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

• Published in: IEEE transactions on information theory Vol. 69; no. 4; p. 1
• Main Authors: Liu, Hongwei, Pan, Xu
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.04.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">b -symbol MDS codes; Codes; Fields (mathematics); generalized <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">b -symbol weights; Generators; Hamming codes; Hamming weight; Isomorphism; Linear codes; linear isomorphisms preserving <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">b -symbol weights; MacWilliams extension theorem; Mathematical analysis; Mathematics; Parity check codes; Symbols; Wiretapping
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2022.3223729

Generalized pair weights of linear codes are generalizations of minimum symbol-pair weights, which were introduced by Liu and Pan [16] 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 generalized 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.