Details

Autor(en) / Beteiligte

• Liu, Hongwei
• Pan, Xu

Titel

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

Ist Teil von

• IEEE transactions on information theory, 2023-04, Vol.69 (4), p.2311-2323

Ort / Verlag

IEEE

Erscheinungsjahr

2023

Link zum Volltext

• IEEE_Xplore

Quelle

IEEE/IET Electronic Library (IEL)

Beschreibungen/Notizen

• 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 <inline-formula> <tex-math notation="LaTeX">b </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">b </tex-math></inline-formula>-symbol weights which are called Singleton-like bounds for generalized <inline-formula> <tex-math notation="LaTeX">b </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">b </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">b </tex-math></inline-formula>-symbol weights between two linear codes is obtained. As a corollary, we get the classical MacWilliams extension theorem when <inline-formula> <tex-math notation="LaTeX">b=1 </tex-math></inline-formula>.