Random Reed-Solomon Codes Achieve the Half-Singleton Bound for Insertions and Deletions over Linear-Sized Alphabets
Roni Con, Zeyu Guo, Ray Li, Zihan Zhang
TL;DR
The paper proves that random Reed-Solomon codes over alphabets of size near linear in the code length can, with high probability, correct insdel errors up to (1−ε)n−2k+1, approaching the half-Singleton bound in the insertions-deletions setting. The authors develop a novel probabilistic framework that blends random matrix techniques from list-decoding with structural properties of longest common subsequences, introducing V-matrices and chain-based decompositions to bound rank under partial assignments. They achieve this with two regimes: quadratic alphabets via probabilistic certificates and linear alphabets via chain decomposition and banked certificates, yielding the near-optimal tradeoff with q = n + poly(1/ε)k and ultimately q = n + Θ(k) for linear alphabets. This work substantially reduces alphabet-size requirements compared to prior existential results and opens pathways to explicit constructions and decoding algorithms for insdel-correcting RS codes in practical settings.
Abstract
In this paper, we prove that with high probability, random Reed-Solomon codes approach the half-Singleton bound - the optimal rate versus error tradeoff for linear insdel codes - with linear-sized alphabets. More precisely, we prove that, for any $ε>0$ and positive integers $n$ and $k$, with high probability, random Reed--Solomon codes of length $n$ and dimension $k$ can correct $(1-\varepsilon)n-2k+1$ adversarial insdel errors over alphabets of size $n+2^{\mathsf{poly}(1/\varepsilon)}k$. This significantly improves upon the alphabet size demonstrated in the work of Con, Shpilka, and Tamo (IEEE TIT, 2023), who showed the existence of Reed--Solomon codes with exponential alphabet size $\widetilde O\left(\binom{n}{2k-1}^2\right)$ precisely achieving the half-Singleton bound. Our methods are inspired by recent works on list-decoding Reed-Solomon codes. Brakensiek-Gopi-Makam (STOC 2023) showed that random Reed-Solomon codes are list-decodable up to capacity with exponential-sized alphabets, and Guo-Zhang (FOCS 2023) and Alrabiah-Guruswami-Li (STOC 2024) improved the alphabet-size to linear. We achieve a similar alphabet-size reduction by similarly establishing strong bounds on the probability that certain random rectangular matrices are full rank. To accomplish this in our insdel context, our proof combines the random matrix techniques from list-decoding with structural properties of Longest Common Subsequences.