On binary alphabetical codes

Binary alphabetical codes, which are prefix free, fixed-to-variable binary codes for discrete memoryless sources, in which the lexicographic order of the codewords agrees with the alphabet order of the respective source letters, are studied. A necessary and sufficient condition on the sequence of codeword lengths of any such code is proved. A new upper bounds on the redundancy of alphabetical codes relative to the optimal prefix free, fixed-to-variable codes - the Huffman codes - is proved. An adaptation of the Ziv-Lempel algorithm making it lexicographic order preserving, without any additional redundancy, is presented.