Due to their practical applications, hulls of linear codes have been of interest and extensively studied. In this paper, we focus on constructions and bounds on quaternary linear codes with Hermitian hull dimension one. Optimal \([n,2]_4\) codes with Hermitian hull dimension one are constructed for all lengths \(n\ge 3\), such that \(n \equiv 1,2,4 \ (\mathrm{mod}\ 5)\). For positive integers \(n \equiv 0,3 \ (\mathrm{mod}\ 5)\), good lower and upper bounds on the minimum weight of quaternary \([n,2]_4\) codes with Hermitian hull dimension one are given.

中文翻译：

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厄米壳尺寸为一的四元线性码的构造和界

由于它们的实际应用，线性代码的外壳已经引起人们的兴趣并进行了广泛的研究。在本文中，我们重点研究Hermitian船体尺寸为1的四元线性代码的构造和边界。对于所有长度\（n \ ge 3 \）构造具有最佳Hermitian船体尺寸为1的最优\（[n，2] _4 \）代码，使得\（n \ equiv 1,2,4 \（\ mathrm {mod} \ 5）\）。对于正整数\（n \ equiv 0,3 \（\ mathrm {mod} \ 5）\），具有厄米壳尺寸的四元\（[n，2] _4 \）最小权重的良好上下限给出一个。