Linear and Reed Solomon Codes Against Adversarial Insertions and Deletions

In this work, we study linear error-correcting codes against adversarial insertion-deletion (insdel) errors. We focus on two different settings: Linear codes over small fields: We construct linear codes over $\mathbb{F}_q$, for $q=\text{poly}(1/\varepsilon)$, that can efficiently decode from a $\delta$ fraction of insdel errors and have rate $(1-4\delta)/8-\varepsilon$. We also show that by allowing codes over $\mathbb{F}_{q^2}$ that are linear over $\mathbb{F}_q$, we can improve the rate to $(1-\delta)/4-\varepsilon$ while not sacrificing efficiency. Using this latter result, we construct fully linear codes over $\mathbb{F}_2$ that can efficiently correct up to $\delta < 1/54$ fraction of deletions and have rate $R = (1 - 54 \delta)/1216$. Cheng, Guruswami, Haeupler, and Li [CGHL21] constructed codes with (extremely small) rates bounded away from zero that can correct up to a $\delta < 1/400$ 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 $\Omega(n^3)$.