When Variable-Length Codes Meet the Field of Error Detection

TL;DR

This paper explores the intersection of variable-length codes and error detection by analyzing the decidability of certain conditions related to quasi-metrics and relations over codes, with implications for error detection capabilities.

Contribution

It introduces a framework connecting variable-length codes, quasi-metrics, and error detection, and investigates the decidability of related conditions for regular codes.

Findings

Decidability results for error detection conditions under specific metrics

Analysis of prefix and automorphism-based quasi-metrics

Framework for assessing error detection in variable-length codes

Abstract

Given a finite alphabet $A$ and a binary relation $\tau\subseteq A^*\times A^*$, a set $X$ is $\tau$-{\it independent} if $ \tau(X)\cap X=\emptyset$. Given a quasi-metric $d$ over $A^*$ (in the meaning of \cite{W31}) and $k\ge 1$, we associate the relation $\tau_{d,k}$ defined by $(x,y)\in\tau_{d,k}$ if, and only if, $d(x,y)\le k$ \cite{CP02}.In the spirit of \cite{JK97,N21}, the error detection-correction capability of variable-length codes can be expressed in term of conditions over $\tau_{d,k}$. With respect to the prefix metric, the factor one, and every quasi-metric associated to (anti-)automorphisms of the free monoid, we examine whether those conditions are decidable for a given regular code.