论文标题
具有一维船体的最佳二进制线性代码的表征和构建
Characterization and construction of optimal binary linear codes with one-dimensional hull
论文作者
论文摘要
有限字段上线性代码的船体是代码及其双重的相交，并且与小船体的线性代码在计算复杂性和信息保护中具有应用。最小船体的线性代码是LCD代码，已被广泛研究。最近，几篇论文专用于具有一维或更高维船体的线性代码大于3的有限字段上的相关LCD代码。因此，一个有趣且非平凡的问题是与一维船体一起研究与二进制LCD代码相关的二进制线性代码。本文的目的是研究具有一维船体的二进制线性代码的某些特性，并与二元LCD代码建立关系。因此，获得了一些有趣的不平等。使用这样的特征，我们研究了所有二进制hull的最小最小距离$ d_ {one}（n，k）$中的最小距离$ d_ {n，k）$。我们确定最大的最小距离$ d_ {one}（n，n-k）$ for $ k \ leq 5 $和$ d_ {one}（n，k）$ for $ k \ leq 4 $或$ 14 \ leq n \ leq n \ leq 24 $。我们部分确定$ d_ {one}（n，k）$的确切值，$ k = 5 $或$ 25 \ leq n \ leq 30 $。
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 $25\leq n\leq 30$.
