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On the minimum size of binary codes with length 2R +  4 and covering radius R

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Abstract

The minimum size of a binary code with length n and covering radius R is denoted by K(n, R). For arbitrary R, the value of K(n, R) is known when n ≤  2R +  3, and the corresponding optimal codes have been classified up to equivalence. By combining combinatorial and computational methods, several results for the first open case, K(2R +  4, R), are here obtained, including a proof that K(10, 3) =  12 with 11481 inequivalent optimal codes and a proof that if K(2R +  4, R) <  12 for some R then this inequality cannot be established by the existence of a corresponding self-complementary code.

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References

  1. Bertolo R., Östergård P.R.J., Weakley W.D. (2004). An updated table of binary/ternary mixed covering codes. J. Combin. Des. 12, 157–176

    Article  MathSciNet  MATH  Google Scholar 

  2. Blass U., Litsyn S. (1999). The smallest covering code of length 8 and radius 2 has 12 words. Ars Combin. 52, 309–318

    MathSciNet  MATH  Google Scholar 

  3. Bollobás B. (1973). Sperner systems consisting of pairs of complementary subsets. J. Combin. Theory Ser. A 15, 363–366

    Article  MATH  Google Scholar 

  4. Brace A., Daykin D.E.: Sperner type theorems for finite sets. In: Welsh, D.J.A., Woodall, D.R. (eds.) Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford), Inst. Math. Appl., Southend-on-Sea, pp. 18–37 (1972).

  5. Cohen G., Honkala I., Litsyn S., Lobstein A. (1997). Covering Codes. Elsevier, Amsterdam

    MATH  Google Scholar 

  6. Cohen G.D., Lobstein A.C., Sloane N.J.A. (1986). Further results on the covering radius of codes. IEEE Trans. Inform. Theory 32, 680–694

    Article  MathSciNet  MATH  Google Scholar 

  7. Habsieger L., Plagne A. (2000). New lower bounds for covering codes. Discrete Math. 222, 125–149

    Article  MathSciNet  MATH  Google Scholar 

  8. Kaski P., Östergård P.R.J. (2006). Classification Algorithms for Codes and Designs. Springer, Berlin

    MATH  Google Scholar 

  9. Katona G.O.H. (1973). Two applications (for search theory and truth functions) of Sperner type theorems. Period. Math. Hungar. 3, 19–26

    Article  MathSciNet  MATH  Google Scholar 

  10. Kéri G., Östergård P.R.J.: The number of inequivalent (2R +  3, 7)R optimal covering codes. J. Integer Seq. 9, Article 06.4.7 (electronic) (2006).

  11. Kéri G., Östergård P.R.J. (2007). Further results on the covering radius of small codes. Discrete Math. 307, 69–77

    Article  MathSciNet  MATH  Google Scholar 

  12. Kleitman D.J., Spencer J. (1973). Families of k-independent sets. Discrete Math. 6, 255–262

    Article  MathSciNet  MATH  Google Scholar 

  13. MacWilliams F.J., Sloane N.J.A. (1977). The Theory of Error-Correcting Codes. North-Holland, Amsterdam

    MATH  Google Scholar 

  14. Östergård P.R.J., Weakley W.D. (2000). Classification of binary covering codes. J. Combin. Des. 8, 391–401

    Article  MathSciNet  MATH  Google Scholar 

  15. Stanton R.G., Kalbfleisch J.G. (1968). Covering problems for dichotomized matchings. Aequationes Math. 1, 94–103

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Computer and Automation Research Institute, Hungarian Academy of Sciences, Kende u. 13–17, 1111, Budapest, Hungary

    Gerzson Kéri

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

    Patric R. J. Östergård

Authors
  1. Gerzson Kéri
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  2. Patric R. J. Östergård
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Correspondence to Patric R. J. Östergård.

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Dedicated to Professor Torleiv Kløve on the occasion of his 65th birthday.

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Kéri, G., Östergård, P.R.J. On the minimum size of binary codes with length 2R +  4 and covering radius R . Des. Codes Cryptogr. 48, 165–169 (2008). https://doi.org/10.1007/s10623-007-9156-4

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  • DOI: https://doi.org/10.1007/s10623-007-9156-4

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