Topologies for Error-Detecting Variable-Length Codes

TL;DR

This paper introduces a framework for analyzing error detection in variable-length codes using quasi-metrics and proves decidability of error detection conditions for certain classes of codes.

Contribution

It defines a new relation based on quasi-metrics to characterize error detection and proves decidability results for regular variable-length codes under specific metrics.

Findings

Decidability of error detection conditions for regular codes

Introduction of the relation _{d,k} for error detection analysis

Application to prefix metric and automorphism-based quasi-metrics

Abstract

Given a finite alphabet $A$, a quasi-metric $d$ over $A^*$, and a non-negative integer $k$, we introduce the relation $\tau_{d,k}\subseteq A^*\times A^*$ such that $(x,y)\in\tau_{d,k}$ holds whenever $d(x,y)\le k$. The error detection capability of variable-length codes is expressed in term of conditions over $\tau_{d,k}$. With respect to the prefix metric, the factor one, and any quasi-metric associated with some free monoid (anti-)automorphism, we prove that one can decide whether a given regular variable-length code satisfies any of those error detection constraints.