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Codes on Graphs: Duality and MacWilliams Identities

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1 Author(s)
Forney, G.D., Jr. ; Lab. for Inf. & Decision Syst., Massachusetts Inst. of Technol., Cambridge, MA, USA

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.

Published in:

Information Theory, IEEE Transactions on  (Volume:57 ,  Issue: 3 )

Date of Publication:

March 2011

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