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 Krawtchouk Polynomials, 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：摘要：近年来，由于极小线性码在秘密共享方案、安全两方计算等领域的应用，其研究得到了广泛关注。构造违反Ashikhmin-Barg条件的极小线性码并确定其权重分布，已成为编码理论与密码学中的有趣课题。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），并推导出该三元线性码为极小码的充要条件。基于该条件与Krawtchouk多项式，得到了一类违反Ashikhmin-Barg条件的新极小三元线性码，进而确定了其完整重量枚举子。