ideal ring and $S$ the Galois extension of $R$ of degree $m$. For $k$ and $k_0$, positive integers we determine the number of free $S$-linear codes $B$ of length $l$ with the property $k = rank_S(B)$ and $k_0 = rank_R (B\cap R^l)$. This corrects a wrong result which was given in the case of finite fields.