Abstract
In this work, we introduce the harmonic generalization of the m-tuple weight enumerators of codes over finite Frobenius rings. A harmonic version of the MacWilliams-type identity for m-tuple weight enumerators of codes over finite Frobenius ring is also given. Moreover, we define the demi-matroid analogue of well-known polynomials from matroid theory, namely Tutte polynomials and coboundary polynomials, and associate them with a harmonic function. We also prove the Greene-type identity relating these polynomials to the harmonic m-tuple weight enumerators of codes over finite Frobenius rings. As an application of this Greene-type identity, we provide a simple combinatorial proof of the MacWilliams-type identity for harmonic m-tuple weight enumerators over finite Frobenius rings. Finally, we provide the structure of the relative invariant spaces containing the harmonic m-tuple weight enumerators of self-dual codes over finite fields.
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Acknowledgements
The authors would also like to thank the anonymous reviewers for their beneficial comments on an earlier version of the manuscript. The fourth named author is supported by JSPS KAKENHI (22K03277).
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Communicated by I. Landjev.
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Britz, T., Chakraborty, H.S., Ishikawa, R. et al. Harmonic Tutte polynomials of matroids II. Des. Codes Cryptogr. (2023). https://doi.org/10.1007/s10623-023-01343-0
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DOI: https://doi.org/10.1007/s10623-023-01343-0