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 the future, 5G radio access and support for the internet of things (IoT) is becoming more important, which is called machine type communications. Different from current mobile communication systems, machine type communications generates relatively small packets. In order to support such small packets with high reliability, channel coding techniques are inevitable. One of the most effective channel codes in such conditions is the tail-biting convolutional code, since it is used in LTE systems due to its good performance for small packet sizes. By employing a list Viterbi algorithm for the tail-biting convolutional code, the block error rate (BLER) performances is further improved. Therefore, this paper evaluates the BLER performances of several list Viterbi algorithms, i.e., circular parallel list Viterbi algorithm (CPLVA), per stage CPLVA (PSCPLVA), and successive state and sequence estimation (SSSE). In the evaluation, computational complexity is also taken into account. It is shown that the performance of the CPLVA is better in the wide range of computational complexity defined in this paper.

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.

In this letter, we discussed some properties of characteristic generators for a finite Abelian group code, proved that any two characteristic generators can not start (end) at the same position and have the same order of the starting (ending) components simultaneously, and that the number of all characteristic generators can be directly computed from the group code itself. These properties are exactly the generalization of the corresponding trellis properties of a linear code over a field.

We investigated the truncated convolutional code with the characteristics of a block code for block-based communication systems. Three truncation methods (direct truncation, tail-terminating, and tail-biting method) were introduced by other researchers. Each of the three methods has a weakness: the direct truncation method decreases the minimum distance, the tail-terminating method uses tail bits, and the tail-biting method can only be applied by using a complicated decoder. Although the tail-biting method gives a better BER performance than the other two methods, we cannot apply the tail-biting method in all situations. Occasionally, the tail-biting convolutional code does not exist. Wang et al. presented two necessary conditions for the existence of the tail-biting convolutional code of the rate-1/2 recursive systematic convolutional code. In this paper, we analyze the encoder of the convolutional code as a linear time invariant system, and present two theorems and six corollaries on the existence of the tail-biting convolutional code. These existence conditions are adaptable to all convolutional codes. In the communication system using the truncated convolutional code, these results are applicable to determining the truncation method.

The circular decoding algorithm for tail-biting convolutional codes is executed using a fixed number of computations and is suitable for DSP/ASIC implementations. This letter presents the performance and complexity trade-off in the circular decoding algorithm using an analytic bound on the error probability. An incremental performance improvement is shown as the complexity increases from O(L) to O(L+10K) where L is the length of the decoding trellis and K is the constraint length. The decoding complexity required to produce the maximum-likelihood performance is presented, which is applicable to many codes of practical interest.

Tail-biting trellises of linear and nonlinear block codes are addressed. We refine the information-theoretic approach of a previous work on conventional trellis representation, and show that the same ideas carry over to tail-biting trellises. We present lower bounds on the state and branch complexity profiles of these representations. These bounds are expressed in terms of mutual information between different portions of the code, and they introduce the notions of superstates and superbranches. For linear block codes, our bounds imply that the total number of superstates, and respectively superbranches, of a tail-biting trellis of the code cannot be smaller than the total number of states, and respectively branches, of the corresponding minimal conventional trellis, though the total number of states and branches of a tail-biting trellis is usually smaller than that of the conventional trellis. We also develop some improved lower bounds on the state complexity of a tail-biting trellis for two classes of codes: the first-order Reed-Muller codes and cyclic codes. We show that the superstates and superbranches determine the Viterbi decoding complexity of a tail-biting trellis. Thus, the computational complexity of the maximum-likelihood decoding of linear block codes on a tail-biting trellis, using the Viterbi algorithm, is not smaller than that of the conventional trellis of the code. However, tail-biting trellises are beneficial for suboptimal and iterative decoding techniques.