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Siberian Mathematical Journal

, Volume 49, Issue 2, pp 375–382 | Cite as

Intersections of q-ary perfect codes

  • F. I. Solov’evaEmail author
  • A. V. Los’
Article

Abstract

The intersections of q-ary perfect codes are under study. We prove that there exist two q-ary perfect codes C 1 and C 2 of length N = qn + 1 such that |C 1C 2| = k · |P i |/p for each k ∈ {0,..., p · K − 2, p · K}, where q = p r , p is prime, r ≥ 1, \(n = \tfrac{{q^{m - 1} - 1}}{{q - 1}}\), m ≥ 2, |P i | = p nr(q−2)+n , and K = p n(2r−1)−r(m−1). We show also that there exist two q-ary perfect codes of length N which are intersected by p nr(q−3)+n codewords.

Keywords

q-ary perfect codes intersection of codes switching of components Hamming code 

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References

  1. 1.
    Etzion T. and Vardy A., “Perfect binary codes and tilings: problems and solutions,” SIAM J. Discrete Math., 11, No. 2, 205–223 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bar-Yahalom S. E. and Etzion T., “Intersection of isomorphic linear codes,” J. Combin. Theory Ser. A, 80, No. 1, 247–256 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Avgustinovich S. V., Heden O., and Solov’eva F. I., “On intersections of perfect binary codes,” Bayreuth. Math. Schr., No. 74, 1–6 (2005).Google Scholar
  4. 4.
    Avgustinovich S. V., Heden O., and Solov’eva F. I., “On intersection problem for perfect binary codes,” Des. Codes Cryptogr., 39, No. 3, 317–322 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Phelps K. T. and Villanueva M., “Intersection of Hadamard codes,” IEEE Trans. Inform. Theory, 53, No. 5, 1924–1928 (2008).CrossRefMathSciNetGoogle Scholar
  6. 6.
    Rifá J., Solov’eva F. I., and Villanueva M., “On the intersection of additive perfect codes,” IEEE Trans. Inform. Theory, 54, No. 3, 1346–1356 (2008).CrossRefMathSciNetGoogle Scholar
  7. 7.
    Rifá J., Solov’eva F. I., and Villanueva M., “On the intersection of additive extended and non-extended perfect codes,” Proc. Intern. Workshop on Coding and Cryptography, Versailles, France, April 16–20, pp. 333–341 (2007).Google Scholar
  8. 8.
    Zinov’ev V. A. and Leont’ev V. K., A Theorem on Nonexistence of Perfect Codes over Galois Fields [Preprint] [in Russian], Inst. Probl. Peredachi Inform., Moscow (1972).Google Scholar
  9. 9.
    Zinov’ev V. A. and Leont’ev V. K., “Nonexistence of perfect codes over Galois fields,” Problems Control Inform. Theory, No. 2, 123–132 (1973).Google Scholar
  10. 10.
    Tietäväinen A., “On the nonexistence of perfect codes over finite fields,” SIAM J. Appl. Math., 24, 88–96 (1973).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    MacWilliams F. J. and Sloane N. J. A., The Theory of Error-Correcting Codes, North-Holland, Amsterdam (1977).zbMATHGoogle Scholar
  12. 12.
    Schönheim J., “On linear and nonlinear single-error-correcting q-nary perfect codes,” Inform. Control., 12, No. 1, 23–26 (1968).zbMATHCrossRefGoogle Scholar
  13. 13.
    Phelps K. T. and Villanueva M., “Ranks of q-ary 1 perfect codes,” Des. Codes Cryptogr., 27, No. 1–2, 139–144 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Los’ A. V., “Construction of perfect q-ary codes by switchings of simple components,” Problems Inform. Transmission, 42, No. 1, 30–37 (2006).CrossRefMathSciNetGoogle Scholar
  15. 15.
    Solov’eva F. I. and Los’ A. V., “On intersections of q-ary perfect codes,” Proc. Tenth Int. Workshop “Algebraic and Combinatorial Coding Theory,” Zvenigorod, Russia, September 3–9, 2006, pp. 244–247.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

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