A Universal Threshold Theory for Local Properties of Random Linear Codes
Abstract
Thresholds for local properties tell us which guarantees to expect from random linear codes (RLCs) and help us carry those guarantees over to structured codes. A monotone property is b-local if a collection of at most b codewords can witness its satisfaction or failure, as in list decoding and list recovery. The threshold of π« is the code dimension at which the probability that an RLC in π½qn satisfies π« crosses 1/2 in a sharp transition. We show that the threshold of a b-local property π« is within O(b) dimensions of the expectation threshold. This is the dimension predicted by the best union bound over covers of π«: families of subspaces such that the span of every witness contains at least one member. This gives a general derandomization statement: Any decreasing local property satisfied by RLCs at a given rate holds at a similar rate for pseudorandom ensembles that fool the cover union bound. In addition to previously studied ensembles, these include codes whose parity-check matrix is sampled from a small-bias distribution. Because our threshold characterization applies to all local properties, this derandomization also reaches properties beyond the scope of earlier reductions, including maximum loads in linear hashing for arbitrary input sets, rank-metric list decoding, and list decoding from insertion-deletion errors and burst errors. Our theorem unifies and strengthens prior works on row-symmetric and coordinate-wise linear characterizations, which located the threshold to within O(q2blogq n) and O(b2+nb/log2 q) dimensions, respectively. We obtain O(b) error for every b-local property, independently of alphabet size. Under suitable pseudorandomness bounds, the rate loss vanishes for every b=o(n), compared to b=O(log n) for earlier frameworks.
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BibTeX
@misc{kumar2026universal,
title = {{A Universal Threshold Theory for Local Properties of Random Linear Codes}},
author = {Vinayak M. Kumar and Ray Li and Jonathan Mosheiff and Gilad Shavit},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07661},
eprint = {2611.07661},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07661},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Universal Threshold Theory for Local Properties of Random Linear Codes
AU - Vinayak M. Kumar
AU - Ray Li
AU - Jonathan Mosheiff
AU - Gilad Shavit
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07661
UR - https://arxiv.org/abs/2611.07661
AB - Thresholds for local properties tell us which guarantees to expect from random linear codes (RLCs) and help us carry those guarantees over to structured codes. A monotone property is b-local if a collection of at most b codewords can witness its satisfaction or failure, as in list decoding and list recovery. The threshold of π« is the code dimension at which the probability that an RLC in π½_q^n satisfies π« crosses 1/2 in a sharp transition. We show that the threshold of a b-local property π« is within O(b) dimensions of the expectation threshold. This is the dimension predicted by the best union bound over covers of π«: families of subspaces such that the span of every witness contains at least one member. This gives a general derandomization statement: Any decreasing local property satisfied by RLCs at a given rate holds at a similar rate for pseudorandom ensembles that fool the cover union bound. In addition to previously studied ensembles, these include codes whose parity-check matrix is sampled from a small-bias distribution. Because our threshold characterization applies to all local properties, this derandomization also reaches properties beyond the scope of earlier reductions, including maximum loads in linear hashing for arbitrary input sets, rank-metric list decoding, and list decoding from insertion-deletion errors and burst errors. Our theorem unifies and strengthens prior works on row-symmetric and coordinate-wise linear characterizations, which located the threshold to within O(q^2blog_q n) and O(b^2+nb/log_2 q) dimensions, respectively. We obtain O(b) error for every b-local property, independently of alphabet size. Under suitable pseudorandomness bounds, the rate loss vanishes for every b=o(n), compared to b=O(log n) for earlier frameworks.
ER -