\mathbb Z2\mathbb Z2[u] ?Cyclic and Constacyclic Codes.

Following the very recent studies on \mathbb Z2\mathbb Z4 -additive codes, \mathbb Z2\mathbb Z2[u] -linear codes have been introduced by Aydogdu et al. In this paper, we introduce and study the algebraic structure of cyclic, constacyclic codes and their duals over the R -module \mathbb {Z}_{2}^\alpha R^\beta where R=\mathbb {Z}_{2}+u\mathbb {Z}_{2}=\left \{{0,1,u,u+1}\right \} is the ring with four elements and u^2=0 . We determine the generating independent sets and the types and sizes of both such codes and their duals. Finally, we present a bound and an optimal family of codes attaining this bound and also give some illustrative examples of binary codes that have good parameters which are obtained from the cyclic codes in \mathbb Z_2^\alpha R^\beta . [ABSTRACT FROM AUTHOR]