Designs, Codes and Cryptography

, Volume 46, Issue 1, pp 45–56 | Cite as

On perfect p-ary codes of length p + 1



Let p be a prime number and assume p ≥ 5. We will use a result of L. Redéi to prove, that every perfect 1-error correcting code C of length p + 1 over an alphabet of cardinality p, such that C has a rank equal to p and a kernel of dimension p − 2, will be equivalent to some Hamming code H. Further, C can be obtained from H, by the permutation of the symbols, in just one coordinate position.


Perfect codes Redei Theorem 

AMS Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Best M.R. (1983). Perfect codes hardly exist. IEEE Trans. Inform. Theory 29(3): 349–351MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Blokhuis A., Lam C.W.H. (1984). More coverings by rook domains. J. Combin. Theory Ser. A 36: 240–244MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Etzion T. (1996). Nonequivalent q-ary perfect codes. SIAM J. Discrete Math. 9(3): 413–423MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Golay M.J.E.: Notes on digital coding. Proc. IRE 37 (1949), Correspondence 657.Google Scholar
  5. 5.
    Hamming, Error detecting and error correcting codes. Bell Syst. Tech. J. 29, 147–160 (1950).Google Scholar
  6. 6.
    Heden O. (1975). A generalised Lloyd Theorem and mixed perfect codes. Math. Scand. 37: 13–26MATHMathSciNetGoogle Scholar
  7. 7.
    Heden O.: Perfect codes from the dual point of view I. submitted to Discrete Math.Google Scholar
  8. 8.
    Heden O. (2006). A full rank perfect code of length 31. Des. Codes Crypthogr. 38, 125–129MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lang S. (1965). Algebra. Addison-Wesley Publishing Company, ReadingMATHGoogle Scholar
  10. 10.
    Leontiev V.K., Zinoviev V.A. (1973). Nonexistence of perfect codes over galois fields. Problems Inform. Theory 2(2): 123–132Google Scholar
  11. 11.
    Lindström B. (1969). On group and nongroup perfect codes in q symbols. Math. Scand. 25: 149–158MATHGoogle Scholar
  12. 12.
    Phelps K.T. (1983). A general product construction of perfect codes. SIAM J. Algebra Discrete Method. 4, 224–228MathSciNetGoogle Scholar
  13. 13.
    Phelps K.T., Villanueva M. (2002). Ranks of q-ary 1-perfect codes. Des. Codes Cryptogr. 27(1–2): 139–144MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Phelps K.T., Rifà J., Villanueva M. (2005). Kernels and p-kernels of p r-ary 1-perfect codes. Des. Codes Cryptogr. 37(2): 243–261CrossRefMathSciNetGoogle Scholar
  15. 15.
    Rédei L.: Lückenhafte Polynome über endlischen Körpern. Birkhäuser verlag Basel und Stuttgart (1970).Google Scholar
  16. 16.
    Schönheim J. (1968). On linear and nonlinear single-error-correcting q-ary perfect codes. Inform. Control 12: 23–26MATHCrossRefGoogle Scholar
  17. 17.
    Solov’eva F.I.: On Perfect Codes and Related Topics. Com2Mac Lecture Note Series 13, Pohang (2004).Google Scholar
  18. 18.
    Tietäväinen A. (1973). On the nonexistence of perfect codes over finite fields. SIAM J. Appl. Math. 24, 88–96CrossRefMathSciNetGoogle Scholar
  19. 19.
    van der Waerden B.L. (1966). Algebra, Erster Teil, Siebte Auflage der Modernen Algebra. Springer-Verlag, BerlinGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsKTHStockholmSweden

Personalised recommendations