# On the Structure of Rate 1/n Convolutional Codes

## Abstract

We show what choice there is in assigning output digits to transitions of a binary rate l/n code trellis so that the latter will correspond to a convolutional code. We then prove that in any rate 1/2 non-catastrophic code of constraint length v each binary sequence of length 2j (1<j: < v - 1) is associated with exactly 2 v- j - 1 distinct paths j branches long. As a consequence of the above properties nondegenerate codes with branch complementarity are fully determined by the topological relationship of the trellis transitions associated with output pairs 00. Finally, we derive a new upper bound on free distance of rate 1/ n convolutional codes and use our results to determine the length of the largest input sequence that can conceivably result in an output whose weight is equal to the free distance of a code of rate 1/2. © 1972, IEEE. All rights reserved.