Minimal Binary Linear Codes

Abstract

In addition to their applications in data communication and storage, linear codes also have nice applications in combinatorics and cryptography. Minimal linear codes, a special type of linear codes, are preferred in secret sharing. In this paper, a necessary and sufficient condition for a binary linear code to be minimal is derived. This condition enables us to obtain three infinite families of minimal binary linear codes with $w_{\min }/w_{\max } \leq 1/2$ from a generic construction, where $w_{\min }$ and $w_{\max }$ , respectively, denote the minimum and maximum nonzero weights in a code. The weight distributions of all these minimal binary linear codes are also determined.