Threshold rates for properties of random codes

TL;DR

This paper characterizes the threshold rates for a broad class of symmetric properties in random codes, establishing that these thresholds match simple first-moment bounds and providing bounds and algorithms for list-recovery thresholds.

Contribution

It introduces a framework to determine threshold rates for symmetric properties of random codes, proving they equal first-moment bounds and extending results to list-recovery scenarios.

Findings

Threshold rates match first-moment bounds for the studied properties.

Sharp bounds on list-recovery threshold rates are established.

An efficient algorithm for estimating list-recovery thresholds is provided.

Abstract

Suppose that $P$ is a property that may be satisfied by a random code $C \subset \Sigma^n$. For example, for some $p \in (0,1)$, ${P}$ 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 $R^* + \epsilon$ is very likely to satisfy ${P}$, while a random code of rate $R^* - \epsilon$ 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.