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Linear codes with covering radius 3

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Abstract

The shortest possible length of a q-ary linear code of covering radius R and codimension r is called the length function and is denoted by ℓ q (r, R). Constructions of codes with covering radius 3 are here developed, which improve best known upper bounds on ℓ q (r, 3). General constructions are given and upper bounds on ℓ q (r, 3) for q = 3, 4, 5, 7 and r ≤ 24 are tabulated.

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References

  1. Baicheva T.S., Velikova E.D.: Covering radii of ternary linear codes of small dimensions and codimensions. IEEE Trans. Inform. Theory 43, 2057–2061 (1997); and 44, 2032 (1998).

    Google Scholar 

  2. 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 

  3. Brualdi R.A., Pless V.S., Wilson R.M.: Short codes with a given covering radius. IEEE Trans. Inform. Theory 35, 99–109 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cock J.C., Östergård P.R.J.: Ternary covering codes derived from BCH codes. J. Combin. Theory. Ser. A 80, 283–289 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cohen G., Honkala I., Litsyn S., Lobstein A.: Covering Codes. North-Holland, Amsterdam (1997)

    MATH  Google Scholar 

  6. Davydov A.A.: Constructions and families of covering codes and saturated sets of points in projective geometry. IEEE Trans. Inform. Theory 41, 2071–2080 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Davydov A.A.: Constructions and families of nonbinary linear codes with covering radius 2. IEEE Trans. Inform. Theory 45, 1679–1686 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Davydov A.A., Faina G., Marcugini S., Pambianco F.: Locally optimal (nonshortening) linear covering codes and minimal saturating sets in projective spaces. IEEE Trans. Inform. Theory 51, 4378–4387 (2005)

    Article  MathSciNet  Google Scholar 

  9. Davydov A.A., Giulietti M., Marcugini S., Pambianco F.: Linear covering codes of radius 2 and 3. In: Proc. Workshop Coding Theory Days in St. Petersburg, pp. 12–17. St. Petersburg, Russia (2008).

  10. Davydov A.A., Giulietti M., Marcugini S., Pambianco F.: Linear covering codes over nonbinary finite fields. In: Proc. XI Int. Workshop Algebraic Comb. Coding Theory, ACCT2008, pp. 70–75. Pamporovo, Bulgaria (2008).

  11. Davydov A.A., Marcugini S., Pambianco F.: Linear codes with covering radius 2, 3 and saturating sets in projective geometry. IEEE Trans. Inform. Theory 50, 537–541 (2004)

    Article  MathSciNet  Google Scholar 

  12. Davydov A.A., Östergård P.R.J.: New linear codes with covering radius 2 and odd basis. Des. Codes Cryptogr. 16, 29–39 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Davydov A.A., Östergård P.R.J.: New quaternary linear codes with covering radius 2. Finite Fields Appl. 6, 164–174 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Davydov A.A., Östergård P.R.J.: On saturating sets in small projective geometries. Eur. J. Combin. 21, 563–570 (2000)

    Article  MATH  Google Scholar 

  15. Davydov A.A., Östergård P.R.J.: Linear codes with covering radius R = 2, 3 and codimension t R. IEEE Trans. Inform. Theory 47, 416–421 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hirschfeld J.W.P.: Maximum sets in finite projective spaces. In: Lloyd, E.K. (eds) Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., vol. 82, pp. 55–76. Cambridge University Press, Cambridge (1983)

    Google Scholar 

  17. Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  18. Hirschfeld J.W.P., Storme L.: The packing problem in statistics, coding theory and finite projective spaces. J. Stat. Plan. Inference 72, 355–380 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Östergård P.R.J.: Classifying subspaces of Hamming spaces. Des. Codes Cryptogr. 27, 297–305 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Alexander A. Davydov

  2. Department of Communications and Networking, Helsinki University of Technology TKK, P.O. Box 3000, 02015, TKK, Finland

    Patric R. J. Östergård

Authors
  1. Alexander A. Davydov
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  2. Patric R. J. Östergård
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Correspondence to Alexander A. Davydov.

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Communicated by Pascale Charpin.

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Davydov, A.A., Östergård, P.R.J. Linear codes with covering radius 3. Des. Codes Cryptogr. 54, 253–271 (2010). https://doi.org/10.1007/s10623-009-9322-y

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  • DOI: https://doi.org/10.1007/s10623-009-9322-y

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