Proximity Gaps Conjecture Fails Near Capacity over Prime Fields
arXiv:2604.0972443.4
AI Analysis
This result refutes a key conjecture in coding theory for a class of codes, showing that proximity gaps do not hold near capacity over prime fields.
The authors prove that for a specific family of Reed-Solomon codes, proximity gaps fail at radii O(1/log n) below capacity, contradicting the proximity gaps conjecture near capacity over prime fields.
In this report we flesh out a sketch by Krachun and Kazanin to prove that for a certain family of Reed-Solomon codes, proximity gaps fail at radii that are $O(1/\log n)$ below the capacity rate of the code, where $n$ is the length of the code.