We deal with ACD (additive complementary dual) codes and cyclic codes over the mixed alphabet \({\mathbb {Z}}_2{\mathcal {R}}_k\) , where \({\mathcal {R}}_{k}:= {\mathbb {Z}}_{2}[y]/\langle y^{k}\rangle \) , \(k\ge 2\) . First, we establish a few criteria for \({\mathbb {Z}}_{2}{\mathcal {R}}_{k}\) -additive codes to be ACD codes. We also present conditions for separable codes and a class of additive codes (not necessarily separable) over \({\mathbb {Z}}_{2}{\mathcal {R}}_{k}\) to be ACD codes that are both necessary and sufficient. With the help of a Gray map, binary LCD codes are obtained from \({\mathbb {Z}}_{2}{\mathcal {R}}_{k}\) -additive codes. Moreover, we describe the generator polynomial of the dual of an additive cyclic code over \({\mathbb {Z}}_2{\mathcal {R}}_k\) . Finally, we construct examples of optimal binary codes as the Gray image of certain additive cyclic codes over \({\mathbb {Z}}_{2}{\mathcal {R}}_{k}\) .
