  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.
