Gallager and Van Voorhis have found optimal prefix-free codes ΞΊ(K) for
a random variable K that is geometrically distributed: Pr[K=k]=p(1βp)k
for kβ₯0. We determine the asymptotic behavior of the expected length Ex[#ΞΊ(K)] of these codes as pβ0: Ex[#ΞΊ(K)]=log2βp1β+log2βlog2+2+f(log2βp1β+log2βlog2)+O(p), where f(z)=4β 2β21β{z}β{z}β1, and
{z}=zββzβ 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).Comment: i+5 p