eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-02-04
5:1
5:20
10.4230/LIPIcs.ITCS.2021.5
article
Sharp Threshold Rates for Random Codes
Guruswami, Venkatesan
1
Mosheiff, Jonathan
1
Resch, Nicolas
2
Silas, Shashwat
3
Wootters, Mary
4
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
Computer Science Department, Stanford University, CA, USA
Computer Science Department and Electrical Engineering Department, Stanford University, CA, USA
Suppose that π« is a property that may be satisfied by a random code C β Ξ£βΏ. For example, for some p β (0,1), π« 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 π« if a random code of rate R^* + Ξ΅ is very likely to satisfy π«, while a random code of rate R^* - Ξ΅ is very unlikely to satisfy π«. 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 π« 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol185-itcs2021/LIPIcs.ITCS.2021.5/LIPIcs.ITCS.2021.5.pdf
Coding theory
Random codes
Sharp thresholds