Substring Edit Correcting Codes and Optimal Single Burst-Deletion Correcting Codes

2026-10-05

AI Summary

This study addresses the challenge of excessive redundancy in error-correcting codes designed for substring edits and burst deletions. Building upon sequence coding theory, this work proposes a novel positioning algorithm to construct q-ary k-substring edit-correcting codes, while systematically optimizing redundancy analysis techniques for the binary setting. The primary contribution is the first achievement of optimal redundancy design—up to an additive constant—for burst-deletion and localized-deletion codes across the entire parameter space, yielding codes with an optimal redundancy of log n + O(1). This result overcomes the limitation of Levenshtein’s classical construction, which attains optimality only under specific parameter regimes, and significantly improves upon prior results established by Li et al.

Abstract

A $k$-substring edit in a sequence first deletes a substring of length at most $k$, and then inserts a sequence of length at most $k$ at the same position. A code that can correct a $k$-substring edit is called a $k$-substring edit code. In this paper, we develop a new localization method and use it to construct a $q$-ary $k$-substring edit correcting code with $\log n+8\log\log n+o(\log\log n)$ bits of redundancy for any fixed $q\ge2$ and $k\ge1$, where $n$ is the code length. For the binary alphabet, this improves upon the redundancy $\log n+16k\log\log n+o(\log\log n)$ obtained by Li \emph{et al}. When the deleted substring and the inserted sequence have different lengths, we further construct codes with redundancy $\log n+O_{q,k}(1)$, which is optimal up to an additive constant. As corollaries, for all fixed $q\ge2$ and $1\le t\le T$, we obtain $q$-ary $(\le t)$-burst-deletion correcting codes and $(t,T)$-localized deletion correcting codes with redundancies $\log n+O_{q,t}(1)$ and $\log n+O_{q,T}(1)$, respectively. To the best of our knowledge, these are the first constructions attaining optimal redundancy up to an additive constant for these two deletion models over the full range of fixed parameters. For $(\le t)$-burst-deletion correction with $t\ge2$, such redundancy had previously been achieved only for $q=t=2$ by Levenshtein in 1967.

Problem

• substring edit correcting codes
• burst-deletion correcting codes
• localized deletion correcting codes
• redundancy

Innovation

• Substring Edit Correcting Codes
• Localization Method
• Burst-Deletion Correcting Codes
• Optimal Redundancy
• Localized Deletion