Towards a MacWilliams identity for convolutional codes

The weight adjacency matrix associated with a convolutional code collects in a detailed manner information about the weight distribution of the code. We will formulate a MacWilliams Identity Conjecture stating that the adjacency matrix of a code fully determines the adjacency matrix of the dual code. An explicit formula for the transformation will be stated as well; it 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 or its dual is a unit memory code. For the general case the conjecture is backed up by many examples, and a weaker version will be established.