论文标题
在二进制自动执行代码的最小最小距离上，有两个猜想的尺寸5
Two conjectures on the largest minimum distances of binary self-orthogonal codes with dimension 5
论文作者
论文摘要
本文的目的是在最大的最小距离$ d_ {so}（n，5）$的最小距离上解决这两个猜想的二进制自动式$ [n，5] $代码，由Kim and Choi（IEEETrans。Inf。Theolay，2022）提出。 $ d_ {so}（n，k）$的确定在编码理论中一直是一个基本和困难的问题，因为随着尺寸$ k $的增加，二进制自动执行代码太多。最近，Kim等人。 （2021）认为是二进制线性代码的最短自动嵌入，以及许多以$ k = 4,5 $构建的二进制最佳自动式$ [n，k] $代码。 Kim and Choi（2022）改善了Kim等人的一些结果。 （2021）并在$ d_ {so}（n，5）$上做出了两个猜想。在本文中，我们开发了一种通用方法，以确定$ d_ {so}（n，k）$的确切值，$ k = 5,6 $，并表明Kim and Choi（2022）做出的两个猜想是正确的。
The purpose of this paper is to solve the two conjectures on the largest minimum distance $d_{so}(n,5)$ of a binary self-orthogonal $[n,5]$ code proposed by Kim and Choi (IEEE Trans. Inf. Theory, 2022). The determination of $d_{so}(n,k)$ has been a fundamental and difficult problem in coding theory because there are too many binary self-orthogonal codes as the dimension $k$ increases. Recently, Kim et al. (2021) considered the shortest self-orthogonal embedding of a binary linear code, and many binary optimal self-orthogonal $[n,k]$ codes were constructed for $k=4,5$. Kim and Choi (2022) improved some results of Kim et al. (2021) and made two conjectures on $d_{so}(n,5)$. In this paper, we develop a general method to determine the exact value of $d_{so}(n,k)$ for $k=5,6$ and show that the two conjectures made by Kim and Choi (2022) are true.
