Thresholds for Local Coordinate-Wise Linear Properties of Random One-Point AG Codes
Abstract
Local coordinate-wise linear (LCL) properties provide a unified framework for studying list decoding, list recovery and other local properties of linear codes. We prove that random one-point algebraic geometry (AG) codes have the same LCL threshold rates as random linear codes under explicit alphabet and sampling conditions. More precisely, for a family of local profiles with random linear code threshold R𝒫, a random one-point AG code avoids all such profiles below R𝒫−ε and contains one above R𝒫+ε, with explicit failure bounds on both sides. Our proof also removes an auxiliary lower bound on the number of available rational places that arises in a direct adaptation of the Reed--Solomon argument and requires no additional genus-dependent rate gap. We also observe that the previously claimed above threshold proof for random Reed--Solomon codes does not establish the required probability lower bound. The containment relation used in that argument gives the probability comparison in the opposite direction from what is needed. We give a different above threshold argument, which in particular also yields an alternative proof in the genus zero case. Combining our threshold theorem with recent results for random linear codes gives a collection of new list decoding, average weight list decoding and list recovery guarantees for random one-point AG codes. We further obtain new correlated agreement and proximity gap results for polynomial curves. These results extend the LCL threshold framework from random linear and Reed--Solomon codes to random one-point AG codes.
Related
- A Universal Threshold Theory for Local Properties of Random Linear Codes
- From Random Quantum Codes to Explicit qLDPC Codes via Local Properties
- Marginal Coordinate Test for Fréchet Regression with Random Objects
- Optimization of ReaxFF parameters for the Mo−S system using random optimization and coordinate search
- On randomized batch-sampling coordinate descent methods for solving linear least-squares problems
BibTeX
@misc{yuan2026thresholds,
title = {{Thresholds for Local Coordinate-Wise Linear Properties of Random One-Point AG Codes}},
author = {Chen Yuan and Ruiqi Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12019},
eprint = {2610.12019},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12019},
}
RIS
TY - UNPB
TI - Thresholds for Local Coordinate-Wise Linear Properties of Random One-Point AG Codes
AU - Chen Yuan
AU - Ruiqi Zhu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12019
UR - https://arxiv.org/abs/2610.12019
AB - Local coordinate-wise linear (LCL) properties provide a unified framework for studying list decoding, list recovery and other local properties of linear codes. We prove that random one-point algebraic geometry (AG) codes have the same LCL threshold rates as random linear codes under explicit alphabet and sampling conditions. More precisely, for a family of local profiles with random linear code threshold R_𝒫, a random one-point AG code avoids all such profiles below R_𝒫−ε and contains one above R_𝒫+ε, with explicit failure bounds on both sides. Our proof also removes an auxiliary lower bound on the number of available rational places that arises in a direct adaptation of the Reed--Solomon argument and requires no additional genus-dependent rate gap. We also observe that the previously claimed above threshold proof for random Reed--Solomon codes does not establish the required probability lower bound. The containment relation used in that argument gives the probability comparison in the opposite direction from what is needed. We give a different above threshold argument, which in particular also yields an alternative proof in the genus zero case. Combining our threshold theorem with recent results for random linear codes gives a collection of new list decoding, average weight list decoding and list recovery guarantees for random one-point AG codes. We further obtain new correlated agreement and proximity gap results for polynomial curves. These results extend the LCL threshold framework from random linear and Reed--Solomon codes to random one-point AG codes.
ER -