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arXiv · 2210.16490

Harmonic Tutte polynomials of matroids II

Thomas Britz·Himadri Shekhar Chakraborty·Reina Ishikawa·Tsuyoshi Miezaki·Hopein Christofen Tang
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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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Thomas Britz, Himadri Shekhar Chakraborty, Reina Ishikawa, Tsuyoshi Miezaki, Hopein Christofen Tang. 2022-10-29. Harmonic Tutte polynomials of matroids II. https://arxiv.org/abs/2210.16490

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