One generator quasi-cyclic codes over F2+uF2

In this paper, we study quasi-cyclic codes over the ring R=F2+uF2={0,1,u,u+1} where u2=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 F2+uF2+vF2+uvF2. Finally, using the Gray map we obtain a family of optimal binary linear codes as the images of quasi-cyclic codes over R.