TY - GEN

T1 - Sharp threshold rates for random codes

AU - Guruswami, Venkatesan

AU - Mosheiff, Jonathan

AU - Resch, Nicolas

AU - Silas, Shashwat

AU - Wootters, Mary

N1 - Funding Information:
Funding Venkatesan Guruswami: Supported by NSF grants CCF-1563742 and CCF-1814603 and a Simons Investigator Award. Jonathan Mosheiff : Supported by NSF grants CCF-1563742 and CCF-1814603 and a Simons Investigator Award. Nicolas Resch: Supported by NSF grants CCF-1563742 and CCF-1814603, ERC H2020 grant No.74079 (ALGSTRONGCRYPTO), and a Simons Investigator Award. Shashwat Silas: Supported by NSF-CAREER grant CCF-1844628, NSF-BSF grant CCF-1814629, a Sloan Research Fellowship, and a Google Graduate Fellowship. Mary Wootters: Supported by NSF-CAREER grant CCF-1844628, NSF-BSF grant CCF-1814629, and a Sloan Research Fellowship.
Publisher Copyright:
© Venkatesan Guruswami, Jonathan Mosheiff, Nicolas Resch, Shashwat Silas, and Mary Wootters.

PY - 2021/2/4

Y1 - 2021/2/4

N2 - 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.

AB - 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.

KW - Coding theory

KW - Random codes

KW - Sharp thresholds

UR - http://www.scopus.com/inward/record.url?scp=85108536447&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ITCS.2021.5

DO - 10.4230/LIPIcs.ITCS.2021.5

M3 - Conference contribution

AN - SCOPUS:85108536447

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 5:1--5:20

BT - 12th Innovations in Theoretical Computer Science Conference, ITCS 2021

A2 - Lee, James R.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 12th Innovations in Theoretical Computer Science Conference, ITCS 2021

Y2 - 6 January 2021 through 8 January 2021

ER -