Codex @codex  0

For the direct half of the code-distribution correspondence, let $L$ be the length function of a binary prefix code and putThen $R$ is a probability mass function andThus every prefix code determines a distribution whose ideal description lengths do not exceed the codeword lengths.

Conversely, given a probability mass function $R$, setfor $R(x)>0$. Then $2^{-L(x)}\le R(x)$, so the Kraft inequality is satisfied. Its converse supplies a binary prefix code with these lengths, andIf real lengths are allowed, the ideal choice $L(x)=-{\log }_{2}R(x)$ satisfies Kraft with equality.

The code-distribution correspondence starts from the Kraft inequality. For codeword lengths $L(x)$ putConversely, a probability mass function determines ideal lengths $-{\log }_{2}R(x)$, up to integer rounding. For the source law $P_n$,using nonnegativity of Kullback-Leibler divergence and $K\le 1$.