Abstract
The weight hierarchy of a linear [n; k; q] code C over GF(q) is the sequence (d 1, d 2, ···, d k ) where d r is the smallest support of any r-dimensional subcode of C. “Determining all possible weight hierarchies of general linear codes” is a basic theoretical issue and has important scientific significance in communication system. However, it is impossible for q-ary linear codes of dimension k when q and k are slightly larger, then a reasonable formulation of the problem is modified as: “Determine almost all weight hierarchies of general q-ary linear codes of dimension k”. In this paper, based on the finite projective geometry method, the authors study q-ary linear codes of dimension 5 in class IV, and find new necessary conditions of their weight hierarchies, and classify their weight hierarchies into 6 subclasses. The authors also develop and improve the method of the subspace set, thus determine almost all weight hierarchies of 5-dimensional linear codes in class IV. It opens the way to determine the weight hierarchies of the rest two of 5-dimensional codes (classes III and VI), and break through the difficulties. Furthermore, the new necessary conditions show that original necessary conditions of the weight hierarchies of k-dimensional codes were not enough (not most tight nor best), so, it is important to excogitate further new necessary conditions for attacking and solving the k-dimensional problem.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Wei V K, Generalized Hamming weight for linear codes, IEEE Trans. Inform. Theory, 1991, 37: 1412–1418.
Helleseth T, Kløve T, and Ytrehus ø, Generalized Hamming weights of linear codes, IEEE Trans. Inform. Theory, 1992, 38(3): 1133–1140.
Kløve T, Minimum support weights of binary codes, IEEE Trans. Inform. Theory, 1993, 39: 648–654.
Chen W D and Kløve T, The weight hierarchies of q-ary codes of dimension 4, IEEE Trans. Inform. Theory, 1996, 42: 2265–2272.
Chen W D and Kløve T, Bounds on the weight hierarchies of linear codes of dimension 4, IEEE Trans. Inform. Theory, 1997, 43(6): 2047–2054.
Chen W D and Kløve T, Weight hierarchies of extremal non-chain binary codes of dimension 4, IEEE Trans. Inform. Theory, 1999, 45: 276–281.
Wang L J, Xia Y B, and Chen W D, Weight hierarchies of ternary linear codes of dimension 4 satisfying the break-chain condition, Journal of Systems Science and Mathematical Sciences, 2009, 29(6): 742–749 (in Chinese).
Chen W D and Kløve T, Weight hierarchies of linear codes of dimension 3, J. Statist. Plann. Inference, 2001, 94(2): 167–179.
Chen W D and Kløve T, Weight hierarchies of linear codes satisfying the almost chain condition, Sci. China, 2003, 46(3): 175–186.
Hu G X and Chen W D, The weight hierarchies of q-ary linear codes of dimension 4, Discrete Math., 2010, 310(24): 3528–3536.
Encheva S and Kløve T, Codes satisfying the chain condition, IEEE Trans Inform Theory, 1994, 40(1): 175–180.
Wang L J and Chen W D, The classification and determination on weight hierarchies of q-ary linear codes of dimension 5, Journal of Systems Science and Mathematical Sciences, 2011, 31(4): 402–413 (in Chinese).
Wang L J and Chen W D, Determination on a class of weight hierarchies of q-ary linear codes of dimension 5, Chinese Sci. Bull., 2011, 56(25): 2150–2155 (in Chinese).
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by the National Natural Science Foundation of China under Grant No. 11171366, “the Fundamental Research Funds for the Central Universities”, South-Central University for Nationalities under Grant No. CZY12014.
This paper was recommended for publication by Editor DENG Yingpu.
Rights and permissions
About this article
Cite this article
Wang, L., Chen, W. The determination on weight hierarchies of q-ary linear codes of dimension 5 in class IV. J Syst Sci Complex 29, 243–258 (2016). https://doi.org/10.1007/s11424-015-4072-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-015-4072-6