  In this paper, we first introduce the notion of generalized pair weights of
an $[n, k]$-linear code over the finite field $\mathbb{F}_q$ and the notion of
pair $r$-equiweight codes, where $1\le r\le k-1$. Some basic properties of
generalized pair weights of linear codes over finite fields are derived. Then
we obtain a necessary and sufficient condition for an $[n,k]$-linear code to be
a pair equiweight code, and we characterize pair $r$-equiweight codes for any
$1\le r\le k-1$. Finally, a necessary and sufficient condition for a linear
isomorphism preserving pair weights between two linear codes is obtained.
