A note on the minimal binary linear code

Abstract

Due to the wide applications in communications, data storage and cryptography, linear codes have received much attention in the past decades. As a subclass of linear codes, minimal linear codes can be used to construct secret sharing with nice access structure. The objective of this paper is to construct new classes of minimal binary linear codes with $w_{\min \limits }/w_{\max \limits }\leq 1/2$ from preferred binary linear codes, where $w_{\min \limits }$ and $w_{\max \limits }$ denote the minimum and maximum nonzero Hamming weights in $\mathcal {C}$ respectively. Firstly, we introduce a concept called preferred binary linear codes and a class of minimal binary linear codes with $w_{\min \limits }/w_{\max \limits }\leq 1/2$ can be deduced from preferred binary linear codes. As an application of preferred binary linear codes, we get a new class of six-weight minimal binary linear codes with $w_{\min \limits }/w_{\max \limits }< 1/2$ from a known class of five-weight preferred binary linear codes. Secondly, by employing vectorial Boolean functions, we construct two new classes of preferred binary linear codes and, consequently, these two new classes of preferred binary linear codes can generate two new classes of minimal binary linear codes with $w_{\min \limits }/w_{\max \limits }\leq 1/2$ and large minimum distance.