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.

在这项工作中，我们介绍了有限 Frobenius 环上代码的m元组权重枚举器的调和推广。还给出了有限 Frobenius 环上码的m元组权重枚举器的 MacWilliams 型恒等式的调和版本。此外，我们定义了拟阵理论中著名多项式的半拟阵类似物，即塔特多项式和共界多项式，并将它们与调和函数相关联。我们还证明了将这些多项式与有限 Frobenius 环上的代码的调和m元组权重枚举器联系起来的格林型恒等式。作为格林型恒等式的应用，我们为有限 Frobenius 环上的调和m元组权重枚举器提供了 MacWilliams 型恒等式的简单组合证明。最后，我们提供了包含有限域上自对偶码的调和m元组权重枚举器的相对不变空间的结构。