🤖 AI Summary
This work addresses the construction of minimal ternary linear codes that violate the Ashikhmin–Barg condition by proposing a general method to generate ternary linear codes of dimension $m+2$. By integrating algebraic coding theory with exponential sum analysis, the authors derive necessary and sufficient conditions for the constructed codes to be minimal and, for the first time, provide their complete weight enumerator. The resulting family of codes strictly transcends the limitations imposed by the classical Ashikhmin–Barg sufficient condition, thereby expanding the known parameter space of minimal codes and offering new candidate constructions for cryptographic applications such as secret sharing.
📝 Abstract
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.