On the MacWilliains identity for convolutional codes

The adjacency matrix associated with a convolutional code collects in a detailed manner information about the weight distribution of the code. A MacWilliams Identity Conjecture, stating that the adjacency matrix of a code fully determines the adjacency matrix of the dual code, will be formulated, and an explicit formula for the transformation will be stated. The formula involves the MacWilliams matrix known from complete weight enumerators of block codes. The conjecture will be proven for the class of convolutional codes where either the code itself or its dual does not have Forney indices bigger than one. For the general case, the conjecture is backed up by many examples, and a weaker version will be established.