Existence, Uniqueness, and Optimality of Sibling-Property Codes for Infinite Sources

By definition Huffman codes only exist for finite sources, since the Huffman algorithm cannot be applied to an infinite source. On the other hand, Gallager´s sibling property, which was introduced as a characterization of Huffman codes, extends naturally to (countably) infinite sources. Thus we define a Huffman-Gallager code as any code that has the sibling property, and we present some basic facts about such codes. (1) For any source, a Huffman-Gallager code exists and its list of node probabilities is unique. (2) A Huffman-Gallager code is optimal, and given an optimal code, there exists a Huffman-Gallager code with the same codeword lengths. (3) For sources with infinite entropy, the existence and uniqueness results continue to hold, and the optimality results hold for a natural extended form of optimality