Fourier-reflexive partitions and MacWilliams identities for additive codes

Heide Gluesing-Luerssen

doi:10.1007/s10623-014-9940-x

Fourier-reflexive partitions and MacWilliams identities for additive codes

Heide Gluesing-Luerssen

Mathematics

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

A partition of a finite abelian group gives rise to a dual partition on the character group via the Fourier transform. Properties of this dualization are investigated, and a convenient test is given for when the bidual partition coincides with the primal partition. Such partitions permit MacWilliams identities for the partition enumerators of additive codes. It is shown that dualization commutes with taking products and symmetrized products of partitions on cartesian powers of the given group. After translating the results to Frobenius rings, which are identified with their character module, the approach is applied to partitions that arise from poset structures.

Original language	English
Pages (from-to)	543-563
Number of pages	21
Journal	Designs, Codes, and Cryptography
Volume	75
Issue number	3
DOIs	https://doi.org/10.1007/s10623-014-9940-x
State	Published - Jun 1 2015

Bibliographical note

Publisher Copyright:
© 2014, Springer Science+Business Media New York.

Funding

The author was partially supported by the National Science Foundation grants #DMS-0908379 and #DMS-1210061. I would like to thank Marcus Greferath and Navin Kashyap for very inspiring suggestions concerning this research project. A major part of the final write-up took place during a research stay at the University of Z\u00FCrich, and I am grateful to Joachim Rosenthal and his research group for the generous hospitality. I also would like to thank the anonymous reviewers for very helpful suggestions, in particular with respect to the proof of Theorem 3.3(b).

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	1210061, 0908379

Keywords

Dualization
Partitions
Poset structures
Weight enumerators

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science Applications
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.1007/s10623-014-9940-x

Cite this

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Fourier-reflexive partitions and MacWilliams identities for additive codes

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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