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New covering codes of radius R, codimension tR and \(tR+\frac{R}{2}\), and saturating sets in projective spaces

  • Alexander A. DavydovEmail author
  • Stefano Marcugini
  • Fernanda Pambianco
Article
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Abstract

The length function \(\ell _q(r,R)\) is the smallest length of a q-ary linear code of codimension r and covering radius R. In this work we obtain new constructive upper bounds on \(\ell _q(r,R)\) for all \(R\ge 4\), \(r=tR\), \(t\ge 2\), and also for all even \(R\ge 2\), \(r=tR+\frac{R}{2}\), \(t\ge 1\). The new bounds are provided by infinite families of new covering codes with fixed R and increasing codimension. The new bounds improve upon the known ones. We propose a general regular construction (called “Line+Ovals”) of a minimal \(\rho \)-saturating \(((\rho +1)q+1)\)-set in the projective space \(\mathrm {PG}(2\rho +1,q)\) for all \(\rho \ge 0\). Such a set corresponds to an \([Rq+1,Rq+1-2R,3]_qR\) locally optimal code of covering radius \(R=\rho +1\). Basing on combinatorial properties of these codes regarding to spherical capsules, we give constructions for code codimension lifting and obtain infinite families of new surface-covering codes with codimension \(r=tR\), \(t\ge 2\). In addition, we obtain new 1-saturating sets in the projective plane \(\mathrm {PG}(2,q^2)\) and, basing on them, construct infinite code families with fixed even radius \(R\ge 2\) and codimension \(r=tR+\frac{R}{2}\), \(t\ge 1\).

Keywords

Covering codes Saturating sets The length function Projective spaces 

Mathematics Subject Classification

51E21 51E22 94B05 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and suggestions.

References

  1. 1.
    Bacsó G., Héger T., Szőnyi T.: The 2-blocking number and the upper chromatic number of \(\text{ PG }(2, q)\). J. Comb. Des. 21(12), 585–602 (2013).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bartoli D., Davydov A.A., Giulietti M., Marcugini S., Pambianco F.: New bounds for linear codes of covering radii 2 and 3, Cryptography and Communications, to appear,  https://doi.org/10.1007/s12095-018-0335-0. Accessed 26 May 2019.
  3. 3.
    Blokhuis A., Lovász L., Storme L., Szőnyi T.: On multiple blocking sets in Galois planes. Adv. Geom. 7(1), 39–53 (2007).MathSciNetzbMATHGoogle Scholar
  4. 4.
    Boros E., Szőnyi T., Tichler K.: On defining sets for projective planes. Discret. Math. 303(1–3), 17–31 (2005).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brualdi R.A., Pless V.S., Wilson R.M.: Short codes with a given covering radius. IEEE Trans. Inf. Theory 35(1), 99–109 (1989).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brualdi R.A., Litsyn S., Pless V.S.: Covering radius. In: Pless V.S., Huffman W.C., Brualdi R.A. (eds.) Handbook of Coding Theory, vol. 1, pp. 755–826. Elsevier, Amsterdam (1998).Google Scholar
  7. 7.
    Cohen G., Honkala I., Litsyn S., Lobstein A.: Covering Codes, vol. 54. Elsevier, Amsterdam (1997).zbMATHGoogle Scholar
  8. 8.
    Csajbók B., Héger T.: Double blocking sets of size \(3q-1\) in \(\text{ PG }(2, q)\). Eur. J. Comb. 78, 73–89 (2019).MathSciNetGoogle Scholar
  9. 9.
    Davydov A.A.: Construction of codes with covering radius 2. In: Cohen G., Litsyn S., Lobstein A., Zemor G. (eds.) Algebraic Coding. Lect. Notes Comput. Science, vol. 573, pp. 23–31. Springer, New–York (1992).Google Scholar
  10. 10.
    Davydov A.A.: Construction of linear covering codes. Probl. Inf. Transm. 26(4), 317–331 (1990).MathSciNetzbMATHGoogle Scholar
  11. 11.
    Davydov A.A.: Constructions and families of covering codes and saturated sets of points in projective geometry. IEEE Trans. Inf. Theory 41(6), 2071–2080 (1995).MathSciNetzbMATHGoogle Scholar
  12. 12.
    Davydov A.A.: Constructions and families of nonbinary linear codes with covering radius 2. IEEE Trans. Inf. Theory 45(5), 1679–1686 (1999).MathSciNetzbMATHGoogle Scholar
  13. 13.
    Davydov A.A., Östergård P.R.J.: On saturating sets in small projective geometries. Eur. J. Comb. 21(5), 563–570 (2000).MathSciNetzbMATHGoogle Scholar
  14. 14.
    Davydov A.A., Östergård P.R.J.: Linear codes with covering radius \(R=2,3\) and codimension \(tR\). IEEE Trans. Inf. Theory 47(1), 416–421 (2001).MathSciNetzbMATHGoogle Scholar
  15. 15.
    Davydov A.A., Östergård P.R.J.: Linear codes with covering radius 3. Des. Codes Cryptogr. 54(3), 253–271 (2010).MathSciNetzbMATHGoogle Scholar
  16. 16.
    Davydov A.A., Marcugini S., Pambianco F.: On saturating sets in projective spaces. J. Comb. Theory Ser. A 103(1), 1–15 (2003).MathSciNetzbMATHGoogle Scholar
  17. 17.
    Davydov A.A., Faina G., Marcugini S., Pambianco F.: Locally optimal (nonshortening) linear covering codes and minimal saturating sets in projective spaces. IEEE Trans. Inf. Theory 51(12), 4378–4387 (2005).MathSciNetzbMATHGoogle Scholar
  18. 18.
    Davydov A.A., Giulietti M., Marcugini S., Pambianco F.: Linear covering codes over nonbinary finite fields. In: Proc. XI Int. Workshop on Algebraic and Combinatorial Coding Theory, ACCT2008. pp. 70–75. Pamporovo, Bulgaria (2008) http://www.moi.math.bas.bg/acct2008/b12.pdf. Accessed 26 May 2019.
  19. 19.
    Davydov A.A., Giulietti M., Marcugini S., Pambianco F.: Linear nonbinary covering codes and saturating sets in projective spaces. Adv. Math. Commun. 5(1), 119–147 (2011).MathSciNetzbMATHGoogle Scholar
  20. 20.
    De Beule J., Héger T., Szőnyi T., Van de Voorde G.: Blocking and double blocking sets in finite planes. Electron. J. Comb. 23(2), 6 (2016).MathSciNetzbMATHGoogle Scholar
  21. 21.
    Etzion T., Storme L.: Galois geometries and coding theory. Des. Codes Cryptogr. 78(1), 311–350 (2016).MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ezerman M.F., Grassl M., Sole P.: The weights in MDS codes. IEEE Trans. Inf. Theory 57(1), 392–396 (2011).MathSciNetzbMATHGoogle Scholar
  23. 23.
    Giulietti M.: The geometry of covering codes: small complete caps and saturating sets in Galois spaces. In: Blackburn S.R., Holloway R., Wildon M. (eds.) Surveys in Combinatorics 2013, London Math. Soc. Lect. Note Series, vol. 409, pp. 51–90. Cambridge Univ Press, Cambridge (2013).Google Scholar
  24. 24.
    Hirschfeld J.W.P.: Projective Geometries Over Finite Fields. Oxford Mathematical Monographs, 2nd edn. Clarendon Press, Oxford (1998).Google Scholar
  25. 25.
    Hirschfeld J.W.P., Storme L.: The packing problem in statistics, coding theory and finite projective spaces. J. Stat. Plan. Infer. 72(1), 355–380 (1998).MathSciNetzbMATHGoogle Scholar
  26. 26.
    Hirschfeld J.W.P., Storme L.: The packing problem in statistics, coding theory and finite geometry: update 2001. In: Blokhuis A., Hirschfeld J.W.P. et al. (eds.) Finite Geometries, Developments of Mathematics, vol. 3, Proc. of the Fourth Isle of Thorns Conf., Chelwood Gate, 2000, pp. 201–246. Kluwer Academic Publisher, Boston (2001).Google Scholar
  27. 27.
    Janwa H.: Some optimal codes from algebraic geometry and their covering radii. Eur. J. Comb. 11(3), 249–266 (1990).MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kiss G., Kóvacs I., Kutnar K., Ruff J., Šparl P.: A note on a geometric construction of large Cayley graphs of given degree and diameter. Stud. Univ. Babes-Bolyai Math. 54(3), 77–84 (2009).MathSciNetzbMATHGoogle Scholar
  29. 29.
    Klein A., Storme L.: Applications of Finite Geometry in Coding Theory and Cryptography. In: Crnković D., Tonchev V. (eds.) NATO Science for Peace and Security, Ser. - D: Information and Communication Security, vol. 29, Information Security, Coding Theory and Related Combinatorics, pp. 38–58 (2011).Google Scholar
  30. 30.
    Landjev I., Storme L.: Galois geometry and coding theory. In: De Beule J., Storme L. (eds.) Current Research Topics in Galois Geometry, Chapter 8, pp. 187–214, NOVA Academic Publisher, New York (2012).Google Scholar
  31. 31.
    Lobstein A.: Covering radius, an online bibliography. https://www.lri.fr/~lobstein/bib-a-jour.pdf. Accessed 26 May 2019.
  32. 32.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes, 3rd edn. Elsevier, Amsterdam (1981).zbMATHGoogle Scholar
  33. 33.
    Ughi E.: Saturated configurations of points in projective Galois spaces. Eur. J. Comb. 8(3), 325–334 (1987).MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institute for Information Transmission Problems (Kharkevich Institute)Russian Academy of SciencesMoscowRussian Federation
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly

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