Optimality of Huffman Code in the Class of 1-bit Delay Decodable Codes

For a given independent and identically distributed (i.i.d.) source, Huffman code achieves the optimal average codeword length in the class of instantaneous code with a single code table. However, it is known that there exist time-variant encoders, which achieve a shorter average codeword length than the Huffman code, using multiple code tables and allowing at most k-bit decoding delay for k = 2, 3, 4, . . .. On the other hand, it is not known whether there exists a 1-bit delay decodable code, which achieves a shorter average length than the Huffman code. This paper proves that for a given i.i.d. source, a Huffman code achieves the optimal average codeword length in the class of 1-bit delay decodable codes with a finite number of code tables

General form of almost instantaneous fixed-to-variable-length codes

A general class of the almost instantaneous fixed-to-variable-length (AIFV) codes is proposed, which contains every possible binary code we can make when allowing finite bits of decoding delay. The contribution of the paper lies in the following. (i) Introducing $N$-bit-delay AIFV codes, constructed by multiple code trees with higher flexibility than the conventional AIFV codes. (ii) Proving that the proposed codes can represent any uniquely-encodable and uniquely-decodable variable-to-variable length codes. (iii) Showing how to express codes as multiple code trees with minimum decoding delay. (iv) Formulating the constraints of decodability as the comparison of intervals in the real number line. The theoretical results in this paper are expected to be useful for further study on AIFV codes.Comment: submitted to IEEE Transactions on Information Theory. arXiv admin note: text overlap with arXiv:1607.07247 by other author