Abstract
Finite projective geometry method is effectively used to study the relative generalized Hamming weights of 4-dimensional linear codes, which are divided into 9 classes in order to get much more information about the relative generalized Hamming weights, and part of the relative generalized Hamming weights of a 4-dimensional linear code with a 1-dimensional subcode are determined.
Similar content being viewed by others
References
Y. Luo, C. Mitrpant, A. J. Han Vinck, and K. F. Chen, Some new characters on the wire-tap channel of type II, IEEE Trans. Inform. Theory, 2005, 51(3): 1222–1229.
L. H. Ozarow and A. D. Wyner, Wire-tap channel II, AT & T Bell Laboratories Technical Journal, 1984, 63(10): 2135–2157.
V. K. Wei, Generalized Hamming weight for linear codes, IEEE Trans. Inform. Theory, 1991, 37(5): 1412–1418.
W. D. Chen and T. Kløve, The weight hierarchies of q-ary codes of dimension 4, IEEE Trans. Inform. Theory, 1996, 42(6): 2265–2272.
T. Helleseth, T. Kløve and Ø. Ytrehus, Generalized Hamming weights of linear codes, IEEE Trans. Inform. Theory, 1992, 38(3): 1133–1140.
W. D. Chen and T. Kløve, Weight hierarchies of extremal non-chain binary codes of dimension 4, IEEE Trans. Inform. Theory, 1999, 45(1): 276–281.
W. D. Chen and T. Kløve, Weight hierarchies of linear codes of dimension 3, J. Statist. Plann. Inference, 2001, 94(2): 167–179.
W. D. Chen and T. Kløve, Finite projective geometries and classification of the weight hierarchies of codes (I), Acta Mathematica Sinica, English Series, 2004, 20(2): 333–348.
Z. H. Liu, W. D. Chen and X. W. Wu, On the relative generalized Hamming weights of linear codes and their subcodes, SIAM Journal on Discrete Mathematics, 2010, 24(4): 1234–1241.
Z. H. Liu, W. D. Chen, and Y. Luo, The relative generalized Hamming weight of linear q-ary codes and their subcodes, Des. Codes Cryptogr., 2008, 48(2): 111–123.
M. A. Tsfasman and S. Vladuts, Geometric approach to higher weights, IEEE Trans. Inform. Theory, 1995, 41(6): 1564–1588.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by the National Natural Science Foundation of China under Grant Nos. 11171366 and 61170257, and the Special Training Program of Beijing Institute of Technology.
This paper was recommended for publication by Editor Xiao-Shan GAO.
Rights and permissions
About this article
Cite this article
Liu, Z., Chen, W. On the relative generalized hamming weights of a 4-dimensional linear code and a subcode with dimension one. J Syst Sci Complex 25, 821–832 (2012). https://doi.org/10.1007/s11424-012-0192-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-012-0192-4