Generalized $b$-symbol weights of Linear Codes and $b$-symbol MDS Codes

TL;DR

This paper introduces generalized $b$-symbol weights for linear codes, extending previous concepts, and explores their properties, bounds, and conditions for maximum distance separability, with applications to code equivalence and classical theorems.

Contribution

It defines generalized $b$-symbol weights, establishes bounds, characterizes $b$-symbol MDS codes, and connects these weights to code isomorphisms and classical theorems.

Findings

Derived Singleton-like bounds for generalized $b$-symbol weights.

Provided necessary and sufficient conditions for $b$-symbol MDS codes.

Established a $b$-symbol weight preserving isomorphism criterion.

Abstract

Generalized pair weights of linear codes are generalizations of minimum symbol-pair weights, which were introduced by Liu and Pan \cite{LP} 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 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 $b=1$.