Generalized Pair Weights of Linear Codes and Linear Isomorphisms Preserving Pair Weights

In this paper, we 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. 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
$1\le r\le k-1$
. A necessary and sufficient condition for a linear isomorphism to preserve pair weights between two linear codes is obtained. At the end of this paper, an application of generalized pair weights of linear codes to symbol-pair read wire-tap channels of type II is introduced.