Doob图中的可加完美码
Additive perfect codes in Doob graphs
摘要：
Doob图$D(m,n)$是Shrikhande图的$m>0$副本和$4$阶完全图的$n$副本的笛卡尔积。自然地，$D(m,n)$可以表示为加法群$(Z_4^2)^m \times (Z_2^2)^{n'} \times Z_4^{n''}$上的凯莱图，其中$n'+n''=n$。如果$D(m,n)$的一组顶点构成该组的一个子群，则称该组顶点为加性码。我们构造了Doob图中的可加完美码的$3$ -参数类，并证明了在$D(m,n'+n'')$中存在可加$1$ -完美码的已知必要条件是充分的.另外，在$D(155,0+31)$和$D(2667,0+127)$中构造了两个准循环加性$1$ -完美码.
英文摘要：
The Doob graph $D(m,n)$ is the Cartesian product of $m>0$ copies of the Shrikhande graph and $n$ copies of the complete graph of order $4$. Naturally, $D(m,n)$ can be represented as a Cayley graph on the additive group $(Z_4^2)^m \times (Z_2^2)^{n'} \times Z_4^{n''}$, where $n'+n''=n$. A set of vertices of $D(m,n)$ is called an additive code if it forms a subgroup of this group. We construct a $3$-parameter class of additive perfect codes in Doob graphs and show that the known necessary conditions of the existence of additive $1$-perfect codes in $D(m,n'+n'')$ are sufficient. Additionally, two quasi-cyclic additive $1$-perfect codes are constructed in $D(155,0+31)$ and $D(2667,0+127)$.
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