Generalized b-Symbol Weights of Linear Codes and b-Symbol MDS Codes

Generalized pair weights of linear codes are generalizations of minimum symbol-pair weights, which were introduced by Liu and Pan (2022) recently. Generalized pair weights can be used to characterize the ability of protecting information in the symbol-pair read wire-tap channels of type II. In this paper, we introduce the notion of generalized
$b$
-symbol weights of linear codes over finite fields, which is a generalization of generalized Hamming weights and generalized pair weights. We obtain some basic properties and bounds of generalized
$b$
-symbol weights which are called Singleton-like bounds for generalized
$b$
-symbol weights. As examples, we calculate the generalized weight matrices for simplex codes and Hamming codes. We provide a necessary and sufficient condition for a linear code to be a
$b$
-symbol MDS code by using the generator matrix and the parity check matrix of this linear code. Finally, a necessary and sufficient condition of a linear isomorphism preserving
$b$
-symbol weights between two linear codes is obtained. As a corollary, we get the classical MacWilliams extension theorem when
$b=1$
.