论文标题
约翰逊图中的代码和设计来自二次形式的符号动作
Codes and Designs in Johnson Graphs From Symplectic Actions on Quadratic Forms
论文作者
论文摘要
Johnson Graph $ j（v，k）$作为$ \ Mathcal {v} = \ {1，\ ldots，v \} $的$ k $ -subsets，如果它们的交叉点具有尺寸$ k-1 $，则边缘将两个顶点涂在边缘。 $ j（v，k）$中的一个\ emph {$ x $ -stronglongly发病代码}是一个适当的顶点子集$γ$，使得留出$γ$不变性的图形$ x $在每个编码$ $ unize $ unize $ unize $ unize $ unize $ unize $ unize $ unize $Δ$Δ$δ$Δ$上，以及noce $δ$δ$δ$Δ $δ\ times（\ Mathcal {V} \setMinusΔ）$。 We classify the \emph{$X$-strongly incidence-transitive codes} in $J(v,k)$ for which $X$ is the symplectic group $\mathrm{Sp}_{2n}(2)$ acting as a $2$-transitive permutation group of degree $2^{2n-1}\pm 2^{n-1}$, where the stabiliser代码字$δ$的$x_δ$包含在$ x $的最大亚组中。特别是，我们构建了两个与$ \ mathrm {sp} _ {2n}（2）$的可降低最大亚组相关的强烈发病率代码。
The Johnson graph $J(v, k)$ has as vertices the $k$-subsets of $\mathcal{V}=\{1,\ldots, v\}$, and two vertices are joined by an edge if their intersection has size $k-1$. An \emph{$X$-strongly incidence-transitive code} in $J (v, k)$ is a proper vertex subset $Γ$ such that the subgroup $X$ of graph automorphisms leaving $Γ$ invariant is transitive on the set $Γ$ of `codewords', and for each codeword $Δ$, the setwise stabiliser $X_Δ$ is transitive on $Δ\times (\mathcal{V}\setminus Δ)$. We classify the \emph{$X$-strongly incidence-transitive codes} in $J(v,k)$ for which $X$ is the symplectic group $\mathrm{Sp}_{2n}(2)$ acting as a $2$-transitive permutation group of degree $2^{2n-1}\pm 2^{n-1}$, where the stabiliser $X_Δ$ of a codeword $Δ$ is contained in a \emph{geometric} maximal subgroup of $X$. In particular, we construct two new infinite families of strongly incidence-transitive codes associated with the reducible maximal subgroups of $\mathrm{Sp}_{2n}(2)$.
