## Length of Phase-in Codes

September 23, 2008

In a previous post, I presented the phase-in codes. I stated that they were very efficient given the distribution is uniform. I also stated that, given the distribution is uniform, they never do worse than $\approx 0.086$ bits worse than the theoretical lower bound of $\lg N$, where $N$ is the number of distinct symbols to code and $\lg N$ is a convenient shorthand for $\log_2$. In this post, I derive that result.