On the MacWilliams Identity for Classical and Quantum Convolutional Codes
Lai, Ching-Yi1; Hsieh, Min-Hsiu1,2; Lu, Hsiao-Feng3
AbstractThe 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.
KeywordConvolutional codes Fourier transforms Hamming weight MacWilliams identity
Funding ProjectAustralian Research Council (ARC)[DP120103776] ; UTS Chancellors postdoctoral research fellowship ; UTS Early Career Researcher Grants Scheme ; ARC Future Fellowship[FT140100574] ; Taiwan Ministry of Science and Technology[MOST 103-2221-E-009-043-MY3]
WOS Research AreaEngineering ; Telecommunications
WOS SubjectEngineering, Electrical & Electronic ; Telecommunications
WOS IDWOS:000384312200002
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Document Type期刊论文
Corresponding AuthorLai, Ching-Yi
Affiliation1.Univ Technol Sydney, Ctr Quantum Computat & Intelligent Syst, Fac Engn & Informat Technol, Ultimo, NSW 2007, Australia
2.Chinese Acad Sci, UTS Acad Math & Syst Sci AMSS, Joint Res Lab Quantum Computat & Quantum Informat, AMSS, Beijing 100190, Peoples R China
3.Natl Chiao Tung Univ, Dept Elect & Comp Engn, Hsinchu 300, Taiwan
Recommended Citation
GB/T 7714
Lai, Ching-Yi,Hsieh, Min-Hsiu,Lu, Hsiao-Feng. On the MacWilliams Identity for Classical and Quantum Convolutional Codes[J]. IEEE TRANSACTIONS ON COMMUNICATIONS,2016,64(8):3148-3159.
APA Lai, Ching-Yi,Hsieh, Min-Hsiu,&Lu, Hsiao-Feng.(2016).On the MacWilliams Identity for Classical and Quantum Convolutional Codes.IEEE TRANSACTIONS ON COMMUNICATIONS,64(8),3148-3159.
MLA Lai, Ching-Yi,et al."On the MacWilliams Identity for Classical and Quantum Convolutional Codes".IEEE TRANSACTIONS ON COMMUNICATIONS 64.8(2016):3148-3159.
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