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Published in:
20-02-2017 | Original Paper
1-Generator quasi-cyclic and generalized quasi-cyclic codes over the ring \(\frac{{{\mathbb {Z}_4}[u]}}{{\left\langle {{u^2} - 1}\right\rangle }}\)
Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 6/2017
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Abstract
In this paper, we study 1-generator quasi-cyclic and generalized quasi-cyclic codes over the ring \(R=\frac{{{\mathbb {Z}_4}[u]}}{{\left\langle {{u^2} - 1} \right\rangle }}\). We determine the structure of the generators and the minimal generating sets of 1-generator QC and GQC codes. We also give a lower bound for the minimum distance of free 1-generator quasi-cyclic and generalized quasi-cyclic codes over this ring, respectively. Furthermore, some new \(\mathbb {Z}_4\)-linear codes via the Gray map which have better parameters than the best known \(\mathbb {Z}_4\)-linear codes are presented.