Existence condition for tail-biting convolutional codes

Abstract

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 butter 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 neccessary 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.