Coding Over Finite Chain Rings

The finite chain ringF₂+uF₂shares some good properties of the Galois fieldF4and the ringZ₄. There has been much research work on the coding over the finitechain ring. Some persons like Bonnecaze[12], Dougherty[13]and Abualrub[23]obtaingood results.In this correspondence, we mainly study the linear codes, cyclic codes and quasicyclic codes overF₂+uF₂.On the base of Bonnecaze’s[12]study, we complement and prove some conclusionsabout linear codes and cyclic codes. Most results of quasi cyclic codes are too complex,which is on the base of Grobo¨ner theory. We study the structure and enumeration ofquasi cyclic codes with module theory, and consider quasi cyclic codes as submodulesof S^m, where S=R[x]/（xⁿ-1）. We obtain that the quasi cyclic codes C withlength mn can be written as the direct sum of some irreducible cyclic submodules,What’s more, the number of irreducible cyclicsubmodules is uniquely determined by C.