  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 using shifted error-subsequences.
In this case, if the identical shifts occur in the 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.
