Designs, Codes and Cryptography

, Volume 13, Issue 3, pp 271–284

New Extremal Type II Codes Over ℤ4

  • Masaaki Harada


Recently Type II codes over ℤ4 have been introduced as self-dual codes containing the all-one vector with the property that all Euclidean weights are divisible by eight. The notion of extremality for the Euclidean weight has been also given. In this paper, we give two methods for constructing Type II codes over ℤ4. By these methods, new extremal Type II codes of lengths 16, 24, 32 and 40 are constructed from weighing matrices.

self-dual codes over ℤ4 External codes and Type II codes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Bonnecaze, P. Solé and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory, Vol. 41 (1995) pp. 366-377.CrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over ℤ4, IEEE Trans. Inform. Theory, Vol. 43 (1997) pp. 969-976.MathSciNetGoogle Scholar
  3. 3.
    A. R. Calderbank and N. J. A. Sloane, Double circulant codes over ℤ4 and even unimodular lattices, J. Alg. Combin., Vol. 6 (1997) pp. 119-131.CrossRefMathSciNetGoogle Scholar
  4. 4.
    H. C. Chan, C. A. Rodger and J. Seberry, On inequivalent weighing matrices, Ars Combin., Vol. 21 (1986) pp. 299-333.MathSciNetGoogle Scholar
  5. 5.
    R. Chapman and P. Solé, Universal codes and unimodular lattices, J. de th. des nombres de Bordeaux, Vol. 8 (1996) pp. 369-376.Google Scholar
  6. 6.
    C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton (1996).Google Scholar
  7. 7.
    J. H. Conway and N. J. A. Sloane, Self-dual codes over the integers modulo 4, J. Combin. Theory Ser. A, Vol. 62 (1993) pp. 30-45.CrossRefMathSciNetGoogle Scholar
  8. 8.
    J. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices and Groups (2nd ed.), Springer-Verlag, New York (1993).Google Scholar
  9. 9.
    J. H. Conway, V. Pless and N. J. A. Sloane, The binary self-dual codes of length up to 32: a revised enumeration, J. Combin. Theory Ser. A, Vol. 60 (1992) pp. 183-195.CrossRefMathSciNetGoogle Scholar
  10. 10.
    A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The ℤ4-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, Vol. 40 (1994) pp. 301-319.CrossRefMathSciNetGoogle Scholar
  11. 11.
    M. Harada and H. Kimura, New extremal doubly-even [64, 32, 12] codes, Des. Codes and Cryptogr., Vol. 6 (1995) pp. 91-96.MathSciNetGoogle Scholar
  12. 12.
    M. Klemm, Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4, Arch. Math., Vol. 53 (1989) pp. 201-207.MATHMathSciNetGoogle Scholar
  13. 13.
    F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam (1977).Google Scholar
  14. 14.
    H. Ohmori, On the classifications of weighing matrices of order 12, J. Combin. Math. Combin. Comput., Vol. 5 (1989) pp. 161-216.MATHMathSciNetGoogle Scholar
  15. 15.
    V. Pless, P. Solé and Z. Qian, Cyclic self-dual ℤ4-codes, Finite Fields and Their Appl., Vol. 3 (1997) pp. 48-69.Google Scholar
  16. 16.
    V. Pless and Z. Qian, Cyclic codes and quadratic residue codes over ℤ4, IEEE Trans. Inform. Theory, Vol. 42 (1996) pp. 1594-1600.CrossRefMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Masaaki Harada
    • 1
  1. 1.Department of MathematicsOkayama UniversityOkayamaJapan

Personalised recommendations