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In this paper, we introduce self-dual cyclic codes over the ring ℤ 2 × (ℤ 2 + u ℤ 2) = ℤ 2 ℤ 2 [ u ]. We determine the conditions for any ℤ 2 ℤ 2 [ u ] -cyclic code to be self-dual, that is, 𝒞 = 𝒞 ⊥ . Since the binary image of a self-orthogonal ℤ 2 ℤ 2 [ u ] -linear code is also a self-orthogonal binary linear code, we introduce quantum codes over ℤ 2 ℤ 2 [ u ]. Finally, we present some examples of self-dual cyclic and quantum codes that have good parameters. [ABSTRACT FROM AUTHOR]

ℤ 2 ℤ 2 s -additive codes for any integer s > 1 are considered as codes over mixed alphabets. They are a generalization of binary linear codes and linear codes over ℤ 2 s . In this paper, we are interested in studying ℤ 2 ℤ 2 s -additive cyclic codes. We will give the generator polynomials of these codes. We will also give the minimal spanning sets for these codes. We will define separable ℤ 2 ℤ 2 s -additive codes and provide conditions on the generator polynomials for a ℤ 2 ℤ 2 s -additive cyclic code to be separable. Finally, we present some examples of optimal parameter binary codes obtained as images of ℤ 2 ℤ 8 -additive cyclic codes. [ABSTRACT FROM AUTHOR]