Threshold Rates for Properties of Random Codes

• Published in: IEEE transactions on information theory Vol. 68; no. 2; pp. 905 - 922
• Main Authors: Guruswami, Venkatesan, Mosheiff, Jonathan, Resch, Nicolas, Silas, Shashwat, Wootters, Mary
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.02.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Codes; Coding theory; Computer science; Decoding; Linear codes; list-decoding and recovery; Lower bounds; Pins; Probabilistic logic; random codes; Recovery; Technological innovation; threshold rates
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2021.3123497

Suppose that <inline-formula> <tex-math notation="LaTeX">\mathcal {P} </tex-math></inline-formula> is a property that may be satisfied by a random code <inline-formula> <tex-math notation="LaTeX">C \subset \Sigma ^{n} </tex-math></inline-formula>. For example, for some <inline-formula> <tex-math notation="LaTeX">p \in (0,1) </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">\mathcal {P} </tex-math></inline-formula> might be the property that there exist three elements of <inline-formula> <tex-math notation="LaTeX">C </tex-math></inline-formula> that lie in some Hamming ball of radius <inline-formula> <tex-math notation="LaTeX">pn </tex-math></inline-formula>. We say that <inline-formula> <tex-math notation="LaTeX">R^{\ast} </tex-math></inline-formula> is the threshold rate for <inline-formula> <tex-math notation="LaTeX">\mathcal {P} </tex-math></inline-formula> if a random code of rate <inline-formula> <tex-math notation="LaTeX">R^{\ast} + \varepsilon </tex-math></inline-formula> is very likely to satisfy <inline-formula> <tex-math notation="LaTeX">\mathcal {P} </tex-math></inline-formula>, while a random code of rate <inline-formula> <tex-math notation="LaTeX">R^{\ast} - \varepsilon </tex-math></inline-formula> is very unlikely to satisfy <inline-formula> <tex-math notation="LaTeX">\mathcal {P} </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">\mathcal {P} </tex-math></inline-formula> 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.