TY - GEN

T1 - A complete MacWilliams theorem for convolutional codes

AU - Lai, Ching-Yi

AU - Hsieh, Min Hsiu

AU - Lu, Francis

PY - 2014/12/1

Y1 - 2014/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84929308661&partnerID=8YFLogxK

U2 - 10.1109/ITW.2014.6970812

DO - 10.1109/ITW.2014.6970812

M3 - Conference contribution

AN - SCOPUS:84929308661

T3 - 2014 IEEE Information Theory Workshop, ITW 2014

SP - 157

EP - 161

BT - 2014 IEEE Information Theory Workshop, ITW 2014

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2014 IEEE Information Theory Workshop, ITW 2014

Y2 - 2 November 2014 through 5 November 2014

ER -