There are four commutative unital rings of order four. Among them, we consider the ring \(R=\mathbb {F}_{2}+u\mathbb {F}_{2}= \left\{ 0,1,u,\bar{u}=u+1\right\} \) where \(u^2=0\) , which is a commutative ring with characteristic 2. In this paper, we study linear complementary dual (LCD) codes over the ring R. We first define \({\text{ LCD }}[n,k]_{R}\) , which denotes the maximum of possible values of d among free [n, k, d] LCD codes over R, and obtain a Griesmer type bound for linear codes over R. We get an upper bound for \({\text{ LCD }}[n,2]_{R}\) , and further show that \({\text{ LCD }}[n,2]_{R}\) with the exception of \(n\equiv 0, -1 \; (\textrm{mod} \; \) 6) meets the upper bound exactly. For \(k=3\) , we also get an upper bound for \({\text{ LCD }}[n,3]_{R}\) . Then we show that \({\text{ LCD }}[n,3]_{R}\) meets the upper bound exactly for \(n\equiv 3, 5\) (mod 7). We also derive bounds of \({\text{ LCD }}[n,k]_{R}\) for \(k=4, 5\) from the binary cases. Furthermore, we obtain the exact value of \({\text{ LCD }}[n,n-i]_{R}\) for i greater than or equal to two using the sphere packing bound.
