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The 2-nd generalized Hamming weight of double-error correcting binary BCH codes and their dual codes

  • Habong Chung
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 539)

Abstract

The generalized Hamming weight of a linear code is a new notion of higher dimensional Hamming weights, first defined by V.K. Wei as follows: Let C be an [n, k] linear code and D be a subcode. The support of D is the cardinality of the set of not-always-zero bit positions of D. The r th generalized Hamming weight of C, denoted by d r (C), is defined as the minimum support of r-dimensional subcode of C. The first generalized Hamming weight, d1(C) is just the minimum Hamming distance of the code C. It was shown that the generalized Hamming weight hierarchy of a linear code completely characterizes the performance of the code on the type II wire-tap channel defined by Ozarow and Wyner.

In this paper, the second generalized Hamming weight of a double-error correcting BCH code and its dual code is derived. It is shown that d2(C) = 8 for all binary primitive double-error-correcting BCH codes. Also, we prove that the second generalized Hamming weight of [2 m – 1,2m]-dual BCH codes satisfies the Griesmer bound for m ≡ 1,2,3 (mod 4) and 0 (mod 12).

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References

  1. [1]
    V. K. Wei, “Generalized Hamming Weights for Linear Codes,” to appear in IEEE Transactions on Information Theory.Google Scholar
  2. [2]
    L. H. Ozarow and A. D. Wyner, “Wire-Tap Channel II,” AT&T Bell Labs. Technical Journal, vol. 63, pp. 2135–2157, 1984.Google Scholar
  3. [3]
    T. Kasami, “Weight Distributions of Bose-Chaudhuri-Hocquenghem Codes,” Proc. Conf. Combinatorial Mathematics and Its Applications,” R. C. Bose and T. A. Dowling, Eds. Chapel Hill, N.C.: University of North Carolina Press, 1968.Google Scholar
  4. [4]
    G. L. Feng, K. K. Tzeng, and V. K. Wei, “On the Generalized Hamming Weights of Several Classes of Cyclic Codes,” to appear in IEEE Transactions on Information Theory.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Habong Chung
    • 1
  1. 1.Department of Electrical and Computer EngineeringState University of New York at BuffaloBuffalo

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