Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 509–516 | Cite as

Local distributions for eigenfunctions and perfect colorings of q-ary Hamming graph

  • Anastasia Vasil’evaEmail author
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography


We study eigenfunctions and perfect colorings of the n-dimensional q-ary Hamming graph. We obtain the formulae of interdependence of local distributions for an eigenfunction in two orthogonal faces. We prove an analogous result for perfect colorings.


Completely regular code Perfect coloring Equitable partition Local distribution Eigenfunction q-ary Hamming graph 

Mathematics Subject Classification




  1. 1.
    Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Springer, New York (1989).CrossRefzbMATHGoogle Scholar
  2. 2.
    Choi S., Hyun J.Y., Kim H.K.: Local duality theorem for q-ary 1-perfect codes. Des. Codes Cryptogr. 70(3), 305–311 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10:vi+-97 (1973).Google Scholar
  4. 4.
    Hyun J.Y.: Local duality for equitable partitions of a Hamming space. J. Comb. Theor. Ser. A 119, 476–482 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Krotov D.S.: On weight distributions of perfect colorings and completely regular codes. Des. Codes Cryptogr. 61(3), 315–329 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Vasil’eva A.Yu.: Local spectra of perfect binary codes. Discret. Appl. Math. 135(1–3), 301–307 (2004) [Translated from Discretn. Anal. Issled. Oper. Ser. 1 6(1), 3–11 (1999)].Google Scholar
  7. 7.
    Vasil’eva A.Yu.: Local and interweight spectra of completely regular codes and of perfect colorings. Probl. Inf. Transm. 45(2), 151–157 (2009) [Translated from Probl. Peredachi Inf. 45(2), 84–90 (2009)].Google Scholar
  8. 8.
    Vasil’eva A.Yu.: Local distributions and reconstruction of hypercube eigenfunctions. Probl. Inform. Transm. 49(1), 32–39 (2013) [Translated from Probl. Peredachi Inf. 49(1), 37–45 (2013)].Google Scholar
  9. 9.
    Vasil’eva A.: Local distributions of q-ary Eigen functions and of q-ary perfect colorings. In: Proceedings of Seventh International Workshop on Optimal Codes and Related Topics (OC2013), Institute of Mathematics and Informatics, Sofia, pp. 181–186 (2013).Google Scholar
  10. 10.
    Vasil’eva A.Yu.: Reconstruction of Eigenfunctions of a q-ary n-dimensional hypercube. Probl. Inf. Transm. 51(3), 231–239 (2015) [Translated from Probl. Peredachi Inf. 51(3):31–40 (2015)].Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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