 Prefix code for a rare geometric success

For a success index $J$ with geometric distribution of parameter $q=2^{-n}$, write $J-1=2^{n}Q+R$, $0\le R<2^n$. Encode $Q$ by $Q$ ones followed by zero, then encode $R$ in exactly $n$ bits. This prefix code has length $n+Q+1$. Since $\mathbb{P}(Q\ge k)=(1-q{)}^{k{2}^n}$, its expected length is the displayed expression, at most $n+e/(e-1)$. Thus a very rare success can be communicated in approximately the logarithm of its inverse probability, rather than one bit per trial.

// Target: quantum-theory.bigb