A MacWilliams Identity for Convolutional Codes: The General Case

A MacWilliams Identity for convolutional codes will be established. It makes use of the weight adjacency matrices of the code and its dual, based on state space realizations (the controller canonical form) of the codes in question. The MacWilliams Identity applies to various notions of duality appearing in the literature on convolutional coding theory.Comment: 20 page

MacWilliams Identities for Terminated Convolutional Codes

Shearer and McEliece [1977] showed that there is no MacWilliams identity for the free distance spectra of orthogonal linear convolutional codes. We show that on the other hand there does exist a MacWilliams identity between the generating functions of the weight distributions per unit time of a linear convolutional code C and its orthogonal code C^\perp, and that this distribution is as useful as the free distance spectrum for estimating code performance. These observations are similar to those made recently by Bocharova, Hug, Johannesson and Kudryashov; however, we focus on terminating by tail-biting rather than by truncation.Comment: 5 pages; accepted for 2010 IEEE International Symposium on Information Theory, Austin, TX, June 13-1

Codes on Graphs: Duality and MacWilliams Identities

A conceptual framework involving partition functions of normal factor graphs is introduced, paralleling a similar recent development by Al-Bashabsheh and Mao. The partition functions of dual normal factor graphs are shown to be a Fourier transform pair, whether or not the graphs have cycles. The original normal graph duality theorem follows as a corollary. Within this framework, MacWilliams identities are found for various local and global weight generating functions of general group or linear codes on graphs; this generalizes and provides a concise proof of the MacWilliams identity for linear time-invariant convolutional codes that was recently found by Gluesing-Luerssen and Schneider. Further MacWilliams identities are developed for terminated convolutional codes, particularly for tail-biting codes, similar to those studied recently by Bocharova, Hug, Johannesson, and Kudryashov

A complete MacWilliams theorem for convolutional codes

© 2014 IEEE. In this paper, we prove a MacWilliams identity for the weight adjacency matrices based on the constraint codes of a convolutional code (CC) and its dual. Our result improves upon a recent result by Gluesing-Luerssen and Schneider, where the requirement of a minimal encoder is assumed. We can also establish the MacWilliams identity for the input-parity weight adjacency matrices of a systematic CC and its dual. Most importantly, we show that a type of Hamming weight enumeration functions of all codewords of a CC can be derived from the weight adjacency matrix, which thus provides a connection between these two very different notions of weight enumeration functions in the convolutional code literature. Finally, the relations between various enumeration functions of a CC and its dual are summarized in a diagram. This explains why no MacWilliams identity exists for the free-distance enumerators

The MacWilliams identity for quantum convolutional codes

In this paper, we propose a definition of the dual code of a quantum convolutional code, with or without entanglement assistance. We then derive a MacWilliams identity for quantum convolutional codes. Along the way, we obtain a direct proof of the MacWilliams identity, first found by Gluesing-Luerssen and Schneider, in the setting of classical convolutional codes. © 2014 IEEE

On the weight distribution of convolutional codes

Detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a controller canonical form of the code. We will show that this matrix is an invariant of the code. Moreover, it will be proven that codes with the same adjacency matrix have the same dimension and the same Forney indices and finally that for one-dimensional binary convolutional codes the adjacency matrix determines the code uniquely up to monomial equivalence