  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.
