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

作者：Liu, Hongwei; Pan, Xu^*

来源：IEEE Transactions on Information Theory, 2023, 69(4): 2311-2323.

DOI：10.1109/TIT.2022.3223729

• Generalized b-symbol weights
• b-symbol MDS codes
• linear isomorphisms preserving b-symbol weights
• MacWilliams extension theorem

pair weights of linear codes are gen-eralizations of minimum symbol-pair weights, which were intro-duced 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 gen-eralized 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.