Cyclic Codes And Quadratic Residue Codes Over Ring F₂+uF₂

In 1994,Hammons et.al proved that some important binary nonlinear codes such as Kerdock, Preparata, and Goethals codes are images under the Gray map of linear codes over Z₄. Since then people all around the world are interested in studying codes over finite rings with four elements. Many important results have been obtained.In 1996 and 1997, Pless, Qian and Sole gave the structures of the liner cyclic codes over ring Z₄. Motivated by the works of these people, we study linear cyclic codes over the ring F2 + uF2 = {0, 1,u,u+ 1|u² = 0} on the basis of codes over the ring Z₄.Firstly, we review some definitions and several basic results of ring F₂+ uF₂ ,then we construct Galois rings over F₂ + uF₂.Next, we focus on our attention to the structure of cyclic codes over F₂ + uF₂ , and we prove that any F₂ + uF₂ -cyclic code C has generators of the form (fh,ufg), where fgh = xⁿ - 1 over F₂ + uF₂ , and |C| = 4deg(g)2deg(h). After that,we discuss some properties of the dual codes over F₂ + uF₂ ,we also prove that the dual code C^⊥ = (g*h*,uf*g*).Finally, we show that idempotent generators exist on the certain condition over F₂ + uF₂ through studying the idempotent generators of cyclic codes over F₂ + uF₂ . A particularly interesting family of F₂ + uF₂— cyclic codes are quadratic residue codes. we define such codes in term of their idempotent generators and show that these codes have many good properties.