Explicit and Efficient Constructions of linear Codes Against Adversarial Insertions and Deletions

TL;DR

This paper presents new explicit linear code constructions over small fields that efficiently correct a significant fraction of insertion-deletion errors, improving previous bounds and approaching theoretical limits.

Contribution

The authors develop explicit linear codes over small fields with improved rates for correcting insdel errors, advancing towards the half-Singleton bound.

Findings

Codes over _q with rate _{1-4 ext{delta}}-_ ext{epsilon} can correct _ ext{delta} fraction of insdel errors efficiently.

Using codes over _{q^2} linear over _q, the rate improves to _{(1- ext{delta})/4-_ ext{epsilon}}.

Constructed fully linear codes over _2 that correct up to _ ext{delta}<1/54 with rate _{(1-54 ext{delta})/1216}.

Abstract

In this work, we study linear error-correcting codes against adversarial insertion-deletion (insdel) errors, a topic that has recently gained a lot of attention. We construct linear codes over $\mathbb{F}_q$, for $q=\text{poly}(1/\varepsilon)$, that can efficiently decode from a $\delta$ fraction of insdel errors and have rate $(1-4\delta)/8-\varepsilon$. We also show that by allowing codes over $\mathbb{F}_{q^2}$ that are linear over $\mathbb{F}_q$, we can improve the rate to $(1-\delta)/4-\varepsilon$ while not sacrificing efficiency. Using this latter result, we construct fully linear codes over $\mathbb{F}_2$ that can efficiently correct up to $\delta < 1/54$ fraction of deletions and have rate $R = (1-54\cdot \delta)/1216$. Cheng, Guruswami, Haeupler, and Li [CGHL21] constructed codes with (extremely small) rates bounded away from zero that can correct up to a $\delta < 1/400$ fraction of insdel errors. They also posed the problem of constructing linear codes that get close to the half-Singleton bound (proved in [CGHL21]) over small fields. Thus, our results significantly improve their construction and get much closer to the bound.