On the MacWilliams Identity for Classical and Quantum Convolutional Codes

The usual weight generating functions associating with a convolutional code (CC) are based on state space realizations or the weight adjacency matrices (WAMs). The MacWilliams identity for CCs on the WAMs was first established by Gluesing-Luerssen and Schneider in the case of minimal encoders, and generalized by Forney using the normal factor graph theorem. We define the dual of a convolutional code in the viewpoint of constraint codes and obtain a simple and direct proof of the MacWilliams identity for CCs. By considering the weight enumeration functions over infinite stages, i.e. all codewords of a CC, we establish additional relations between a CC and its dual. Relations between various notions of weight generating function are also clarified, and the reason that no MacWilliams identity exists for free-distance enumerators is clear now. Hence the MacWilliams theorem for CCs can be considered complete. For our purpose, we choose a different representation for the exact weight generating function (EWGF) of a block code, by defining it as a linear combination of orthonormal vectors in Dirac bra-ket notation, rather than the standard polynomial representation. Within this framework, the MacWilliams identity for the EWGFs can be derived with simply a Fourier transform. This representation provides great flexibility so that various notions of weight generating functions and their MacWilliams identities can be easily obtained from the MacWilliams identity for the EWGFs. As a result, we also obtain the MacWilliams identity for the input-output weight adjacency matrices (IOWAMs) of a systematic convolutional code and its dual, which cannot be obtained from previous approaches. Finally, paralleling the development of the classical case, we establish the MacWilliams identity for quantum convolutional codes.