Divergence and the construction of variable-to-variable-length lossless codes by source-word extensions

Such codes are described using dual leaf-linked trees: one specifying the parsing of the source symbols into source words, and the other specifying the formation of code words from code symbols. Compression exceeds entropy by the amount of the informational divergence, between source words and code words, divided by the expected source-word length. The asymptotic optimality of Tunstall or Huffman codes derives from the bounding of divergence while the expected source-word length is made arbitrarily large. A heuristic extension scheme is asymptotically optimal but also acts to reduce the divergence by retaining those source words which are well matched to their corresponding code words