Optimal maximal encoding different from Huffman encoding

Abstract

Novel maximal encoding, encoding, and maximal prefix encoding different from Huffman encoding are introduced. It is proven that for finite source alphabets all Huffman codes are optimal maximal codes, codes, and maximal prefix codes. Conversely, the above three types optimal codes need not to be the Huffman codes. Completely similar to Huffman codes, we prove that for every random variable with a countably infinite set of outcomes and with finite entropy there exists an optimal maximal code (code, maximal prefix code) which can be constructed from optimal maximal codes (codes, maximal prefix codes) for truncated versions of the random variable, and furthermore, that the average code word lengths of any sequence of optimal maximal codes (codes, maximal prefix codes) for the truncated versions converge to that of the optimal maximal code (cone, maximal prefix code).