Reed Solomon Codes Against Adversarial Insertions and Deletions

Abstract

In this work, we study linear error-correcting codes against adversarial insertiondeletion (insdel) errors, a topic that has recently gained a lot of attention. We focus on two different settings -codes over small alphabets and Reed-Solomon codes.Linear codes over small fields: 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.Reed-Solomon codes: We prove that over fields of size n O(k) there are [n, k] Reed-Solomon codes that can decode from n − 2k + 1 insdel errors and hence attain the half-Singleton bound. We also give a deterministic construction of such codes over much larger fields (of size n k O(k) ). Nevertheless, for k = O(log n/ log log n) our construction runs in polynomial time. For the special case k = 2, which received a lot of attention in the literature, we construct an [n, 2] Reed-Solomon code over a field of size O(n 4 ) that can decode from n − 3 insdel errors. Earlier construction required an exponential field size. Lastly, we prove that any such construction requires a field of size Ω(n 3 ).

Explicit and Efficient Construction of (nearly) Optimal Rate Codes for Binary Deletion Channel and the Poisson Repeat Channel

“…In the adversarial model, Guruswami and Wang showed that there are codes with rate 1 − O √ δ that can correct up to δn errors, and a recent result by Con, Shpilka and Tamo [CST21] provides error correction for adversarial InsDel channels as well.…”

Explicit Good Subspace-metric Codes and Subset-metric Codes

“…For each positive integer k satisfying k < n 2 , explicit k dimension Sidon spaces were given in [67]. These Sidon spaces were also used in [18] for constructing explicit two dimensional Reed-Solomon codes attaining the half-Singleton bound.…”

“…We refer to [56,57,58,79,77,59,19,22,1,64,63,71,51,5,4,31,33,23,61] for the historic development of insertion-deletion error-correcting codes. For the recent breakthroughs and constructions we refer to [37,38,39,43,13,33,15,28,70,69,73,74,75,55,78,13,45,34,18] and a nice latest survey [42].…”

“…Their proof was based on the above half-Singleton bound in Section 5 of [15] or their Half-Plotkin bound. For better constructions of linear insertion-deletion codes and Reed-Solomon insertion-deletion codes we refer to the recent paper [18].…”

“…Most insertion-deletion codes in [32,37,38,42,18] have been only given algorithmically. They are not explicit codes, though sometimes these nice insertion-deletion codes can be constructed from highly efficient polynomial time algorithms.…”