The hull of a linear code over finite fields is the intersection of the code and its dual, and linear codes with small hulls have applications in computational complexity and information protection. Linear codes with the smallest hull are LCD codes, which have been widely studied. Recently, several papers were devoted to related LCD codes over finite fields with size greater than 3 to linear codes with one-dimensional or higher dimensional hull. Therefore, an interesting and non-trivial problem is to study binary linear codes with one-dimensional hull with connection to binary LCD codes. The objective of this paper is to 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. Using such a characterization, we study the largest minimum distance $d_{one}(n,k)$ among all binary linear $[n,k]$ codes with one-dimensional hull. We determine the largest minimum distances $d_{one}(n,n-k)$ for $ k\leq 5$ and $d_{one}(n,k)$ for $k\leq 4$ or $14\leq n\leq 24$. We partially determine the exact value of $d_{one}(n,k)$ for $k=5$ or $25\leq n\leq 30$.

翻译：有限域上线性码的对偶码定义为该码与其对偶码的交集，具有小对偶码的线性码在计算复杂度和信息保护中有应用。具有最小对偶码的线性码是LCD码，已被广泛研究。近年来，多篇论文致力于研究大于3阶有限域上的相关LCD码与具有一维或高维对偶码的线性码。因此，一个有趣且非平凡的问题是在二元LCD码的背景下研究具有一维对偶码的二元线性码。本文旨在研究具有一维对偶码的二元线性码的若干性质，并建立其与二元LCD码的联系，进而获得一些有趣的不等式。利用这一表征，我们研究了所有具有一维对偶码的二元$[n,k]$线性码中最大最小距离$d_{one}(n,k)$。对于$k\leq 5$，我们确定了$d_{one}(n,n-k)$的值；对于$k\leq 4$或$14\leq n\leq 24$，我们确定了$d_{one}(n,k)$的值；对于$k=5$或$25\leq n\leq 30$，我们部分确定了$d_{one}(n,k)$的精确值。