In this paper, we study linear and cyclic codes over the ring F2 + uF2 + vF2. The ring F2 + uF2 + vF2 is the smallest non-Frobenius ring. We characterize the structure of cyclic codes over the ring R = F2 + uF2 + vF2 using of the work Abualrub and Saip (Des Codes Cryptogr 42:273–287, 2007).We study the rank and dual of cyclic codes of odd length over this ring. Specially, we show that the equation |C||C⊥| = |R|n does not hold in general for a cyclic code C of length n over this ring. We also obtain some optimal binary codes as the images of cyclic codes over the ring F2 + uF2 + vF2 under a Gray map, which maps Lee weights to Hamming weights. Finally, we give a condition for cyclic codes over R that contains its dual and find quantum codes over F2 from cyclic codes over the ring F2 + uF2 + vF2.
