Explicit and Efficient Constructions of linear Codes Against Adversarial Insertions and Deletions
share
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 𝔽_q, for q=poly(1/ε), that can efficiently decode from a δ fraction of insdel errors and have rate (1-4δ)/8-ε. We also show that by allowing codes over 𝔽_q^2 that are linear over 𝔽_q, we can improve the rate to (1-δ)/4-ε while not sacrificing efficiency. Using this latter result, we construct fully linear codes over 𝔽_2 that can efficiently correct up to δ < 1/54 fraction of deletions and have rate R = (1-54·δ)/1216. Cheng, Guruswami, Haeupler, and Li [CGHL21] 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 [CGHL21]) over small fields. Thus, our results significantly improve their construction and get much closer to the bound.
READ FULL TEXT