Two Constructions for Minimal Ternary Linear Codes.

Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and determining their weight distributions have been an interesting research topic in coding theory and cryptography. In this paper, basing on exponential sums and Krawtchouk polynomials, we first prove that $g_{(m,k)}$ in \cite{Heng-Ding-Zhou}, which is the characteristic function of some subset in $\mathbb{F}_3^m$, can be generalized to be $f{(m,k)}$ for obtaining a minimal linear code violating the Ashikhmin-Barg condition; secondly, we employ $\overline{g}_{(m,k)}$ to construct a class of ternary minimal linear codes violating the Ashikhmin-Barg condition, whose minimal distance is better than that of codes in \cite{Heng-Ding-Zhou}.