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 \((Z_4^2)^m \times (Z_2^2)^{n'} \times Z_4^{n''}\), 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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The authors thank Tatsuro Ito, Jack Koolen, and Patrick Solé for the consulting concerning the last remark and the anonymous referees for useful comments.
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Communicated by J. H. Koolen.
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This research is supported by National Natural Science Foundation of China (61672036), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (No. 1808085J20), and the Program of fundamental scientific researches of the Siberian Branch of the Russian Academy of Sciences No. I.1.1. (No. 0314-2016-0016).
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Shi, M., Huang, D. & Krotov, D.S. Additive perfect codes in Doob graphs. Des. Codes Cryptogr. 87, 1857–1869 (2019). https://doi.org/10.1007/s10623-018-0586-y
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DOI: https://doi.org/10.1007/s10623-018-0586-y