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}.

翻译：最近,由于在秘密共享计划、两党计算等应用中应用了最低线性代码,因此对最低线性代码进行了广泛研究。 建立违反Ashikhmin-Barg条件的最低限度线性代码并确定其重量分布,是编码理论和加密学中一个有趣的研究课题。 在本论文中,我们首先证明,在\cite{Heng-Ding-Zhou}中,以指数和Krawtchouk多元数字为基础,美元(m,k)$(m,k)$($)($-cite{Heng-Ding-Zhou})是某些子子($_mathb{F}3 ⁇ (m,k)$($)($)($)($)($)(m,k)($)($)(m)(m)(k)($)($)(g)(m)($)($_(m)(m)(k)($)($-cite{Heng-Darg)($)($)($)($)($)($-Barg)($)($($)($)($)($)($)(t)($)($)($)($)(t)(t)($)($)($)(t)($)($)($)($)($)($)($)($)(t)(t)(t)(t)($)(t)($)($)($)($)($)($)($)($)(t)($)($)($))($)($)($)($))))($)($)($)($)($)($)($)(t)($)(t)($)($)))($)($)($)($)($)($)($)($)($)($)($)($)($)($)($)($)($)($)($)($)($)($)($)(