A class of constacyclic codes over \({\mathbb {Z}}_{4}[u]/\langle u^{k}\rangle \). (English) Zbl 1468.94441
Summary: In this paper, we study \(\lambda\)-constacyclic and skew \(\lambda \)-constacyclic codes over the ring \({\mathbb {Z}}_{4}[u]/\langle u^{k}\rangle \) where \(u^{k}=0 \) with \(\lambda =(1+2u^{k-1})\) and \((3+2u^{k-1})\). It is shown that the Gray images of \(\lambda \)-constacyclic and skew \(\lambda \)-constacyclic codes over the ring \({\mathbb {Z}}_{4}[u]/\langle u^{k}\rangle \) are cyclic, quasi-cyclic, permutation equivalent to a \(QC\) code over \({\mathbb {Z}}_{4}\). Further, the generators of these \(\lambda \)-constacyclic codes over \({\mathbb {Z}}_{4}[u]/\langle u^{k}\rangle \) are obtained.
MSC:
| 94B15 | Cyclic codes |
References:
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