Explicit and Efficient Constructions of linear Codes Against Adversarial Insertions and Deletions · arXivDesk
2201.06130Jan 16, 2022The content of this paper appeared in a previous version of arXiv:2107.05699. As that version was split, this paper contains the part about efficient linear codes against insertions and deletions
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 (1−4δ)/8−ε
Nearby in the stack
. 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, 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.