Cyclic Codes And Constacyclic Codes Over Z₄+uZ₄+u²Z₄

With a more deeply study in encoding over finite chain rings,cyclic codes and constacyclic codes over several finite rings which are not finite chain rings also have been considered by some authors.In this paper,cyclic codes for an arbitrary length over the ring Z₄+uZ₄+u²Z₄ and constacyclic codes and negacyclic codes of length 2k over the ring Z₄+uZ₄+u²Z₄ are studied.The detail are given as follows:1、In this paper,cyclic codes over R of an arbitrary length is studied.A Gray map from R to Z4 is defmed.It is proved that the Gray map of a cyclic code over R of length n is a quasi-cyclic code of length 3n and index 3 over Z4.2、Isomorphic mapping is constructed, and proves that constacyclic codes over R of an odd length and cyclic codes are isomorphic. A Gray map from R to Z4 is defined.It is proved that the Gray map of a （1+2u2）-cyclic code over R of length n is a quasi-cyclic code of length 4n and index 4 over Z4.3、We study negacyclic codes of length 2k over the ring Z₄+uZ₄+u²Z₄.This aspect makes the study a step better.It provides a wealth of resources to construct good codes.