Weights on finite fields and failures of the MacWilliams identities

In the 1960s, MacWilliams proved that the Hamming weight enumerator of a linear code over a finite field completely determines, and is determined by, the Hamming weight enumerator of its dual code. In particular, if two linear codes have the same Hamming weight enumerator, then their dual codes have the same Hamming weight enumerator. In contrast, there is a wide class of weights on finite fields whose weight enumerators have the opposite behavior: there exist two linear codes having the same weight enumerator, but their dual codes have different weight enumerators.

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