Simultaneous code/error-trellis reduction for convolutional codes using shifted code/error-subsequences

In this paper, we show that the code-trellis and the error-trellis for a convolutional code can be reduced simultaneously, if reduction is possible. Assume that the error-trellis can be reduced by shifting particular error-subsequences. In this case, if the identical shifts occur in the corresponding subsequences of each code path, then the code-trellis can also be reduced. First, we obtain pairs of transformations which generate the identical shifts both in the subsequences of the code-path and in those of the error-path. Next, by applying these transformations to the generator matrix and the parity-check matrix, we show that reduction of these matrices is accomplished simultaneously, if it is possible. Moreover, it is shown that the two associated trellises are also reduced simultaneously.