论文标题
关于与几乎简单的顶部组的二进制完全传递代码的分类
On the Classification of Binary Completely Transitive Codes with Almost-Simple Top-Group
论文作者
论文摘要
锤子指标中的代码$ c $是锤子图$ \ vargamma = h（m，q）$的顶点集$ v \ vargamma $的子集，引起了自然距离分区$ \ \ \ {c，c_1，c_1，c_1，\ ldots，c_ρ\} $，$ c $ c $ c $ co $ co $ co $ co $ co $ co $ co $ co $ co $ co $ co $ co $ co $ co $ copt $ copt $ co $。如果自动形态组$ \ rm {aut}（c）$在每个集合$ c $，$ c_1 $，\ ldots，$c_ρ$上施用，则称这种代码$ c $被称为完全传递的。如果$ρ\ geq 2 $和$ \ rm {aut}（c）$在$ c $，$ c $，$ c_1 $和$ c_2 $上施用代码$ c $，则称为$ 2 $ -neighbour-transitive-geq 2 $和$ \ rm {aut}（c）$。 令$ c $为二进制（$ q = 2 $）中的完全传递代码，其hamming图具有完整的自动形态组$ \ rm {aut}（c）$和最小距离$δ\ geq 5 $。然后知道$ \ rm {aut}（c）$诱导了$ 2 $ - 均匀的动作，对锤子图的顶点的坐标。本文的主要结果将$ C $分类为$ c $，该$ c $该诱使$ 2 $均匀的动作不是仿射，线性或符号组。我们发现有13美元的代码，其中$ 4 $是非线性代码。尽管大多数代码都是众所周知的，但我们获得了几个新结果。首先，构建了一个新的非线性完全传递代码，以及相关的非线性代码为$ 2 $ -neighbour的传输，但不是完全传递的。此外，给出了几种代码完全传递性的新证明。此外，我们回答了与我们主要结果中出现的完全传递代码相关的距离规则图的存在的问题。
A code $C$ in the Hamming metric, that is, is a subset of the vertex set $V\varGamma$ of the Hamming graph $\varGamma=H(m,q)$, gives rise to a natural distance partition $\{C,C_1,\ldots,C_ρ\}$, where $ρ$ is the covering radius of $C$. Such a code $C$ is called completely transitive if the automorphism group $\rm{Aut}(C)$ acts transitively on each of the sets $C$, $C_1$, \ldots, $C_ρ$. A code $C$ is called $2$-neighbour-transitive if $ρ\geq 2$ and $\rm{Aut}(C)$ acts transitively on each of $C$, $C_1$ and $C_2$. Let $C$ be a completely transitive code in a binary ($q=2$) Hamming graph having full automorphism group $\rm{Aut}(C)$ and minimum distance $δ\geq 5$. Then it is known that $\rm{Aut}(C)$ induces a $2$-homogeneous action on the coordinates of the vertices of the Hamming graph. The main result of this paper classifies those $C$ for which this induced $2$-homogeneous action is not an affine, linear or symplectic group. We find that there are $13$ such codes, $4$ of which are non-linear codes. Though most of the codes are well-known, we obtain several new results. First, a new non-linear completely transitive code is constructed, as well as a related non-linear code that is $2$-neighbour-transitive but not completely transitive. Moreover, new proofs of the complete transitivity of several codes are given. Additionally, we answer the question of the existence of distance-regular graphs related to the completely transitive codes appearing in our main result.
