This paper presents new encoding and decoding methods for Berlekamp-Preparata convolutional codes (BPCCs) based on tail-biting technique. The proposed scheme can correct a single block of n bit errors relative to a guard space of m error-free blocks while no fractional rate loss is incurred. The proposed tail-biting BPCCs (TBBPCCs) can attain optimal complete burst error correction bound. Therefore, they have the optimal phased-burst-error-correcting capability for convolutional codes. Compared with the previous scheme, the proposed scheme can also improve error correcting capability.

In this paper, we clarify the relationship between an initial (final) state in a tail-biting error-trellis and the obtained syndromes. We show that a final state is dependent on the first M syndromes as well, where M is the memory length of the parity-check matrix. Next, we calculate the probability of an initial (final) state conditioned by the syndromes. We also apply this method to concrete examples. It is shown that the initial (final) state in a tail-biting error-trellis is well estimated using these conditional probabilities.