ACD codes and cyclic codes over Z2Rk

We deal with ACD (additive complementary dual) codes and cyclic codes over the mixed alphabet Z2Rk, where Rk := Z2[ y]/yk , k = 2. First, we establish a few criteria for Z2Rk -additive codes to be ACD codes. We also present conditions for separable codes and a class of additive codes (not necessarily separable) over Z2Rk to be ACD codes that are both necessary and sufficient. With the help of a Gray map, binary LCD codes are obtained from Z2Rk -additive codes. Moreover, we describe the generator polynomial of the dual of an additive cyclic code over Z2Rk. Finally, we construct examples of optimal binary codes as the Gray image of certain additive cyclic codes over Z2Rk.