A complete MacWilliams theorem for convolutional codes

Ching-Yi Lai, Min-Hsiu Hsieh, Hsiao-feng Lu
2014 2014 IEEE Information Theory Workshop (ITW 2014)  
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
more » ... 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.
doi:10.1109/itw.2014.6970812 dblp:conf/itw/LaiHL14 fatcat:bgvxkldwafda5nkdhevpgal6fq