Optimal linear codes of dimension 4 over GF(5)

  • Ivan N. Landjev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1255)


We characterize the linear codes with parameters [2q2q−1,3,2q2−3q]q and [2q2q−2,3,2q2−3q−1]q. Using this characterization and the geometry of the plane arcs in PG(2, 5), we prove the nonexistence of codes with parameters [215, 4, 171]5 and [209, 4, 166]5. This determinesthe exact value of n5(4, d) for d=166, 167, 168, 169, 170, 171. There remain 16 d's for which the exact value of n5 (4, d) is not known.


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  1. 1.
    S.M. Ball, On sets of points in finite planes, PhD Thesis, University of Sussex, 1994.Google Scholar
  2. 2.
    I. Boukliev, S. Kapralov, T. Maruta, M. Fukui, Optimal linear codes of dimension 4 over \(\mathbb{F}_5\), IEEE Trans, Inf. Theory, to appear.Google Scholar
  3. 3.
    C.D. Baumert, M.J. McEliece, A Note on the Griesmer Bound, IEEE Trans. Inf. Theory IT-19(1973), 134–135.Google Scholar
  4. 4.
    M. van Eupen, Four New Optimal Ternary Linear Codes, IEEE Trans. Inf. Theory IT-40(1994), 193.Google Scholar
  5. 5.
    P. Greenough, R. Hill, Optimal linear codes over GF(4), Discrete Math. 125(1994), 187–199.Google Scholar
  6. 6.
    J.H. Griesmer, A Bound for Error-Correcting Codes, IBM J. Res. Develop.4(1960), 532–542.Google Scholar
  7. 7.
    N. Hamada, A survey of recent work on characterization of minihypers in PG(t, q) and nonbinary linear codes meeting the Griesmer bound, J. Combin. Inform. Syst. Sci. 18(1993), 161–191.Google Scholar
  8. 8.
    N. Hamada, The nonexistence of some quaternary linear codes meeting the Griesmer bound and the bounds for n4 (5, d), 1≤d ≤256, Mathematica Japonica 43 No 1 (1996), 7–21.Google Scholar
  9. 9.
    N. Hamada, M. Deza. A characterization of {vυ+1+ε,vυ;t,q}-minihypers and its application to error-correcting codes and factorial design, J. Statist. Plann. Inference 22(1989), 323–336.Google Scholar
  10. 10.
    N. Hamada, Y. Watamori, The nonexistence of some ternary linear codes of dimension 6 and the bounds for n3(6, d), 1 ≤d ≤243, Mathematica Japonica, 43 No 3 (1996), 577–593.Google Scholar
  11. 11.
    R. Hill, Optimal Linear Codes, in: C. Mitchell ed., Proc. 2nd IMA Conference on Cryptography and Coding, Oxford Univ. Press, Oxford, 1992, 75–104.Google Scholar
  12. 12.
    R. Hill, I. Landjev, On the nonexistence of some quaternary codes, Applications of Finite Fields (ed. D. Gollmann), IMA Conference Series 59, Clarendon Press, Oxford, 1996, 85–98.Google Scholar
  13. 13.
    R. Hill, P. Lizak, Extensions of linear codes, Proc. Int. Symp. on Inf. Theory, Whistler, Canada, 1995, 345.Google Scholar
  14. 14.
    R. Hill, D.E. Newton, Optimal ternary linear codes, Designs, Codes and Cryptography2(1992), 137–157.Google Scholar
  15. 15.
    J.W.P. Hirschfeld, Projective geometries over finite fields, Clarendon Press, Oxford, 1979.Google Scholar
  16. 16.
    F.J. MacWilliams, N.J.A. Sloane, The theory of error-correcting codes, North-Holland, Amsterdam, 1977.Google Scholar
  17. 17.
    H.C.A. van Tilborg, The smallest length of binary 7-dimensional linear codes with prescribed minmum distance, Discrete Math.33(1981), 197–207.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Ivan N. Landjev
    • 1
  1. 1.Institute of MathematicsBulgarian Academy of SciencesSofiaBulgaria

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