Code Decomposition in the Analysis of a Convolutional Code

A convolutional code can be decomposed into smaller codes if it admits decoupled encoders. In this paper, we show that if a code can be decomposed into smaller codes (subcodes) its column distances are the minimum of the column distances of its subcodes. Moreover, the j-th column distance of a convolutional code C is equal to the j-th column distance of the convolutional codes generated by the truncation of the canonical encoders of C to matrices which entries have degree smaller or equal than j. We show that if one of such codes can be decomposed into smaller codes, so can be all the other codes.