MacWilliams Identity for Codes with the Hamming and Rank Metrics

The MacWilliams identity, which relates the weight distribution of a code to the weight distribution of its dual code, is useful in determining the weight distribution of codes. In this paper, we first provide an alternative proof to the MacWilliams identity for linear codes with the Hamming metric. An intermediate result of our approach is that the Hamming weight enumerator of the dual of any vector depends on only the Hamming weight of the vector and is related to the Hamming weight enumerator of a maximum distance separable (MDS) code. We then extend this approach to derive the MacWilliams identity for linear codes with the rank metric, and our identity has a different form than that by Delsarte. Using our MacWilliams identity, we also derive related identities for rank metric codes.