🤖 AI Summary
This study addresses the efficient computation of the minimum weight and weight enumerator for error-correcting codes generated by integer matrices. By interpreting the weight enumerator as a quasi-polynomial, the work establishes—for the first time—a transformational relationship with the Tutte quasi-polynomial. Leveraging reductions modulo integer residue rings and the characteristic quasi-polynomial of hyperplane arrangements, the counting of maximum-weight codewords is recast as a problem in combinatorial geometry. This approach substantially reduces the computational complexity of determining minimum weights and yields exact counts of maximum-weight codewords for codes associated with the $N_k$ and $Z_k$ matroids. The results provide a novel algebraic–geometric perspective within combinatorial coding theory.
📝 Abstract
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 $Z_k$.