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 – 1,2m]-dual BCH codes satisfies the Griesmer bound for m ≡ 1,2,3 (mod 4) and 0 (mod 12).