On Study Of Linear Codes Over Ring F₂+uF₂

In this thesis, properties of linear codes, (l+u)-cyclic and cyclic codes over discussed. The main contents are as follows:Firstly, Based on the generator matrix of linear codes C over F2+uF2 and Gray map, the generator matrixes of the dual codes C~⊥ and the Gray images φ(c) of the linear codes C are presented. Besides, the proposition that linear codes φ(C~⊥) are the dual codes of φ(c) is showed.Secondly, the concept of Lee weight distribution over the foresaid ring is defined. Then the relationship between Gray image of the linear codes and dual codes is employed to achieve the Mac Willimas identity of all kinds of weight distribution about the linear codes and dual codes over this ring.Thirdly, (1+u)-cyclic codes over this ring are defined. Based on the relationship between (1+u)-cyclic codes and cyclic codes, we discuss the structure of (l+u)-cyclic codes.Fourthly, that Gray images of (l+u)-cyclic codes over F2+uF2 are still cyclic codes over F2 is proved. Furthermore, with the generator polynomial of (l+u)-cyclic codes, the generator polynomial of Gray images of (l+u)-cyclic codes is obtained.Fifthly, with Nechaeve map, the Nechaeve-Graymap is defined, and it is proved that the binary image under the Nechaeve-Graymap of the cyclic codes over this ring is cyclic codes. In doing so, relationships of cyclic codes with their residue codes and torsion codes are further discussedSixthly, the concept of the depth of the codeword is firstly defined and then the algorithm of the depth of codeword is obtained. This part ends with the discussion on the depth distribution of linear codes and cyclic over F2+uF2.