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

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.