One generator quasi-cyclic codes over

In this paper, we study quasi-cyclic codes over the ring R = F 2 + u F 2 = { 0,1 , u , u + 1 } where u 2 = 0 . By exploring their structure, we determine the type of one generator quasi-cyclic codes over R and the size by giving a minimal spanning set. We also determine the rank and introduce a lower bound for the minimum distance of free quasi-cyclic codes over R . We include some examples of quasi-cyclic codes of various lengths over R . In particular, we obtain a family of 2-quasi-cyclic codes from cyclic codes over the ring F 2 + u F 2 + v F 2 + uv F 2 . Finally, using the Gray map we obtain a family of optimal binary linear codes as the images of quasi-cyclic codes over R.