Generalized Hamming Weights For Further Research

Let C be an [n,k] codes,the support of C,denated by Supp(C) {i:there exists a code-word(x1 ,x2, ...... ,Xn) 6 C, withxi 0 } For0 ≤r ≤ k , the r-th generalized Hamm-ing weight of C isdr(C) min {|supp(D)|:D is a subcodes of C of rank r} .while r=1,1-th generalized Hamm-ing weights of C is the Hamming weight of a codeword for the linear code.so generalized Hamming weights is the generalization for the Hamming weight. Now it is used in cryptography and corrspording field, specially, men study wiretap channel of type II with maintain secrecy corresponding. So many researcher study to the extensive and higher the generalized Hamming weight in the internal and the external. It is more difficult to work out the all generalized Hamming weight for the code, at present, the generalized Hamming weights are worked out with some codes in the most of works, or some lower and upper bounds are given with the generalized Hamming weights.In this paper, we will more deeply research on the base of the works, the first part, we give the summarize for the condition and the significance .The second part, we give the preparation knowledge to the whole paper; the third part, we research the lower and upper bound of the generalized Hamming weights for the linear codes; include D(r,n,k) bound, the finite sum representation of the lower and upper bound function of generalized Hamming weights for linear codes, generalized Griesmer bounds; the 4-th part, research the definition, the property of the r-th generalized weights for the non-linear codes and non-linear constant codes, and give the expression of the generalized weight of binary (n,M,d) non-linear codes; the 5-th part, research the weight hierarchy of linear codes and non-linear codes,for example , necessary condition and sufficient condition, the 6-th part, we research the expression of the r-th generalized Hamming weights of reseaval classes codes. Include:Direct sum code, Cartesian product code, Tensor product code, Extend Hamming code; and 7-th part, we summarize the total paper, and put forward some problem which should be solved in perhaps development direction in the future.In the paper, main results gather in the 3-rd,4-th,5-th,6-th part.