The Parameters of Minimal Linear Codes

TL;DR

This paper introduces a new necessary and sufficient condition for minimal linear codes, enabling easier construction and identification of such codes, and presents four new classes that generalize previous binary cases.

Contribution

It provides a novel condition for minimal linear codes, simplifies their construction, and extends known classes to broader cases.

Findings

New necessary and sufficient condition for minimal linear codes

Construction of four new classes of minimal linear codes

Generalization of binary minimal codes to broader cases

Abstract

Let $k\leq n$ be two positive integers and $q$ a prime power. The basic question in minimal linear codes is to determine if there exists an $[n,k]_q$ minimal linear code. The first objective of this paper is to present a new sufficient and necessary condition for linear codes to be minimal. Using this condition, it is easy to construct minimal linear codes or to prove some linear codes are minimal. The second objective of this paper is to use the new sufficient and necessary condition to partially give an answer to the basic question in minimal linear codes. The third objective of this paper is to present four classes of minimal linear codes, which generalize the results about the binary case given in [X. Li and Q. Yue, Des. Codes Cryptogr., 2019]. One can find that our method is much easier and more effective.