Generalized Hamming Weights of Irreducible Cyclic Codes

The generalized Hamming weight (GHW) $d_r(C)$ of linear codes $C$ is a natural generalization of the minimum Hamming distance $d(C)(=d_1(C))$ and has become one of important research objects in coding theory since Wei's originary work [23] in 1991. In this paper two general formulas on $d_r(C)$ for irreducible cyclic codes are presented by using Gauss sums and the weight hierarchy $\{d_1(C), d_2(C), \ldots, d_k(C)\}$ $(k=\dim C)$ are completely determined for several cases.