Reed Solomon Codes Against Adversarial Insertions and Deletions

TL;DR

This paper investigates Reed-Solomon codes' ability to correct adversarial insertion and deletion errors, providing new bounds, constructions, and complexity results for such codes over various field sizes.

Contribution

It introduces Reed-Solomon codes capable of correcting many insdel errors, with explicit constructions over large fields and complexity analysis for special cases.

Findings

Codes can decode from nearly twice the number of insdel errors as the code length.

Explicit constructions over large fields are provided for general and specific cases.

Any such code requires a field of size at least proportional to n^3.

Abstract

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 $\Omega(n^3)$.