A characterization of $$mathbb {Z}_{2}mathbb {Z}_{2}[u]$$-linear codes

We prove that the class of (mathbb {Z}_2mathbb {Z}_2[u])-linear codes is exactly the class of (mathbb {Z}_2)-linear codes with automorphism group of even order. Using this characterization, we give examples of known codes, e.g. perfect codes, which have a nontrivial (mathbb {Z}_2mathbb {Z}_2[u]) structure. Moreover, we exhibit some examples of (mathbb {Z}_2)-linear codes which are not (mathbb {Z}_2mathbb {Z}_2[u])-linear. Also, we state that the duality of (mathbb {Z}_2mathbb {Z}_2[u])-linear codes is the same as the duality of (mathbb {Z}_2)-linear codes. Finally, we prove that the class of (mathbb {Z}_2mathbb {Z}_4)-linear codes which are also (mathbb {Z}_2)-linear is strictly contained in the class of (mathbb {Z}_2mathbb {Z}_2[u])-linear codes.