Researches On Mac Williams Identities And Depth Spectrum Of Linear Codes Over The Ring Z₄+uZ₄

With the development of coding theory,the linear codes over finite rings have become an important research direction.This paper is devoted to the research on linear codes over the ring Z₄+uZ₄,where u²=0.We mainly study the Mac Williams identities on the Hermitian inner product and on the-ply weight enumerator.We also consider the depth distribution and depth spectrum of linear codes over this type of the ring.Meanwhile,we give a construct method on the self-dual codes with optimal parameters over the ring Z₄.Main results are organized as follows:In Chapter 3,we discuss the Mac Williams identities for linear codes on the Hermitian inner product over the ring Z₄+uZ₄,where u²=0.Firstly,the definition of Lee weight,Gray map,Hermitian inner product and Hermitian dual for linear codes are given.Then the complete weight enumerator,the symmetrical weight enumerator,the Lee weight enu-merator and the Hamming weight enumerator of linear codes over the ring are discussed,respectively.We obtain the Mac Williams identities for these weight enumerators on the Hermitian inner product.Finally,we give an example to illustrate the obtained results.In Chapter 4,we study the Mac Williams identities of linear codes on the joint weight enumerator and on the ?-ply weight enumerator over the ring Z₄+uZ₄,where u²=0.The joint weight enumerator and the ?-ply weight enumerator are defined.We obtain the Mac Williams identities of these two weight enumerator.In Chapter 5,we study the depth distribution and depth spectrum of linear codes over the ring Z₄+uZ₄,where u²=0.We firstly introduce the definition of differential operation and the depth of linear codes over the ring.A recursive algorithm for the depth of the codewords are given.Finally,we obtain the depth distribution and the depth spectrum of linear codes over the ring Z₄+uZ₄,.In Chapter 6,as the last part of this paper,we study the costruction of self-dual codes over the ring Z₄.The Gary map from(Z₄+uZ₄)ⁿ to Z₄²ⁿis given,where u²=1.Based on the self-dual codes over the ring Z₄+uZ₄,we construct the self-dual codes with optimal parameters over the ring Z₄.