Perfect 2-colorings of Hamming graphs

Abstract

We consider the problem of existence of perfect $2$-colorings (equitable $2$-partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we observe known constructions of perfect colorings and propose some new ones giving new parameters. At last, we deduce which parameters of colorings are covered by these constructions and give tables of admissible parameters of $2$-colorings in Hamming graphs $H(n,q)$ for small $n$ and $q$. Using the connection with perfect colorings, we construct an orthogonal array OA(2048,7,4,5).

arXiv comment21 pages. Minor improvements

References in corpus (10)

• On the Classification of MDS Codes
• On weight distributions of perfect colorings and completely regular codes
• Perfect colorings of the 12-cube that attain the bound on correlation immunity
• On the binary codes with parameters of doubly-shortened 1-perfect codes
• Equitable partitions of Latin-square graphs
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Cited by in corpus (4)

• On regular sets in Cayley graphs
• On multifold packings of radius-1 balls in Hamming graphs
• Multifold 1-perfect codes
• Generalizing the Bierbrauer-Friedman bound for orthogonal arrays