文章基本信息

标题：Sharp Threshold Rates for Random Codes
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作者：Venkatesan Guruswami ; Jonathan Mosheiff ; Nicolas Resch 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2021
卷号：185
页码：5:1-5:20
DOI：10.4230/LIPIcs.ITCS.2021.5
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Suppose that ð'« is a property that may be satisfied by a random code C âS, Î£â¿. For example, for some p â^^ (0,1), ð'« 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 ð'« if a random code of rate R^* Îµ is very likely to satisfy ð'«, while a random code of rate R^* - Îµ is very unlikely to satisfy ð'«. 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 ð'« 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.
关键词：Coding theory; Random codes; Sharp thresholds