Generalized Hamming Weights for Linear Codes

Error control codes are widely used to increase the reliability of transmission of information over various forms of communications channels. The Hamming weight of a codeword is the number of nonzero entries in the word; the weights of the words in a linear code determine the error-correcting capacity of the code. The rth generalized Hamming weight for a linear code C, denoted by dr(C), is the minimum of the support sizes for r-dimensional subcodes of C. For instance, d1(C) equals the traditional minimum Hamming weight of C. In 1991, Feng, Tzeng, and Wei proved that the second generalized Hamming weight d2(C) = 8 for all double-error correcting BCH(2m, 5) codes. We study d3(C) and higher Hamming weights for BCH(2m, 5) codes by a close examination of the words of weight 5.