Skip to main content
SpringerLink
Log in

New families of completely regular codes and their corresponding distance regular coset graphs

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In this paper three new infinite families of linear binary completely regular codes are constructed. They have covering radius ρ = 3 and 4, and are halves of binary Hamming and binary extended Hamming codes of length n = 2m−1 and 2m, where m is even. There are also shown some combinatorial (binomial) identities which are new, to our knowledge.These completely regular codes induce, in the usual way, i.e., as coset graphs, three infinite families of distance-regular graphs of diameter three and four. This description of such graphs is new.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bassalygo L.A., Zinoviev V.A.: A note on uniformly packed codes. Problems Inf. Transm. 13(3), 22–25 (1977)

    MATH  Google Scholar 

  2. Bassalygo L.A., Zaitsev G.V., Zinoviev V.A.: Uniformly packed codes. Problems Inf. Transm. 10(1), 9–14 (1974)

    MATH  Google Scholar 

  3. Borges J., Rifa J., Phelps K.P., Zinoviev V.A.: On \({\mathbb{Z}_4}\)-linear Preparata-like and Kerdock-like codes. IEEE Trans. Inf. Theory 49(11), 2834–2843 (2003)

    Article  MathSciNet  Google Scholar 

  4. Borges J., Rifa J., Zinoviev V.A.: On non-antipodal binary completely regular codes. Discret. Math. 308, 3508–3525 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Borges J., Rifa J., Zinoviev V.A.: On q-ary linear completely regular codes with ρ = 2 and antipodal dual. Adv. Math. Commun. 4(4), 567–578 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  7. de Caen D., Mathon R., Moorhouse G.E.: A family of antipodal distance-regular graphs related to the classical Preparata codes. J. Algebr. Comb. 4, 317–327 (1995)

    Article  MATH  Google Scholar 

  8. Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, 1–97 (1973)

    MathSciNet  Google Scholar 

  9. Fon-Der-Flaass D.G.: Perfect 2-coloring of hypercube. Sib. Math. J. 48, 923–930 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fon-Der-Flaass D.G.: Perfect 2-coloring of the 12-cube that attain the bound on correlation immunity. Sib. Electron. Math. Rep. 4, 292–295 (2007)

    MATH  MathSciNet  Google Scholar 

  11. Godsil C.D., Hensel A.D.: Distance regular covers of the complete graph. J. Comb. Theory B 56, 205–238 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Goethals J.M., Van Tilborg H.C.A.: Uniformly packed codes. Philips Res. 30, 9–36 (1975)

    MATH  MathSciNet  Google Scholar 

  13. Jurišic A., Koolen J., Terwilliger P.: Tight distance-regular graphs. J. Algebr. Comb. 12, 163–197 (2000)

    Article  MATH  Google Scholar 

  14. Koolen J.H., Lee W.S., Martin W.J.: Characterizing completely regular codes from an algebraic view point. Comb. Graphs. Contemp. Math. 531, 223–242 (2010)

    Article  MathSciNet  Google Scholar 

  15. Neumaier A.: Completely regular codes. Discret. Math. 106(107), 335–360 (1992)

    MathSciNet  Google Scholar 

  16. Rifà J., Zinoviev V.A.: On new completely regular q-ary codes. Problems Inf. Transm. 43(2), 97–112 (2007)

    Article  MATH  Google Scholar 

  17. Rifà J., Zinoviev V.A.: On lifting perfect codes. IEEE Trans. Inf. Theory 57(9), 5918–5925 (2011)

    Article  Google Scholar 

  18. Taylor D.E.: Regular 2-graphs. Proc. Lond. Math. Soc. 35(3), 257–274 (1977)

    Article  MATH  Google Scholar 

  19. Terwilliger P.: Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array. Des. Codes Cryptogr. 34(2–3), 307–332 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Thas J.A.: Two infinite classes of perfect codes in metrically regular graphs. J. Comb. Theory B 23, 236–238 (1977)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    Joaquim Borges & Josep Rifà

  2. Institute for Problems of Information Transmission, Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 127994, Russia

    Victor Zinoviev

Authors
  1. Joaquim Borges
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Josep Rifà
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Victor Zinoviev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Josep Rifà.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding Theory and Applications”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borges, J., Rifà, J. & Zinoviev, V. New families of completely regular codes and their corresponding distance regular coset graphs. Des. Codes Cryptogr. 70, 139–148 (2014). https://doi.org/10.1007/s10623-012-9713-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-012-9713-3

Keywords

Mathematics Subject Classification

Search

Navigation