Universal codes with finite block lengths

Following Root and Varaiya´s compound channel coding theorem, we examine the performance limit of codes with finite block length under a compound channel. Although the theorem promises a uniform convergence of the probability of error for individual channels when the block length approaches infinity, the performance can differ considerably with finite block length. The compound channel theorem is applied to the design of universal codes for periodic erasure channels and MIMO channels. Design criteria are proposed in both cases.