Abstract Constructing minimal linear codes is an interesting research topic due to their applications in coding theory and cryptography. Ashikhmin and Barg pointed out that w min ∕ w max > ( q − 1 ) ∕ q is a sufficient condition for a linear code over the finite field F q to be minimal, where w min and w max respectively denote the minimum and maximum nonzero weights in a code. However, only a few families of minimal linear codes over F q with w min ∕ w max ≤ ( q − 1 ) ∕ q were reported in the literature. In this paper, we obtain several families of minimal q -ary linear codes with w min ∕ w max ≤ ( q − 1 ) ∕ q . The weight distributions of all the constructed minimal linear codes are presented.

中文翻译：

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wmin∕wmax≤(q−1)∕q的几族q元最小线性码

摘要 构建最小线性码因其在编码理论和密码学中的应用而成为一个有趣的研究课题。Ashikhmin 和 Barg 指出 w min ∕ w max > ( q − 1 ) ∕ q 是有限域 F q 上的线性代码为最小的充分条件，其中 w min 和 w max 分别表示非零的最小值和最大值代码中的权重。然而，文献中只报道了少数 F q 上的最小线性码族，其中 w min ∕ w max ≤ ( q − 1 ) ∕ q 。在本文中，我们获得了几个最小 q 元线性码族，其中 w min ∕ w max ≤ ( q − 1 ) ∕ q 。给出了所有构造的最小线性码的权重分布。