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

Abstract

We prove that the class of $$\mathbb {Z}_2\mathbb {Z}_2[u]$$ -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 $$\mathbb {Z}_2\mathbb {Z}_2[u]$$ structure. Moreover, we exhibit some examples of $$\mathbb {Z}_2$$ -linear codes which are not $$\mathbb {Z}_2\mathbb {Z}_2[u]$$ -linear. Also, we state that the duality of $$\mathbb {Z}_2\mathbb {Z}_2[u]$$ -linear codes is the same as the duality of $$\mathbb {Z}_2$$ -linear codes. Finally, we prove that the class of $$\mathbb {Z}_2\mathbb {Z}_4$$ -linear codes which are also $$\mathbb {Z}_2$$ -linear is strictly contained in the class of $$\mathbb {Z}_2\mathbb {Z}_2[u]$$ -linear codes.