The existence of perfect codes in Doob graphs
Denis S. Krotov (Sobolev Institute of Mathematics, Novosibirsk,, Russia)
TL;DR
This paper determines the precise conditions under which perfect codes exist in Doob graphs, showing they occur only when a specific linear combination of parameters is a power of two.
Contribution
It provides a complete characterization of the existence of 1-perfect codes in Doob graphs based on algebraic conditions.
Findings
1-perfect codes exist in Doob graphs if and only if 6m+3n+1 is a power of 2
The existence depends on the divisibility of the size of a 1-ball
The result links perfect codes to properties of Eisenstein-Jacobi integers.
Abstract
We solve the problem of existence of perfect codes in the Doob graph. It is shown that 1-perfect codes in the Doob graph D(m,n) exist if and only if 6m+3n+1 is a power of 2; that is, if the size of a 1-ball divides the number of vertices. Keywords: perfect codes, distance-regular graphs, Doob graphs, Eisenstein-Jacobi integers.
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