Polycyclic Codes Over Finite Rings

Polycyclic code is a special class of linear codes. For its rich algebra structures and efficient decoding algorithm, polycyclic codes over finite chain rings have been gained comprehensive attention in recent years. Cyclic, negacyclic and constacyclic codes are important subclasses of polycyclic codes. In this paper, we mainly research the cyclic and negacyclic codes over finite rings. We also analyse its dual and self-dual codes.In Chapter 1, we mainly introduce the background and the research status of polycyclic codes over finite rings.In Chapter 2, we give some well-known results on coding theory.In Chapter 3, we mainly research the Mac Williams transforms of the distance and weight distribution of q-ary codes. A new expression of the values Kk(i) of Krawtchouk polynomial is given using the primitive p-th root of unity. Then the distance distribution of q-ary codes are obtained by the expression of Kk(i). The Mac Williams transform of the weight distribution of a q-ary code C—υ is given. Then the weight distribution of a q-ary code are obtained by the Mac Williams transform.In Chapter 4, we discuss the cyclic codes of length 2·ps over Galois ring GR(p2,m). We derive a method to represent the negacyclic codes of length ps over GR(p2,m) first. Then the dual and self-dual codes of negacyclic codes of length ps over GR(p2,m) are obtained. As an example, we discuss self-dual ideals over ring GR(32)[u]/<u3k+1>, where p= 3,m=1 and k={1,2,3}. Finally, by the Chinese remainder theorem, the representation of cyclic codes of length 2·ps over GR(p2,m) is given. As an example, the representation of cyclic codes over ring GR(p,m) is discussed, where p=3,m=1,k=1.In Chapter 5, we study the dual codes of cyclic codes of length ps over the finite ring Fpm+uFpm. We give a method to represent the dual codes C’⊥ of cyclic codes C’ of length ps over finite ring R’= Fpm+uFpm The number of codeswords of C’⊥ is obtained. As an example, we give the representation of the cyclic codes of length 2 over ring R’and its dual codes, where p=2,m=1,s=1.