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

## 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, *d*_{1}(*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 *d*_{2}(*C*) = 8 for all binary primitive double-error-correcting BCH codes. Also, we prove that the second generalized Hamming weight of [2^{ m } – 1,2*m*]-dual BCH codes satisfies the Griesmer bound for *m* ≡ 1,2,3 (mod 4) and 0 (mod 12).

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]V. K. Wei, “Generalized Hamming Weights for Linear Codes,” to appear in
*IEEE Transactions on Information Theory*.Google Scholar - [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]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]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