Random Reed--Solomon Codes Correcting Permutations, Insertions, and Deletions over Polynomial-Size Alphabets

2026-06-21

AI Summary

This work addresses the challenge that Reed–Solomon codes, under coordinate permutations and subjected to insertion-deletion (insdel) errors, require an exponentially large alphabet to approach the half-Singleton bound. The authors investigate the insdel error-correcting capability of random Reed–Solomon codes in this setting and demonstrate, for the first time, that the alphabet size can be reduced to polynomial in the block length \(n\)—specifically \(n^{O(1)}\)—at the cost of only an \(\varepsilon n\) loss in performance. With high probability, such codes robustly correct up to \((1 - \varepsilon)n - 2k + 1\) insdel errors. Furthermore, for two-dimensional explicit constructions, they design a decoder operating over an alphabet of size \(O(n^3)\) with average-case linear decoding time \(O(n)\). Key contributions include an existence proof under polynomial alphabet size, a lower bound analysis on alphabet requirements, and an efficient decoding algorithm.

Abstract

We study Reed--Solomon codes against adversarial coordinate permutations followed by insertion-deletion (insdel) errors. It was previously shown by Con (2025) that Reed--Solomon codes can attain the exact half-Singleton bound in this setting, but only over exponentially large alphabets. We prove that, by allowing an additive $εn$ gap from this bound, the alphabet size can be reduced to polynomial. More precisely, for fixed constants $R,ε\in(0,1)$ satisfying $2R+ε<1$ and $k=Rn$, a random Reed--Solomon code of length $n$ and dimension $k$ over an alphabet of size $n^{O_{R,ε}(1)}$ is, with high probability, robust against arbitrary coordinate permutations followed by up to $(1-ε)n-2k+1$ insdel errors. We also prove a complementary alphabet-size lower bound, showing that positive-rate codes, which are robust against linearly many insdel errors in the permutation-insdel setting, require a polynomially superlinear alphabet. Finally, for the explicit two-dimensional Reed--Solomon codes constructed by Con et al. (2024) over alphabet size $O(n^3)$, we give an average $O(n)$-time decoder against arbitrary coordinate permutations followed by $n-3$ insdel errors. Previously, an $O(n)$-time decoder for this code was known only for the deletion setting.

Problem

• Reed-Solomon codes
• insertion-deletion errors
• coordinate permutations
• alphabet size
• error correction

Innovation

• Reed–Solomon codes
• insertion-deletion errors
• coordinate permutations
• polynomial-size alphabet
• efficient decoding