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

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 Fq, for q= poly(1/ϵ), that can efficiently decode from a δ fraction of insdel errors and have rate (1-4\delta)/8-ϵ. We also show that by allowing codes over Fq2 that are linear over Fq, we can improve the rate to (1-δ)/4-ϵ while not sacrificing efficiency. Using this latter result, we construct fully linear codes over F2 that can efficiently correct up to δ < 1/54 fraction of deletions and have rate R = (1-54⊙δ)/1216. Cheng et al. (2021) constructed codes with (extremely small) rates bounded away from zero that can correct up to a δ < 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 Cheng et al. (2021)] over small fields. Thus, our results significantly improve their construction and get much closer to the bound.