The minimum average code for finite memoryless monotone sources

In this paper, the average probability distribution P~^N~ of equiprobable finite monotone sources, where N is the number of symbols, is derived and some properties of this distribution are investigated. The associated Huffman code has the minimum redundancy in the average sense. An efficient suboptimal encoding technique is proposed. We show that log N -H(P~^N~) is a constant asymptotically, and consequently there is a negligible penalty (asymptotically) in using a fixed length code. The length of the shortest codeword should be modified such that lim_N→∞ l_N/l₁=2. This limit cannot be a constant for minimax and universal codes such as the ω and δ codes. These properties are also shown to hold when the number of symbols is unknown but bounded.