Reed Solomon Codes Against Adversarial Insertions and Deletions

In this work, we study the performance of Reed-Solomon codes against adversarial insertion-deletion (insdel) errors. 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 constructions required an exponential field size. Lastly, we prove that any such construction requires a field of size $\Omega (n^{3})$.