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Generalized pair weights of linear codes are generalizations of minimumsymbol-pair weights, which were introduced by Liu and Pan \cite{LP} recently.Generalized pair weights can be used to characterize the ability of protectinginformation in the symbol-pair read wire-tap channels of type II. In thispaper, we introduce the notion of generalized $b$-symbol weights of linearcodes over finite fields, which is a generalization of generalized Hammingweights and generalized pair weights. We obtain some basic properties andbounds of generalized $b$-symbol weights which are called Singleton-like boundsfor generalized $b$-symbol weights. As examples, we calculate generalizedweight matrices for simplex codes and Hamming codes. We provide a necessary andsufficient condition for a linear code to be a $b$-symbol MDS code by using thegenerator matrix and the parity check matrix of this linear code. Finally, anecessary and sufficient condition of a linear isomorphism preserving$b$-symbol weights between two linear codes is obtained. As a corollary, we getthe classical MacWilliams extension theorem when $b=1$.

Generalized $b$-symbol weights of Linear Codes and $b$-symbol MDS Codes