\(\mathbb{Z}_4\mathbb{Z}_4 [u]\)-additive cyclic and constacyclic codes. (English) Zbl 1479.94366
Summary: We study mixed alphabet cyclic and constacyclic codes over the two alphabets \(\mathbb{Z}_4\), the ring of integers modulo \(4\), and its quadratic extension \(\mathbb{Z}_4 [u] = \mathbb{Z}_4 +u\mathbb{Z}_4, u^2 = 0\). Their generator polynomials and minimal spanning sets are obtained. Further, under new Gray maps, we find cyclic, quasi-cyclic codes over \(\mathbb{Z}_4\) as the Gray images of both \(\lambda\)-constacyclic and skew \(\lambda\)-constacyclic codes over \(\mathbb{Z}_4 [u]\). Moreover, it is proved that the Gray images of \(\mathbb{Z}_4 \mathbb{Z}_4 [u]\)-additive constacyclic and skew \(\mathbb{Z}_4 \mathbb{Z}_4 [u]\)-additive constacyclic codes are generalized quasi-cyclic codes over \(\mathbb{Z}_4\). Finally, several new quaternary linear codes are obtained from these cyclic and constacyclic codes.
Keywords:
additive code; Gray map; quasi-cyclic code; skew cyclic code; generalized quasi-cyclic codeReferences:
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