Generalized $b$-weights and $b$-MDS Codes

In this paper, we first introduce the notion of generalized $b$-weights of $[n,k]$-linear codes over finite fields, and obtain some basic properties and bounds of generalized $b$-weights of linear codes which is called Singleton bound for generalized $b$-weights in this paper. Then we obtain a necessary and sufficient condition for an $[n,k]$-linear code to be a $b$-MDS code by using generator matrixes of this linear code and parity check matrixes of this linear code respectively. Next a theorem of a necessary and sufficient condition for a linear isomorphism preserving $b$-weights between two linear codes is obtained, in particular when $b=1$, this theorem is the MacWilliams extension theorem. Then we give a reduction theorem for the MDS conjecture. Finally, we calculate the generalized $b$-weight matrix $D(C)$ when $C$ is simplex codes or two especial Hamming codes.