On The Generalized Hamming Weights Of Some Product Codes

The rth generalized Hamming weight of a linear code is the minimum support size of any r—dimensional subcode defined by V.K.Wei in 1991. It has been found useful in the studies of cryptography including the wire-tap channel of type II, and t-resilient functions. Many interesting results for binary codes have been obtained on this topic. In this thesis, we consider the q-ary codes. We determine the generalized Hamming weights of MDS codes, the product of two MDS codes and the product of a MDS code and a first-order Reed-Muller code over the field F_q in terms of those of component codes, where q is a power of a prime number.Firstly, we give the general expression of the generalized Hamming weights of the q-ary MDS code. Then we derive an equivalent condition of MDS code.Secondly, noticing that the q-ary MDS code satisfies the chain conditions, we derive the generalized Hamming weights of the product of two MDS codes directly in terms of those of component codes.Finally, we calculate the generalized Hamming weights of the product of a MDS code and a first-order Reed-Muller code.