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arXiv · 2312.07109

On multifold perfect codes and some other completely regular codes in the Doob graphs and quaternary Hamming graphs

Evgeny Bespalov
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Abstract

We consider the problem of existence of perfect $2$-colorings in the Doob graphs $D(m,n)$ and $4$-ary Hamming graphs $H(n,4)$. We characterize all parameters for which multifold $1$-perfect code in $D(m,n)$ exists. Also, we prove that for any pair $(b,c)$ that satisfy standard conditions (Lloyd's and sphere-packing conditions) there is perfect $(b,c)$-coloring in Doob graphs and Hamming graphs if diameter of graph and $n$ are sufficiently large ($n$ not less than $0$, $1$ or $8$ for some cases). Also we obtain some completely regular codes with covering radius $2$.

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Evgeny Bespalov. 2023-12-12. On multifold perfect codes and some other completely regular codes in the Doob graphs and quaternary Hamming graphs. https://arxiv.org/abs/2312.07109

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