Abstract:
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 $C$.
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