For infinitely many rates $R \in (0,1)$, we show that full-length Reed-Solomon codes over fields of low characteristic can contain super-polynomially many codewords in a Hamming ball of radius just exceeding the Johnson radius. Over prime fields, we exhibit full-length Reed-Solomon codes of constant rate $R$ with polynomially many codewords in a Hamming ball of relative radius bounded strictly below $1-R$. This refutes a recent conjecture that, over prime fields, list sizes below capacity are always bounded by a constant. Together, these results demonstrate that certain restrictions in recent works on explicit Reed-Solomon codes achieving list-decoding capacity, related to the choice of field or evaluation points or to the list-size guarantee, are inherent.
Rohan Goyal, Venkatesan Guruswami, Yihang Sun, Mary Wootters
Local Coordinate-wise Linear (LCL) properties, introduced by Levi, Mosheiff, and Shagrithaya (2025), capture coding-theoretic properties such as list decoding and list recovery. They showed that every LCL property has the same rate threshold for random linear codes (RLC) and random Reed--Solomon codes (RRS). Brakensiek, Chen, Dhar, and Zhang (2025) extended this to random additive codes. Thus any property of all subspace design codes holds for RLCs and RRS at the same rate. In this work, we improve the probabilistic guarantees in the RRS local equivalence theorem and extend the equivalence to all known subspace design constructions individually. Each such construction, including folded Reed--Solomon codes, has the same LCL threshold as RLC and RRS codes. We apply these results to alphabet-size bounds for curve decoding. Curve decoding, introduced by Goyal and Guruswami (2025) and Carmon et al. (2026), is a row-span-constrained LCL property implying proximity gaps (PG), correlated agreement (CA), and mutual correlated agreement (MCA), relevant to IOPs and SNARKs. Informally, a rate-$R$ code $C \subset \mathbb{F}^n$ is $(\ell,δ,a,b)$-curve decodable if any degree-$\ell$ curve $u:\mathbb{F}\to\mathbb{F}^n$ that is $δ$-close to $C$ on at least $a$ points has at least $b\leq a$ nearby codewords lying on a degree-$\ell$ curve in $C$. Goyal and Guruswami (2025) and Goyal, Guruswami, Sun, and Wootters (2026) established curve decodability for RLC and RRS codes at radius $δ$ approaching $1-R$, but with large alphabets in $\ell$ and $η:=1-R-δ$. Using our improved LCL equivalence and rank-sensitive counting inspired by Goyal and Guruswami (2026), we exponentially improve alphabet requirements while matching the best known values of $(a,b)$. For $\ell=O(1)$, our alphabet-size bounds match the best known for list decoding, and improves results for PG, CA, and MCA.
For any desired rate $R \in (0,1)$, and gap to list-decoding capacity $\varepsilon > 0$, we give an explicit $n^{O_{\varepsilon}(1)}$ time construction of Reed-Solomon codes over extension fields of size polynomial in the block length $n$ that can be list-decoded from a fraction $1-R-\varepsilon$ of errors with the optimal list-size of $O(\frac{1-R}{\varepsilon})$. We also give a simpler, strongly explicit construction of evaluation points for which the associated Reed-Solomon code can be list-decoded from $1-R-\varepsilon$ errors with list size $(1/\varepsilon)^{O(1/\varepsilon)}$. We provide a deterministic list-decoding algorithm for both constructions running in time $O_ε(1)\mathsf{poly}(n)$. In fact, our Reed-Solomon codes are near-optimal subspace-design codes over a subfield. Via known results, this directly implies the combinatorial list-decoding guarantee, as well as near-optimal list-recovery bounds. We also present a new reduction showing that the subspace-design guarantee also implies line decoding over the extension field itself, which then yields proximity gaps and mutual correlated agreement up to proximity parameter approaching the optimal $1-R$ radius. Compared to the recent algorithm of Brakensiek, Chen, Putterman, Zhang, and Zheng (2026), which list decodes every Reed-Solomon code over prime fields up to capacity, our approach achieves constant (in fact, near-optimal) list size rather than a polynomially growing one, at the cost of carefully choosing the code's evaluation points. In a companion paper, we show that these respective restrictions are unavoidable.