Z_2k+1-Quasicyclic Code

With the development of information techniques, error-correcting code becomesa widely used standard technology, no longer an academic subject. At the same time,researchers, especially algebraic experts, pay more attention to error-correcting code,making it attain great progress. Cyclic codes is one of the most important subclassesof error-correcting code. At first , people were interested in its extrinsic characterthat every code of cyclic codes is also a code after cyclic transposition. It is exactlythe property that brings us much convenience to encode and decode the cyclic code.In later practice, people also find many specialities from the view of cyclic group,algebraic structure and error-correcting property control. In this thesis, we study theextending cyclic codes :quasicyclic codes.The thesis generalizes the essential conceptions of algebra, the main researchworks and contributions of this thesis are as follows:Firstly, this thesis introduces the basic conceptions of error-correcting code, es-pecially introduces the conceptions of cyclic codes and its encoding system in order tostudy the specialities of cyclic codes further, which provides us important theorizationto study the quasicyclic codes over Z2k+1 .Secondly, the thesis introduces two important definitions: Gray mapping andhomogeneous weight.Thirdly, the thesis extends the cyclic codes and the quasicyclic codes to the finitering Z2k+1 ,examining the forms of the generators. The thesis also brings up anothernew notion: the generalize quasicyclic codes and obtains some better codes using theGray mapping.