Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then determining their weight distributions have been interesting in coding theory and cryptography. In this paper, a generic construction for ternary linear codes with dimension $m+2$ is presented, where $m$ is an integer, and a necessary and sufficient condition for this ternary linear code to be minimal is derived. Based on this condition and exponential sums, a new class of minimal ternary linear codes violating the Ashikhmin-Barg condition are obtained, and then their complete weight enumerators are determined.

翻译：近年来，极小线性码因其在秘密共享方案、安全两方计算等领域的应用而得到广泛研究。构造违反Ashikhmin-Barg条件的极小线性码并确定其重量分布已成为编码理论与密码学中的热点问题。本文提出了一种维数为$m+2$的三元线性码的通用构造方法（其中$m$为整数），并推导了该三元线性码为极小码的充要条件。基于该条件与指数和，得到了一类新的违反Ashikhmin-Barg条件的极小三元线性码，进而确定了其完备重量枚举器。