Generalized Hamming Weight as a Weight Function
Résumé
We prove that the weight function of a linear code, that is, an integer function defined on the vector space of messages, uniquely determines the code up to equivalence. We propose a natural way to extend the r-th generalized Hamming weight, that is, a function on r-subspaces of a code, to a function on the r-th exterior power of the code. Using this, we show that for any linear code C and any integer r not greater than the dimension of C, another code C' exists whose weight distribution corresponds to a part of the generalized weight spectrum of C from the r-th weights to the k-th. In particular, the minimum distance of C' is proportional to the r-th generalized weight of C.