Cyclic codes over the ring $$F_2+uF_2+vF_2$$ F 2 + u F 2 + v F 2

Abstract

In this paper, we study linear and cyclic codes over the ring $$F_2+uF_2+vF_2$$ . The ring $$F_2+uF_2+vF_2$$ is the smallest non-Frobenius ring. We characterize the structure of cyclic codes over the ring $$R=F_2+uF_2+vF_2$$ 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 $$|C||C^\bot |= |R|^n$$ 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 $$F_2+uF_2+vF_2$$ 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 $$F_2$$ from cyclic codes over the ring $$F_2+uF_2+vF_2$$ .