Generalized b-weights and b-MDS Codes
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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.
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