Additive perfect codes in Doob graphs

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 (Z42)m×(Z22)n′×Z4n′′, 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).