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Fourier-reflexive partitions and MacWilliams identities for additive codes

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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

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Notes

  1. The author is grateful to the anonymous reviewer for suggesting this line of proof.

  2. In [9, 43], the authors use a reversed inner dot product. This results in the partition \({\mathcal P}_{{\mathbf P}}\) being self-dual, and no dual poset is needed.

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Acknowledgments

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 Zürich, 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).

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  1. Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY , 40506-0027, USA

    Heide Gluesing-Luerssen

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Communicated by V. A. Zinoviev.

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Gluesing-Luerssen, H. Fourier-reflexive partitions and MacWilliams identities for additive codes. Des. Codes Cryptogr. 75, 543–563 (2015). https://doi.org/10.1007/s10623-014-9940-x

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