Dual of Codes over Finite Quotients of Polynomial Rings

TL;DR

This paper explores the duality of codes over finite quotients of polynomial rings, correcting previous theorems, and provides algorithms and characterizations for dual and self-dual codes.

Contribution

It corrects a prior theorem on $A$-codes duality, introduces an efficient generator-finding algorithm, and characterizes self-dual $A$-codes of length 2.

Findings

Counterexample and correction to Berger and El Amrani's theorem

Efficient algorithm for dual code generators

Characterization of self-dual $A$-codes of length 2

Abstract

Let $A=\frac{\mathbb{F}[x]}{\langle f(x)\rangle }$, where $f(x)$ is a monic polynomial over a finite field $\mathbb{F}$. In this paper, we study the relation between $A$-codes and their duals. In particular, we state a counterexample and a correction to a theorem of Berger and El Amrani (Codes over finite quotients of polynomial rings, \emph{Finite Fields Appl.} \textbf{25} (2014), 165--181) and present an efficient algorithm to find a system of generators for the dual of a given $A$-code. Also we characterize self-dual $A$-codes of length 2 and investigate when the $\mathbb{F}$-dual of $A$-codes are $A$-codes.