On cosets of weight 4 of binary BCH codes with minimum distance 8 and exponential sums

被引：2
作者：
Zinoviev V.A. [1 ]
Helleseth T. [2 ]
Charpin P. [3 ]
机构：
[1] Institute for Information Transmission Problems, RAS, Moscow
[2] Department of Informatics, University of Bergen, Bergen
[3] INRIA, Domaine de Voluceau-Rocquencourt
来源：
Problems of Information Transmission | 2005年 / 41卷 / 4期
基金：
俄罗斯基础研究基金会;
关键词：
System Theory; Minimum Distance; Weight Distribution; Finite Field; Exact Expression;
D O I：
10.1007/s11122-006-0003-4
中图分类号：
学科分类号：
摘要：
We study coset weight distributions of binary primitive (narrow-sense) BCH codes of length n = 2m (m odd) with minimum distance 8. In the previous paper [1], we described coset weight distributions of such codes for cosets of weight j = 1,2,3,5,6. Here we obtain exact expressions for the number of codewords of weight 4 in terms of exponential sums of three types, in particular, cubic sums and Kloosterman sums. This allows us to find the coset distribution of binary primitive (narrow-sense) BCH codes with minimum distance 8 and also to obtain some new results on Kloosterman sums over finite fields of characteristic 2. © 2005 Pleiades Publishing, Inc.
引用
收藏
页码：331 / 348
页数：17
相关论文
共 18 条
  • [1] Charpin P., Zinoviev V.A., On Coset Weight Distributions of the 3-Error-Correcting BCH Codes, SIAM J. Discrete Math., 10, 1, pp. 128-145, (1997)
  • [2] van der Geer G., van der Vlugt M., The Coset Weight Distributions of Certain BCH Codes and a Family of Curves, Enseign. Math., 48, 1-2, pp. 3-21, (2002)
  • [3] Carlitz L., Kloosterman Sums and Finite Field Extensions, Acta Arith., 15, 2, pp. 179-193, (1969)
  • [4] Charpin P., Helleseth T., Zinoviev V.A., On Binary BCH Codes with Minimum Distance 8 and Kloosterman Sums, Proc. 9th Int. Workshop on Algebraic and Combinatorial Coding Theory, pp. 90-95, (2004)
  • [5] Charpin P., Helleseth T., Zinoviev V.A., The Divisibility Modulo 24 of Kloosterman Sums on GF (2<sup>m</sup>), m Odd, J. Combin. Theory, Ser. A
  • [6] Charpin P., Helleseth T., Zinoviev V.A., The Coset Distribution of Triple-Error-Correcting Binary Primitive BCH Codes, IEEE Trans. Inform. Theory
  • [7] Helleseth T., Zinoviev V.A., On ℤ<sub>4</sub>-Linear Goethals Codes and Kloosterman Sums, Des. Codes Cryptogr., 17, 1-3, pp. 246-262, (1999)
  • [8] Helleseth T., Zinoviev V.A., On CosetWeight Distributions of the ℤ<sub>4</sub>-Linear Goethals Codes, IEEE Trans. Inf. Theory, 49, 5, pp. 1758-1772, (2001)
  • [9] Helleseth T., Zinoviev V.A., On a New Identity for Kloosterman Sums and Nonlinear System of Equations over Finite Fields of Characteristic 2, Discrete Math., 274, 1-3, pp. 109-124, (2004)
  • [10] MacWilliams F.J., Sloane N.J.A., The Theory of Error-Correcting Codes, (1977)
12