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arXiv · 1806.04834

Additive perfect codes in Doob graphs

Minjia Shi·Daitao Huang·Denis S. Krotov
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Abstract

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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Minjia Shi, Daitao Huang, Denis S. Krotov. 2018-06-13. Additive perfect codes in Doob graphs. https://doi.org/10.1007/s10623-018-0586-y

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