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.

翻译：假设 $P$ 是随机码 $C \subset \Sigma^n$ 可能满足的一个性质。例如，对于某个 $p \in (0,1)$，$P$ 可能表示存在 $C$ 中的三个元素位于某个汉明半径为 $pn$ 的球内。若速率为 $R^* + \epsilon$ 的随机码极有可能满足 $P$，而速率为 $R^* - \epsilon$ 的随机码极不可能满足 $P$，则称 $R^*$ 是性质 $P$ 的阈值率。尽管随机码在编码理论中已得到充分研究，但即使是上述这类相对简单性质的阈值率也尚未被充分理解。本文刻画了一类丰富性质的阈值率。这些性质（如上例所示）由包含特定的码字集合所定义，且这些集合具有适当的“对称性”。对于此类性质，我们证明阈值率实际上等于简单一阶矩计算所得到的下界。我们的技术不仅确定了上述性质 $P$ 的阈值率，还为列表恢复在多个参数区域中的阈值率提供了尖锐的界限，并给出了一种有效算法以估计列表恢复在一般情况下的阈值率。