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.

翻译：20世纪60年代，MacWilliams证明了有限域上线性码的汉明权枚举子完全由其偶码的汉明权枚举子决定，反之亦然。特别地，若两个线性码具有相同的汉明权枚举子，则它们的偶码也具有相同的汉明权枚举子。与之相反，存在一类广泛的有限域上的权，其权枚举子表现出相反的性质：存在两个具有相同权枚举子的线性码，但它们的偶码却具有不同的权枚举子。