On the MacWilliams Identity for Classical and Quantum Convolutional Codes

Lai, CY; Hsieh, MH; Lu, HF

On the MacWilliams Identity for Classical and Quantum Convolutional Codes

Lai, CY Hsieh, MH

Lu, HF

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Publication Type:: Journal Article
Citation:: IEEE Transactions on Communications, 2016, 64 (8), pp. 3148 - 3159
Issue Date:: 2016-08-01

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Field	Value	Language
dc.contributor.author	Lai, CY	en_US
dc.contributor.author	Hsieh, MH https://orcid.org/0000-0002-3396-8427	en_US
dc.contributor.author	Lu, HF	en_US
dc.date.issued	2016-08-01	en_US
dc.identifier.citation	IEEE Transactions on Communications, 2016, 64 (8), pp. 3148 - 3159	en_US
dc.identifier.issn	0090-6778	en_US
dc.identifier.uri	http://hdl.handle.net/10453/59431
dc.description.abstract	© 2016 IEEE. The weight generating functions associated with convolutional codes (CCs) 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. We consider this problem in the viewpoint of constraint codes and obtain a simple and direct proof of this MacWilliams identity in the case of minimal encoders. 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. This representation provides great flexibility so that general split weight generating functions and their MacWilliams identities can be easily obtained from the MacWilliams identity for EWGFs. As a result, we also obtain the MacWilliams identity for the input-parity WAMs of a systematic convolutional code and its dual. Finally, paralleling the development of the classical case, we establish the MacWilliams identity for quantum CCs.	en_US
dc.relation	http://purl.org/au-research/grants/arc/FT140100574
dc.relation	http://purl.org/au-research/grants/arc/DP120103776
dc.relation.ispartof	IEEE Transactions on Communications	en_US
dc.relation.isbasedon	10.1109/TCOMM.2016.2585641	en_US
dc.title	On the MacWilliams Identity for Classical and Quantum Convolutional Codes	en_US
dc.type	Journal Article
utslib.citation.volume	8	en_US
utslib.citation.volume	64	en_US
utslib.for	0906 Electrical and Electronic Engineering	en_US
utslib.for	0804 Data Format	en_US
utslib.for	1005 Communications Technologies	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	/University of Technology Sydney/Strength - QSI - Centre for Quantum Software and Information
utslib.copyright.status	recently_added
pubs.issue	8	en_US
pubs.publication-status	Published	en_US
pubs.volume	64	en_US

Abstract:

© 2016 IEEE. The weight generating functions associated with convolutional codes (CCs) 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. We consider this problem in the viewpoint of constraint codes and obtain a simple and direct proof of this MacWilliams identity in the case of minimal encoders. 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. This representation provides great flexibility so that general split weight generating functions and their MacWilliams identities can be easily obtained from the MacWilliams identity for EWGFs. As a result, we also obtain the MacWilliams identity for the input-parity WAMs of a systematic convolutional code and its dual. Finally, paralleling the development of the classical case, we establish the MacWilliams identity for quantum CCs.

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http://hdl.handle.net/10453/59431