F₂+uF₂-Quasicyclic Code

With the development of information techniques, error-correcting codes have become a widely used standard technology, no longer an academic subject. Many researchers, especially algebraists, pay more attention to error-correcting codes. Cyclic code is one of the most important subclasses of error-correcting codes. At first, researchers were interested in its extrinsic character that every code of cyclic codes is also a code after cyclic transposition. This property brings us much convenience to encode and decode the cyclic code. In later practice, researchers also find many specialities in view of cyclic group, algebraic structure and error-correcting property control. Quasicyclic codes are the extention of cyclic codes and are better than cyclic codes in sense of correcting errors.Based on studying and brief listing some essential conceptions of abstract algebra, such as group, ring, field and module, we introduce the background and current development of error-correcting codes. The main contents of this thesis are as follows:Firstly, this thesis introduces the basic conceptions of error-correcting codes and its properties, especially the conceptions of cyclic codes and its encoding system in order to study the specialities of cyclic codes further, which provides us an important theorization to study the quasicyclic codes over F₂ +uF₂. Secondly, the thesis introduces the structure of the ring F₂ + uF₂. Comparing with Z₄ and F₄ Galois field , we give the operations and properties of the ring F₂ + uF₂. Then introduce linear codes and cyclic codes on F₂ + uF₂ .Thirdly, the thesis extends the cyclic codes and the quasicyclic codes to the finite ring F₂ + uF₂, presenting the forms of the generators and bringing up a notion of generalized quasicyclic codes. Finally we find some better codes using the Gray mapping.