Thresholds for Local Coordinate-Wise Linear Properties of Random One-Point AG Codes

2026-10-08

AI Summary

This study investigates the threshold rate of random one-point algebraic geometry (AG) codes with respect to the locally coordinate-wise linear (LCL) property. Addressing deficiencies in existing proofs and the dependence of the threshold on the curve genus, this work integrates combinatorial probabilistic methods with algebraic geometric coding theory to extend the LCL framework to AG codes and correct an error in the upper bound proof for Reed-Solomon codes. The primary contribution is demonstrating that random one-point AG codes achieve the same LCL threshold rate as random linear codes, thereby completely eliminating the genus-dependent gap. Furthermore, this research derives novel theoretical guarantees for list decoding, average-weight list decoding, and proximity gaps for polynomial curves.

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 $R_{\mathcal P}$, a random one-point AG code avoids all such profiles below $R_{\mathcal P}-\varepsilon$ and contains one above $R_{\mathcal P}+\varepsilon$, 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.

Problem

• local coordinate-wise linear properties
• random one-point AG codes
• threshold rates
• list decoding
• Reed-Solomon codes

Innovation

• Local Coordinate-Wise Linear Properties
• Algebraic Geometry Codes
• List Decoding
• Threshold Rates
• Reed-Solomon Codes