CYCLIC CODES AND THE WEIGHT ENUMERATOR OF LINEAR CODES OVER F2 + vF2 v2F2

In this paper, we mainly study cyclic codes and the weight enumerator of linear codes over non-principal ideal ring R = F-2 + vF(2) + v(2)F(2). We first introduce an isometric Gray map Phi : (R-n, Lee distance) to (F-2(3n), Hamming distance), and investigate structures and properties of linear codes over R and their Gray images. Then we define and discuss several weight enumerators of linear codes over R, we also obtain a MacWilliams identity of linear codes with respect to Lee weight. Finally, we prove that cyclic codes and their dual codes over R are principally generated, and give their generator polynomials, respectively. We show that the Gray image of each cyclic code C over R, is a binary linear quasi-cyclic code of index 3 and length 3n. Moreover, we obtain some best-known binary linear quasi-cyclic codes by the Gray map.