Additive perfect codes in Doob graphs
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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 × (Z_2^2)^n'×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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