Correcting Errors Through Partitioning and Burst-Deletion Correction

TL;DR

This paper introduces a novel partitioning method to construct error-correcting codes capable of correcting combined deletion and substitution errors, improving upon existing techniques for certain parameters.

Contribution

It develops a unified framework for correcting deletion and substitution errors by partitioning sequences and leveraging burst-deletion correction methods, with new constructions for specific error parameters.

Findings

Constructed codes for t=1,2 in binary alphabets

Some codes match existing results, others outperform them

Provides new insights into error correction principles

Abstract

In this paper, we propose a partitioning technique that decomposes a pair of sequences with overlapping $t$-deletion $s$-substitution balls into sub-pairs, where the $^{\leq}t$-burst-deletion balls of each sub-pair intersect. This decomposition facilitates the development of $t$-deletion $s$-substitution correcting codes that leverage approaches from $^{\leq}t$-burst-deletion correction. Building upon established approaches in the $^{\leq}t$-burst-deletion correction domain, we construct $t$-deletion $s$-substitution correcting codes for $t\in \{1,2\}$ over binary alphabets and for $t=1$ in non-binary alphabets, with some constructions matching existing results and others outperforming current methods. Our framework offers new insights into the underlying principles of prior works, elucidates the limitations of current approaches, and provides a unified perspective on error correction strategies.