Self-Dual Permutation Codes And M-Weights

This dissertation focuses on linear codes from two aspects: the existence of self-dual permutation codes over finite field and finite chain rings and the m-weight of linear codes over finite fields. It consists of three parts. The preface introduces the backgrounds and status quo on our two topics.In chapter one, generalizing the group codes, we first introduce permutation codes. In the thesis, we focus on the self-dual permutation codes with respect to a given non-degenerate symmetric bilinear form.Self-dual group code is a very important object in coding theory, it has been applied in many other mathematical branches, such as lattice theory. Moreover, some famous codes can be regarded as self-dual group codes, for instance, the binary extended Golay code is a self-dual group code over symmetric group of degree 4.We generalize the group action on itself to the group actions on general sets and introduce the permutation code, which is a sub-module of a permutation module of a group algebra. With the elementary knowledge of coding, group representation and group action, we get some existent and un-existent conditions on the existence of self-dual permutation codes in the case that the group action is transitive. Especially, when the group is the direct product of a 2-group and a 2'-group, we give a necessary and sufficient condition for the existence of the self-dual transitive permutation codes.Furthermore, we generalize the permutation codes over finite fields to finite chain rings. As above, we pay attention to the existence of the self-dual permutation codes over finite chain rings and get the similar results.In chapter two, we discuss the m-weight of linear codes over finite field. m-weight is a new generalization of the classical Hamming weight and a new parameter of the linear codes over finite field. We generalize a classical exis-tence theorem, the Plotkin bound and the Sphere Packing bound on Hamming weight to m-weight.