Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

On the Classification of Completely Regular Codes with Covering Radius Two and Antipodal Duals

  • CODING THEORY
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

We classify all linear completely regular codes which have covering radius \(\rho=2\) and whose dual are antipodal. For this, we firstly show several properties of such dual codes, which are two-weight codes with weights \(d\) and \(n\).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Neumaier, A., Distance Matrices, Dimension, and Conference Graphs, Nederl. Akad. Wetensch. Indag. Math., 1981, vol. 43, no. 4, pp. 385–391. https://doi.org/10.1016/1385-7258(81)90059-7

    Article  MathSciNet  Google Scholar 

  2. Bassalygo, L.A., Zaitsev, G.V., and Zinoviev, V.A., Uniformly Packed Codes, Probl. Peredachi Inf., 1974, vol. 10, no. 1, pp. 9–14 [Probl. Inf. Transm. (Engl. Transl.), 1974, vol. 10, no. 1, pp. 6–9]. https://www.mathnet.ru/eng/ppi1014

    MathSciNet  Google Scholar 

  3. Semakov, N.V., Zinoviev, V.A., and Zaitsev, G.V., Uniformly Packed Codes, Probl. Peredachi Inf., 1971, vol. 7, no. 1, pp. 38–50 [Probl. Inf. Transm. (Engl. Transl.), 1971, vol. 7, no. 1, pp. 30–39]. https://www.mathnet.ru/eng/ppi1621

    MathSciNet  Google Scholar 

  4. Goethals, J.-M. and van Tilborg, H.C.A., Uniformly Packed Codes, Philips Res. Rep., 1975, vol. 30, pp. 9–36.

    MathSciNet  Google Scholar 

  5. Borges, J., Rifà, J. and Zinoviev, V.A., On Completely Regular Codes, Probl. Peredachi Inf., 2019, vol. 55, no. 1, pp. 3–50 [Probl. Inf. Transm. (Engl. Transl.), 2019, vol. 55, no. 1, pp. 1–45]. https://doi.org/10.1134/S0032946019010010

    MathSciNet  Google Scholar 

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

  7. van Dam, E.R., Koolen, J.H., and Tanaka, H., Distance-Regular Graphs, Electron. J. Combin., 2016, Dynamic Surveys, DS22 (156 pp.). https://doi.org/10.37236/4925

  8. Koolen, J., Krotov, D., and Martin, B., Completely Regular Codes (electronic pages), https://sites.google.com/site/completelyregularcodes.

  9. Bonisoli, A., Every Equidistant Linear Code Is a Sequence of Dual Hamming Codes, Ars Combin., 1984, vol. 18, pp. 181–186.

    MathSciNet  Google Scholar 

  10. Borges, J., Rifà, J., and Zinoviev, V.A., On \(q\)-ary Linear Completely Regular Codes with \(\rho=2\) and Antipodal Dual, Adv. Math. Commun., 2010, vol. 4, no. 4, pp. 567–578. https://doi.org/10.3934/amc.2010.4.567

    Article  MathSciNet  Google Scholar 

  11. Boyvalenkov, P., Delchev, K., Zinoviev, V.A., and Zinoviev, D.V., On Codes with Distances \(d\) and \(n\), Probl. Peredachi Inf., 2022, vol. 58, no. 4, pp. 62–83 [Probl. Inf. Transm. (Engl. Transl.), 2022, vol. 58, no. 4, pp. 352–371]. https://doi.org/10.1134/S0032946022040068

    Google Scholar 

  12. Delsarte, P., An Algebraic Approach to the Association Schemes of Coding Theory, Philips Res. Rep. Suppl., 1973, no. 10 (97 pp.).

    MathSciNet  Google Scholar 

  13. Bassalygo, L.A. and Zinoviev, V.A., Remark on Uniformly Packed Codes, Probl. Peredachi Inf., 1977, vol. 13, no. 3, pp. 22–25 [Probl. Inf. Transm. (Engl. Transl.), 1977, vol. 13, no. 3, pp. 178–180]. https://www.mathnet.ru/eng/ppi1091

    MathSciNet  Google Scholar 

  14. Semakov, N.V., Zinoviev, V.A., and Zaitsev, G.V., A Class of Maximum Equidistant Codes, Probl. Peredachi Inf., 1969, vol. 5, no. 2, pp. 84–87 [Probl. Inf. Transm. (Engl. Transl.), 1969, vol. 5, no. 2, pp. 65–68]. http://mi.mathnet.ru/eng/ppi1804

    MathSciNet  Google Scholar 

  15. Bassalygo, L.A., Dodunekov, S.M., Zinoviev, V.A., and Helleseth, T., The Grey–Rankin Bound for Nonbinary Codes, Probl. Peredachi Inf., 2006, vol. 42, no. 3, pp. 37–44 [Probl. Inf. Transm. (Engl. Transl.), 2006, vol. 42, no. 3, pp. 197–203]. https://doi.org/10.1134/S0032946006030033

    MathSciNet  Google Scholar 

  16. Helleseth, T., Kløve, T., and Levenshtein, V.I., A Bound for Codes with Given Minimum and Maximum Distances, in Proc. 2006 IEEE Int. Symp. on Information Theory (ISIT’2006), Seattle, WA, USA, July 9–14, 2006, pp. 292–296. https://doi.org/10.1109/ISIT.2006.261600

  17. Beth, T., Jungnickel, D., and Lenz, H., Design Theory, Cambridge, UK: Cambridge Univ. Press, 1986.

  18. Denniston, R.H.F., Some Maximal Arcs in Finite Projective Planes, J. Combin. Theory, 1969, vol. 6, no. 3, pp. 317–319. https://doi.org/10.1016/S0021-9800(69)80095-5

    Article  MathSciNet  Google Scholar 

  19. Thas, J.A., Projective Geometry over a Finite Field, Ch. 7 of Handbook of Incidence Geometry: Buildings and Foundations, Buekenhout, F., Ed., Amsterdam: Elsevier, 1995, pp. 295–347. https://doi.org/10.1016/B978-044488355-1/50009-8

  20. Bouyukliev, I.G., Classification of Griesmer Codes and Dual Transform, Discrete Math., 2009, vol. 309, no. 12, pp. 4049–4068. https://doi.org/10.1016/j.disc.2008.12.002

    Article  MathSciNet  Google Scholar 

  21. Delsarte, P., Two-Weight Linear Codes and Strongly Regular Graphs, MBLE Res. Lab. Report, Brussels, Belgium, 1971, no. R160.

  22. Boyvalenkov, P., Delchev, K., Zinoviev, D.V., and Zinoviev, V.A., On Two-Weight Codes, Discrete Math., 2021, vol. 344, no. 5, Paper No. 112318 (15 pp.). https://doi.org/10.1016/j.disc.2021.112318

    Article  MathSciNet  Google Scholar 

  23. Farrell, P.G., Linear Binary Anticodes, Electron. Lett., 1970, vol. 6, no. 13, pp. 419–421. https://doi.org/10.1049/el:19700293

    Article  Google Scholar 

  24. Hamada, N. and Helleseth, T., Codes and Minihypers, in Proc. 3rd EuroWorkshop on Optimal Codes and Related Topics (OC’2001), Sunny Beach, Bulgaria, June 10–16, 2001, pp. 79–84.

  25. Delsarte, P., Weights of Linear Codes and Strongly Regular Normed Spaces, Discrete Math., 1972, vol. 3, no. 1–3, pp. 47–64. https://doi.org/10.1016/0012-365X(72)90024-6

    Article  MathSciNet  Google Scholar 

  26. Boyvalenkov, P., Delchev, K., Zinoviev, D.V., and Zinoviev, V.A., On \(q\)-ary Codes with Two Distances \(d\) and \(d+1\), Probl. Peredachi Inf., 2020, vol. 56, no. 1, pp. 38–50 [Probl. Inf. Transm. (Engl. Transl.), 2020, vol. 56, no. 1, pp. 33–44]. https://doi.org/10.1134/S0032946020010044

    MathSciNet  Google Scholar 

  27. Calderbank, R. and Kantor, W.M., The Geometry of Two-Weight Codes, Bull. London Math. Soc., 1986, vol. 18, no. 2, pp. 97–122. https://doi.org/10.1112/blms/18.2.97

    Article  MathSciNet  Google Scholar 

  28. Bush, K.A., Orthogonal Arrays of Index Unity, Ann. Math. Statist., 1952, vol. 23, no. 3, pp. 426–434. https://doi.org/10.1214/aoms/1177729387

    Article  MathSciNet  Google Scholar 

  29. Ball, S., Blokhuis, A., and Mazzocca, F., Maximal Arcs in Desarguesian Planes of Odd Order Do Not Exist, Combinatorica, 1997, vol. 17, no. 1, pp. 31–41. https://doi.org/10.1007/BF01196129

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work has been partially supported by the Spanish Ministerio de Ciencia e Innovación under Grant PID2022-137924NB-I00 (AEI/FEDER UE) and RED2022-134306-T, and by the Catalan AGAUR Grant 2021-SGR-00643. The research of the second and third authors of the paper was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences within the program of fundamental research on the topic “Mathematical Foundations of the Theory of Error-Correcting Codes” and was also supported by the National Science Foundation of Bulgaria under project no. 20-51-18002.

Author information

Authors and Affiliations

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

    J. Borges

  2. Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow, Russia

    V. A. Zinoviev & D. V. Zinoviev

Authors
  1. J. Borges
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. A. Zinoviev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. V. Zinoviev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Borges.

Ethics declarations

The authors of this work declare that they have no conflicts of interest.

Additional information

Publisher’s Note. Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borges, J., Zinoviev, V. & Zinoviev, D. On the Classification of Completely Regular Codes with Covering Radius Two and Antipodal Duals. Probl Inf Transm 59, 204–216 (2023). https://doi.org/10.1134/S003294602303002X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S003294602303002X

Keywords

Search

Navigation