Additive perfect codes in Doob graphs

  • Minjia ShiEmail author
  • Daitao Huang
  • Denis S. Krotov


The Doob graph D(mn) is the Cartesian product of \(m>0\) copies of the Shrikhande graph and n copies of the complete graph of order 4. Naturally, D(mn) 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(mn) 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 \(D(2667,0+127)\).


Distance regular graphs Additive perfect codes Doob graphs Quasi-cyclic codes Tight 2-designs 

Mathematics Subject Classification

94B05 94B25 05B40 



The authors thank Tatsuro Ito, Jack Koolen, and Patrick Solé for the consulting concerning the last remark and the anonymous referees for useful comments.


  1. 1.
    Borges J., Fernández-Córdoba C.: There is exactly one \({Z_2Z_4}\)-cyclic \(1\)-perfect code. Des. Codes Cryptogr. 85(3), 557–566 (2017). Scholar
  2. 2.
    Hammons Jr. A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \(Z_4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994). Scholar
  3. 3.
    Heden O., Güzeltepe M.: On perfect \(1\)-\(\cal{E}\)-error-correcting codes. Math. Commun. 20(1), 23–35 (2015).MathSciNetzbMATHGoogle Scholar
  4. 4.
    Herzog M., Schönheim J.: Linear and nonlinear single-error-correcting perfect mixed codes. Inf. Control 18(4), 364–368 (1971). Scholar
  5. 5.
    Koolen J.H., Munemasa A.: Tight \(2\)-designs and perfect \(1\)-codes in Doob graphs. J. Stat. Plann. Inference 86(2), 505–513 (2000). Scholar
  6. 6.
    Krotov D.S.: Perfect codes in Doob graphs. Des. Codes Cryptogr. 80(1), 91–102 (2016). Scholar
  7. 7.
    Tietäväinen A.: On the nonexistence of perfect codes over finite fields. SIAM J. Appl. Math. 24(1), 88–96 (1973). Scholar
  8. 8.
    Wan Z.X.: Quaternary Codes, Series on Applied Mathematics. World Scientific, Singapore (1997). Scholar
  9. 9.
    Zinoviev V., Leontiev V.: The nonexistence of perfect codes over Galois fields. Probl. Control Inf. Theory 2(2), 123–132 (1973).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesAnhui UniversityHefeiChina
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia

Personalised recommendations