Combinatorial limitations of list-decoding Reed-Solomon codes
Abstract
For infinitely many rates R ∈ (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.
Related
- Algorithmic List Decoding of Reed-Solomon Codes up to Capacity
- Reed-Solomon Codes at Capacity: Algorithmic List-Decoding and Proximity Gaps
- Algorithmic List Decoding at Capacity and Optimal Proximity Gaps for Reed-Solomon Codes
- Explicit Reed-Solomon Subspace Designs: Optimal List and Line Decoding
- List Size of Random Linear Codes: Optimality and the Limits of Average-Radius List Decoding
BibTeX
@misc{alrabiah2026combinatorial,
title = {{Combinatorial limitations of list-decoding Reed-Solomon codes}},
author = {Omar Alrabiah and Rohan Goyal and Venkatesan Guruswami},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07607},
eprint = {2611.07607},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07607},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Combinatorial limitations of list-decoding Reed-Solomon codes
AU - Omar Alrabiah
AU - Rohan Goyal
AU - Venkatesan Guruswami
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07607
UR - https://arxiv.org/abs/2611.07607
AB - For infinitely many rates R ∈ (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.
ER -