Weight Functions and Generalized Weights of Linear Codes

We prove that the weight function $\mathrm{wt}\colon\mathbb F_q^k\to\mathbb Z$ on a set of messages uniquely determines a linear code of dimension $k$ 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 $C$, to a function on $\mathbb F_q^{\binom kr}\cong\Lambda^rC$. Using this, we show that, for each linear code $C$ and any integer $r\le k=\dim C$, a linear code 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 this code is proportional to the $r$th generalized weight of $C$.