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.

翻译：本文引入了有限Frobenius环上码的$m$重权枚举子的调和推广形式，并给出了有限Frobenius环上码的$m$重权枚举子的调和型MacWilliams恒等式。进一步地，我们定义了拟阵理论中著名多项式（即Tutte多项式和上边界多项式）的半拟阵类比，并将其与调和函数相关联。同时证明了将这些多项式与有限Frobenius环上码的调和$m$重权枚举子相联系的Greene型恒等式。作为该Greene型恒等式的应用，我们给出了有限Frobenius环上调和$m$重权枚举子MacWilliams型恒等式的一个简单组合证明。最后，我们给出了包含有限域上自对偶码的调和$m$重权枚举子的相对不变子空间结构。