Weakly Symmetric Fix-Free Codes

Bidirectional decoding improves the error resilience of the fix-free codes, which is important in some applications like video coding. However, in general, two separate decoders must be designed for forward and backward decoding and hence, the cost of decoding is almost doubled. For the symmetric fix-free codes, although the forward and backward decoders are the same, the redundancy could be much greater than the case of asymmetric fix-free codes. In this paper, a class of fix-free codes, called weakly symmetric, is studied. A code is weakly symmetric if the reverse of each codeword itself is a codeword. This imposes a much weaker constraint on the codewords than what is necessitated by the symmetric codes, i.e., each codeword must be equal to its reverse. We show that the forward and backward decoders of a weakly symmetric fix-free code are essentially the same. In addition, it is shown that two significant sufficient conditions for the existence of asymmetric fix-free codes are also applicable to the case of weakly symmetric fix-free codes. As a result, we conclude that for any memoryless source, the redundancy of the optimal weakly symmetric fix-free code is at most 1.0817 bits. Moreover, it can be only 0.8 bit greater than the redundancy of the Huffman code in the worst case. In summary, weakly symmetric fix-free codes can be regarded as an option by which one can to some extent achieve the low redundancy of asymmetric fix-free codes and the low-cost bidirectional decoding of symmetric fix-free codes, simultaneously.