In the theory of error-correcting codes, the minimum weight and the weight enumerator play a crucial role in evaluating the error-correcting performance. In this paper, by viewing the weight enumerator as a quasi-polynomial, we reduce the determination of the minimum weight of the resulting family to finitely many representative values of $q$. We also give a transformation formula between the Tutte quasi-polynomial and the weight enumerator. Furthermore, we compute the number of full-support codewords for the codes related to the special matroids $N_k$ and $Z_k$. This is equivalent to computing the characteristic quasi-polynomials of the hyperplane arrangements related to $N_k$ and $Z_k$.

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