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

中文翻译：

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随机代码的尖锐阈值率

假设 $\mathcal{P}$ 是一个可以由随机代码 $C \subset \Sigma^n$ 满足的性质。例如，对于某些 $p \in (0,1)$，$\mathcal{P}$ 可能是存在 $C$ 的三个元素位于某个半径为 $pn$ 的汉明球中的属性。如果速率 $R^{*} + \varepsilon$ 的随机代码很可能满足 $\mathcal{P}$，我们说 $R^*$ 是 $\mathcal{P}$ 的阈值速率，而速率 $R^{*} - \varepsilon$ 的随机代码不太可能满足 $\mathcal{P}$。尽管在编码理论中对随机码进行了充分研究，但即使是上述相对简单属性的阈值率也没有得到很好的理解。我们表征了一类丰富的属性的阈值率。这些属性，就像上面的例子一样，是通过包含特定的代码字集来定义的，这些代码字也适合“