| This paper generalizes the algebra essential conceptions of group, ring and field, introduces the development and current situations of cyclic code. The main research works and contributions of this paper are as follows:Firstly, this paper introduces concept of K-cyclic code and K-negacyclic code over Z4, defines the corresponding Gary map and Nechaev-Gray map, and gives the property of the Gary image of K-negacyclic code over Z4 and the Nechaev-Gray image of K-cyclic code of length which is an odd integer over Z4 respectively.Secondly, this paper introduces concept of cyclic code and negacyclic code over Z4, defines the corresponding Gary map and Nechaev-Gray map, and gives the property of the Gary image of negacyclic code over Z4 and the Nechaev-Gray image of cyclic code of length which is an odd integer over Z4 respectively. Then from polynomial represent we study the Nechaev-Gray image of a class of special linear code of length which is an odd integer over Z4 and give the generator of its Nechaev-Gray image.Finally, this paper introduces concept of quasi-cyclic code and quasi-negacyclic code over Zp, defines the corresponding Gary map, and gives the property of the Gary image of quasi-cyclic code over Zp and the Gary image of quasi-negacyclic code over Zp respectively. Then we study quasi-cyclic code and quasi-negacyclic code over Z 4, define the corresponding Gary map , shiftσm of Z4nm, negashiftνm of F22 nm and permutationπm, and give the property of the Gary image of quasi-negacyclic code over Z4 and theπmΦimage of quasi-cyclic code over Z4 respectively .The paper has three chapters together .The first chapter is introduction, which consists of the background of research and the original of this paper. The second chapter is basic knowledge, where we give concepts and results which we need in the later chapter. The last chapter is the main part of this paper, and we put forward some results on the binary image of K-cyclic code, cyclic code and quasi-cyclic code over Z4. |
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