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 ).

This work gives an explicit construction of a family of error correcting codes for the binary deletion channel and for the Poisson repeat channel. In the binary deletion channel with parameter p (BDC p ) every bit is deleted independently with probability p. A lower bound of (1 − p)/9 is known on the capacity of the BDC p [MD06], yet no explicit construction is known to achieve this rate. We give an explicit family of codes of rate (1 − p)/16, for every p. This improves upon the work of Guruswami and Li [GL18] that gave a construction of rate (1 − p)/120. The codes in our family have polynomial time encoding and decoding algorithms.Another channel considered in this work is the Poisson repeat channel with parameter λ (PRC λ ) in which every bit is replaced with a discrete Poisson number of copies of that bit, where the number of copies has mean λ. We show that our construction works for this channel as well. As far as we know, this is the first explicit construction of an error correcting code for PRC λ .

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The problem of repairing linear codes and, in particular, Reed Solomon (RS) codes has attracted a lot of attention in recent years due to their extreme importance to distributed storage systems. In this problem, a failed code symbol (node) needs to be repaired by downloading as little information as possible from a subset of the remaining nodes. By now, there are examples of RS codes that have efficient repair schemes, and some even attain the cut-set bound. However, these schemes fall short in several aspects; they require a considerable field extension degree. They do not provide any nontrivial repair scheme over prime fields. Lastly, they are all linear repairs, i.e., the computed functions are linear over the base field. Motivated by these and by a question raised in [GW17] on the power of nonlinear repair schemes, we study the problem of nonlinear repair schemes of RS codes.Our main results are the first nonlinear repair scheme of RS codes with asymptotically optimal repair bandwidth (asymptotically matching the cut-set bound). This is the first example of a nonlinear repair scheme of any code and also the first example that a nonlinear repair scheme can outperform all linear ones. Lastly, we show that the cut-set bound for RS codes is not tight over prime fields by proving a tighter bound, using additive combinatorics ideas.

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