  Gallager and Van Voorhis have found optimal prefix-free codes $\kappa(K)$ for
a random variable $K$ that is geometrically distributed: $\Pr[K=k] = p(1-p)^k$
for $k\ge 0$. We determine the asymptotic behavior of the expected length ${\rm
Ex}[{\#\kappa(K)}]$ of these codes as $p\to 0$: $${\rm Ex}[{\#\kappa(K)}] =
\log_2 {1\over p} + \log_2 \log 2 + 2 + f\left(\log_2 {1\over p} + \log_2 \log
2\right) + O(p),$$ where $$f(z) = 4\cdot 2^{-2^{1-\{z\}}} - \{z\} - 1,$$ and
$\{z\} = z - \lfloor z\rfloor$ is the fractional part of $z$. The function
$f(z)$ is a periodic function (with period $1$) that exhibits small
oscillations (with magnitude less than $0.005$) about an even smaller average
value (less than $0.0005$).
