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Improved Constructions of Linear Codes for Insertions and Deletions

Roee Gross, Roni Con, Eitan Yaakobi

TL;DR

This work advances the study of linear codes for adversarial indel errors by constructing explicit codes that approach the half-Singleton bound. It introduces half-linear codes, built by linearizing CST22, to achieve rates near $\frac{1}{2}$ while efficiently correcting a $\delta$-fraction of indels, with $q=\Theta(\varepsilon^{-4})$. It then shows how to lift these to fully linear codes over $\mathbb{F}_q$ via Flat/Pad transformations, obtaining rates $\frac{1}{2}-2\sqrt{\delta}-\varepsilon$ with efficient decoding. Additionally, the paper generalizes the half-Singleton bound to base-field linear codes and discusses extensions to subfield-linear and additive settings, highlighting open problems and practical implications for synchronization-aware storage systems and related applications.

Abstract

In this work, we study linear error-correcting codes against adversarial insertion-deletion (indel) errors. While most constructions for the indel model are nonlinear, linear codes offer compact representations, efficient encoding, and decoding algorithms, making them highly desirable. A key challenge in this area is achieving rates close to the half-Singleton bound for efficient linear codes over finite fields. We improve upon previous results by constructing explicit codes over \(\mathbb{F}_{q^2}\), linear over \(\mathbb{F}_q\), with rate \(1/2 - δ- \varepsilon\) that can efficiently correct a \(δ\)-fraction of indel errors, where \(q = O(\varepsilon^{-4})\). Additionally, we construct fully linear codes over \(\mathbb{F}_q\) with rate \(1/2 - 2\sqrtδ - \varepsilon\) that can also efficiently correct \(δ\)-fraction of indels. These results significantly advance the study of linear codes for the indel model, bringing them closer to the theoretical half-Singleton bound. We also generalize the half-Singleton bound, for every code \(C \subseteq \mathbb{F}^n\) linear over \(\mathbb{E} \subset \mathbb{F}\) a subfield of $\mathbb{F}$, such that \(C\) has the ability to correct \(δ\)-fraction of indels, the rate is bounded by $(1-δ)/2$.

Improved Constructions of Linear Codes for Insertions and Deletions

TL;DR

This work advances the study of linear codes for adversarial indel errors by constructing explicit codes that approach the half-Singleton bound. It introduces half-linear codes, built by linearizing CST22, to achieve rates near while efficiently correcting a -fraction of indels, with . It then shows how to lift these to fully linear codes over via Flat/Pad transformations, obtaining rates with efficient decoding. Additionally, the paper generalizes the half-Singleton bound to base-field linear codes and discusses extensions to subfield-linear and additive settings, highlighting open problems and practical implications for synchronization-aware storage systems and related applications.

Abstract

In this work, we study linear error-correcting codes against adversarial insertion-deletion (indel) errors. While most constructions for the indel model are nonlinear, linear codes offer compact representations, efficient encoding, and decoding algorithms, making them highly desirable. A key challenge in this area is achieving rates close to the half-Singleton bound for efficient linear codes over finite fields. We improve upon previous results by constructing explicit codes over , linear over , with rate that can efficiently correct a -fraction of indel errors, where \(q = O(\varepsilon^{-4})\). Additionally, we construct fully linear codes over with rate that can also efficiently correct -fraction of indels. These results significantly advance the study of linear codes for the indel model, bringing them closer to the theoretical half-Singleton bound. We also generalize the half-Singleton bound, for every code linear over a subfield of , such that has the ability to correct -fraction of indels, the rate is bounded by .

Paper Structure

This paper contains 18 sections, 14 theorems, 16 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Previous works
  3. Our Results
  4. Structure of the Paper
  5. Preliminaries
  6. Linear codes
  7. Levenshtein distance and self-matching sequences
  8. The Matching Algorithm of haeupler-survey2021synchronization
  9. Half-Linear Codes
  10. The Linearization of CST22
  11. Decoding Algorithm
  12. Proving Theorem \ref{['thm:half-linear-code']}
  13. From Half-Linear Codes to Linear Codes
  14. Construction
  15. Decoding Algorithm
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mathbb{E}\subset\mathbb{F}$ be finite fields and let $\mathcal{C}\subseteq\mathbb{F}^{n}$ be an $\mathbb{E}$-linear code that can correct up to $\delta n$ indels for some fixed $\delta\in[0,1)$. Then,

Figures (1)

  • Figure 1: Code rate $R(\delta)$ vs. indel fraction $\delta$ for different cases.

Theorems & Definitions (33)

  • Theorem 1.1: Half–Singleton bound over a base field
  • Theorem 1.2: Informal, see \ref{['thm:half-lin-code-formal']}
  • Theorem 1.3: Informal, see \ref{['thm:full-linear-code-formal']}
  • Theorem 2.1: AG codes tsfasman1982modularskorobogatov1990decodingkotter1996faststichtenoth2009algebraic
  • Definition 2.1: CST22
  • Definition 2.2
  • Definition 2.3
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.2: Theorem 1.2, DBLP:conf/soda/ChengHLSW19
  • ...and 23 more