In the theory of error-correcting codes, the minimum weight and the weight enumerator play a crucial role in evaluating the error-correcting capacity. In this paper, by viewing the weight enumerator as a quasi-polynomial, we reduce the calculation of the minimum weight to that of a code over a smaller integer residue ring. We also give a transformation formula between the Tutte quasi-polynomial and the weight enumerator. Furthermore, we compute the number of maximum weight codewords for the codes related to the matroids $N_k$ and $Z_k$. This is equivalent to computing the characteristic quasi-polynomial of the hyperplane arrangements related to $N_k$ and $Z_k$.

翻译：在纠错码理论中，最小权与权枚举子对于评估纠错能力起着至关重要的作用。本文将权枚举子视为拟多项式，从而将最小权的计算约化为在更小整数剩余环上的码的计算。我们还给出了Tutte拟多项式与权枚举子之间的变换公式。此外，我们计算了与拟阵$N_k$和$Z_k$相关码的最大权码字个数。这等价于计算与$N_k$和$Z_k$相关的超平面构型的特征拟多项式。