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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]

In this paper, we are interested in finding an algebraic structure of conjucyclic codes of length n over the finite field F 4. We show that conjucyclic codes of length n over F 4 are related to binary cyclic codes of length 2 n and show that there is a canonical bijective correspondence between the two sets. We illustrate how the factorization of the polynomial x 2 n + 1 plays a critical role in each setting. Moreover, we construct the generator and parity check matrices of conjucyclic codes of length n over F 4. [ABSTRACT FROM AUTHOR]