Four families of minimal binary linear codes with $w_{\min }/w_{\max }\le 1/2$

Abstract

As a special type of linear codes, minimal linear codes have important applications in secret sharing. Up to now, only a few infinite families of minimal binary linear codes with $w_{\min }/w_{\max }\le 1/2$ were reported in the literature, while vast knowledge exists on the ones with $w_{\min }/w_{\max }> 1/2$. Herein, $w_{\min }$ and $w_{\max }$ respectively denote the minimum and maximum nonzero Hamming weights in a linear code. Recently, several classes of linear codes with certain properties were constructed by Zhou et al. from a generic construction. The objective of this paper is to obtain four families of minimal binary linear codes with $w_{\min }/w_{\max }\le 1/2$ from those linear codes proposed by Zhou et al. The parameters of our minimal linear codes are quite different from known ones. Based on the properties of Krawtchouk polynomials, the weight distributions of all these four families of binary linear codes are established.