Sharp threshold rates for random codes

Venkatesan Guruswami, Jonathan Mosheiff, Nicolas Resch, Shashwat Silas, Mary Wootters

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations


Suppose that P is a property that may be satisfied by a random code C ⊂ Σn. For example, for some p ∈ (0, 1), P might be the property that there exist three elements of C that lie in some Hamming ball of radius pn. We say that R is the threshold rate for P if a random code of rate R + ε is very likely to satisfy P, while a random code of rate R - ε is very unlikely to satisfy P. While random codes are well-studied in coding theory, even the threshold rates for relatively simple properties like the one above are not well understood. We characterize threshold rates for a rich class of properties. These properties, like the example above, are defined by the inclusion of specific sets of codewords which are also suitably “symmetric.” For properties in this class, we show that the threshold rate is in fact equal to the lower bound that a simple first-moment calculation obtains. Our techniques not only pin down the threshold rate for the property P above, they give sharp bounds on the threshold rate for list-recovery in several parameter regimes, as well as an efficient algorithm for estimating the threshold rates for list-recovery in general.

Original languageEnglish
Title of host publication12th Innovations in Theoretical Computer Science Conference, ITCS 2021
EditorsJames R. Lee
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771771
StatePublished - 4 Feb 2021
Externally publishedYes
Event12th Innovations in Theoretical Computer Science Conference, ITCS 2021 - Virtual, Online
Duration: 6 Jan 20218 Jan 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference12th Innovations in Theoretical Computer Science Conference, ITCS 2021
CityVirtual, Online


  • Coding theory
  • Random codes
  • Sharp thresholds

ASJC Scopus subject areas

  • Software


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