The 2-nd Generalized Hamming Weight of Double-Error Correcting Binary BCH Codes and Their Duals Codes

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.