Research On Cyclic Code And Constacyclic Code Over Ring F₂+uF₂

At present, with the successive development of production technique and the successive deep-going researches on error-correcting coding theory and theory of stream ciphers over finite rings have not only important theoretical significance but also important practical value.In recent decade, researches on error-correcting coding theory over finite rings have drawn intensive attention in the field of error-correcting coding theory. The ring F₂+uF₂ is a four-element ring which shares some good properties of both Z₄ and F₄.At first, the ring F₂ + uF₂ is mostly used to construct modular and lattices. Now there are many papers on the coding theory over the ring F₂ + uF₂. In this paper, we discuss the poperties of (a(x)b(x))₂ⁿ+ u(a(x))₂ⁿ cyclic code over the ring F₂ +uF₂. And we study the structure of the dual of constacyclic code over the ring F₂ + uF₂, whose length is even. The details are given as follows:1. We discuss the properties of (a(x)(b(x) + u))_Rⁿ cyclic code over F₂+uF₂ and the properties of it's image of Nechaev-Gray map.2. We discuss the properties of (a(x)b(x))₂ⁿ+u(a(x))₂ⁿ cyclic code over F₂+uF₂ and the properties of it's image of Nechaev-Gray map.3. We give the sufficient and necessary condition for a cyclic code C of odd length n to be self-dual, where C is of type C = C₁+ uC₂ with C₁,C₂ are binary linear cyclic codes of odd length n respectively.4. We give the structure of the dual of the constacyclic codes over F₂+uF₂ for arbitrary even length, and determine the number of cyclic codes for a given length.