Study On Quaternary Linear Codes And Their Generalization

The theory of quaternary linear codes is one of the important subjects in recent ten years, which is closely related with the study on constructions and properties of binary nonlinear codes. In this paper, several questions about quaternary codes and their generalization are discussed. Solutions presented in this thesis can be summarized as follows:- A sufficient and necessary condition is given for the negacyclic codes whose binary images under the Gray map being cyclic, and the binary images are determined when the condition is satisfied.- The trace representation and the 2-adic representation for the dual of the quaternary Goethals code are given, and its binary image is proved to be the formal dual of the binary Goethals code.- The generalized MacWilliams identities are given for the support weight distribution of a quaternary linear code with type 4k and that of its dual code.- All of the cyclic codes over the Galois ring GR(4m), and the bases for GR(4m) over Z4 are determined, such that with respect to the bases, the 4-ary image of the cyclic codes are still cyclic.- The Hensel's Lemma and Lift over the polynomial ring Zpe [x] are discussed, and it is proved that xn - 1, where (n,p) = 1, can be uniquely factored. Furthermore, all of the cyclic codes over Zpe are determined.- The Berlekamp-Massey algorithm over the Galois ring GR(pe,.Pem) is given.