Abstract
C is an [n, k, d] q linear code over \(\mathbb{F}_q \). And s(C) = n + 1 − k − d is the Singleton defect of C. AnMDS code C with s(C) = 0 has been studied extensively. Recently, a near-MDS code C with s(C) = s(C ⊥) = 1 is studied by many scholars, where C ⊥ denotes the dual code of C. This paper concentrates on the linear code C with s(C) = s(C ⊥) = 2, and the author calls it an NNMDS code. A series of iff conditions of NNMDS codes are presented. And the author gives an upper bound on length of NNMDS codes. In the last, some examples of NNMDS are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Ling and C. P. Xing, Coding Theory, A First Course, Cambridge University Press, 2004.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, NorthHolland, Amsterdam, 1977.
M. A. De Boer, Almost MDS codes, Des. Codes Cryptogr., 1996, 9: 143–155.
V. S. Pless, W. C. Huffman, and R. A. Brualdi, An introduction to algebraic codes, Handbook of Coding Theory, Volume I (ed. by V. S. Pless, W. C. Huffman), Elsevier, Amsterdam/Lausanne/New York/Oxford/Shannon/Singapore/Tokyo, 1998.
S. M. Dodunekov and I. N. Landgev, On near-MDS codes, J. Geom., 1995, 54: 30–43.
S. Henning, Algebraic Function Fields and Codes, Springer, Berlin, 1993.
S. M. Dodunekov and I. N. Landgev, Near-MDS codes over some small fields, Discr. Math., 2000, 213: 55–65.
S. Marcugini, A. Milani, and F. Pambianco, NMDS codes of maximal length over \(\mathbb{F}_q \), 8 ≤ q ≤ 11, IEEE Trans. Inform. Theory, 2002, 48: 963–966.
V. K. Wei, Generelized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 1991, 37: 1412–1418.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by Key Disciplines of Shanghai Municipality under Grant No. S30104.
This paper was recommended for publication by Editor Lei HU.
Rights and permissions
About this article
Cite this article
Tong, H. NNMDS codes. J Syst Sci Complex 25, 617–624 (2012). https://doi.org/10.1007/s11424-012-0041-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-012-0041-5