Abstract
We characterize the linear codes with parameters [2q 2−q−1,3,2q 2−3q]q and [2q 2−q−2,3,2q 2−3q−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 n 5(4, d) for d=166, 167, 168, 169, 170, 171. There remain 16 d's for which the exact value of n 5 (4, d) is not known.
This research was partially supported by the Bulgarian NSF under Contract MM-502/95
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
S.M. Ball, On sets of points in finite planes, PhD Thesis, University of Sussex, 1994.
I. Boukliev, S. Kapralov, T. Maruta, M. Fukui, Optimal linear codes of dimension 4 over \(\mathbb{F}_5\), IEEE Trans, Inf. Theory, to appear.
C.D. Baumert, M.J. McEliece, A Note on the Griesmer Bound, IEEE Trans. Inf. Theory IT-19(1973), 134–135.
M. van Eupen, Four New Optimal Ternary Linear Codes, IEEE Trans. Inf. Theory IT-40(1994), 193.
P. Greenough, R. Hill, Optimal linear codes over GF(4), Discrete Math. 125(1994), 187–199.
J.H. Griesmer, A Bound for Error-Correcting Codes, IBM J. Res. Develop.4(1960), 532–542.
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.
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.
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.
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.
R. Hill, Optimal Linear Codes, in: C. Mitchell ed., Proc. 2nd IMA Conference on Cryptography and Coding, Oxford Univ. Press, Oxford, 1992, 75–104.
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.
R. Hill, P. Lizak, Extensions of linear codes, Proc. Int. Symp. on Inf. Theory, Whistler, Canada, 1995, 345.
R. Hill, D.E. Newton, Optimal ternary linear codes, Designs, Codes and Cryptography2(1992), 137–157.
J.W.P. Hirschfeld, Projective geometries over finite fields, Clarendon Press, Oxford, 1979.
F.J. MacWilliams, N.J.A. Sloane, The theory of error-correcting codes, North-Holland, Amsterdam, 1977.
H.C.A. van Tilborg, The smallest length of binary 7-dimensional linear codes with prescribed minmum distance, Discrete Math.33(1981), 197–207.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Landjev, I.N. (1997). Optimal linear codes of dimension 4 over GF(5). In: Mora, T., Mattson, H. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1997. Lecture Notes in Computer Science, vol 1255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63163-1_17
Download citation
DOI: https://doi.org/10.1007/3-540-63163-1_17
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63163-7
Online ISBN: 978-3-540-69193-8
eBook Packages: Springer Book Archive