The study on minimal linear codes has received great attention due to their significant applications in secret sharing schemes and secure two-party computation. Until now, numerous minimal linear codes have been discovered. However, to the best of our knowledge, no infinite family of minimal ternary linear codes was found from vectorial functions. In this paper, we present a necessary and sufficient condition for a large class of ternary linear codes from vectorial functions such that those codes are minimal. Based on that, we construct several minimal ternary linear codes with three-weight from vectorial regular plateaued functions, and determine their weight distributions. Moreover, we also give a necessary and sufficient condition for a large family of ternary linear codes from vectorial functions such that the codes are minimal and violate the AB condition simultaneously. According to this characterization, we find several minimal ternary linear codes violating the AB condition. Notably, our results show that our method can be applied to solve a problem on minimal linear codes proposed by Li et al.

翻译：关于极小线性码的研究因其在秘密共享方案和安全两方计算中的显著应用而备受关注。迄今为止，已有大量极小线性码被发现。然而，据我们所知，目前尚未有从向量函数构造的无限族极小三元线性码。本文给出了由向量函数构造的一大类三元线性码成为极小码的充要条件。基于此，我们利用向量正则平台函数构造了若干具有三个权重的极小三元线性码，并确定了其权重分布。此外，我们还给出了由向量函数构造的另一大类三元线性码同时满足极小性与违反AB条件的充要条件。根据这一刻画，我们找到了若干违反AB条件的极小三元线性码。值得注意的是，我们的结果表明该方法可解决Li等人提出的关于极小线性码的一个开放问题。