The hull of a linear code over finite fields is the intersection of the code and its dual. Linear codes with small hulls have been widely studied due to their applications in computational complexity and information protection. In this paper, we study some properties of binary linear codes with one-dimensional hull, and establish their relation with binary LCD codes. Some interesting inequalities are thus obtained. We determine the exact value of $d_{one}(n,k)$ for $k\in\{1,3,4,n-5,n-4,n-3,n-2,n-1\}$ or $14\leq n\leq 24$, where $d_{one}(n,k)$ denotes the largest minimum weight among all binary linear $[n,k]$ codes with one-dimensional hull. We partially determine the value of $d_{one}(n,k)$ for $k=5$ or $25\leq n\leq 30$. As an application, we construct some entanglement-assisted quantum error-correcting codes (EAQECCs).
翻译：线性代码的外壳是该代码及其双倍的交汇处。由于对计算复杂度和信息保护的应用,对小型船体的线性代码进行了广泛研究。在本文中,我们研究了单体体双线代码的某些特性,并确定了它们与二体LCD代码的关系。因此,取得了一些有趣的不平等。我们确定了美元=1,3,4,4,n,n,k美元=1,n,n,n,n,4,n,n,3,n,n,n,2,n,n,1美元或14\leq nleq 24美元的确切价值。其中,$done}(n,k)表示所有单体线性($)线性($)代码的最大最低重量。我们部分确定了美元=5美元或25美元=leq nleq 30美元的价值。作为应用,我们制作了一些折叠辅助量误校准代码(EAQECCs)。
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