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A class of constacyclic codes and skew constacyclic codes over \(\pmb {\mathbb {Z}}_{2^s}+u\pmb {\mathbb {Z}}_{2^s}\) and their gray images

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Abstract

In this paper, we study \((1+2^{s-1}u)\)-constacyclic codes and a class of skew \((1+2^{s-1}u)\)-constacyclic codes of odd length over the ring \(R= {\mathbb {Z}}_{2^s}+u{\mathbb {Z}}_{2^s}\), \(u^2=0\), where \(s \ge 3\) is an odd integer. We have obtained the algebraic structure of \((1+2^{s-1}u)\)-constacyclic codes over R. Three new Gray maps from R to \({\mathbb {Z}}_2+u{\mathbb {Z}}_2\) have been defined and it is shown that Gray images of \((1+2^{s-1}u)\)-constacyclic codes and skew \((1+2^{s-1}u)\)-constacyclic codes are cyclic codes, quasi-cyclic codes or codes that are permutation equivalent to quasi-cyclic codes over \({\mathbb {Z}}_2+u{\mathbb {Z}}_2\). Using Magma, some good cyclic codes of length 6 over \({\mathbb {Z}}_2+u{\mathbb {Z}}_2\) are obtained.

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Acknowledgements

The first author would like to thank the Ministry of Human Resource Development (MHRD), India for providing financial support. The authors would also like to thank the anonymous referees for their valuable comments and suggestions.

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  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India

    Raj Kumar & Maheshanand Bhaintwal

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  1. Raj Kumar
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  2. Maheshanand Bhaintwal
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Correspondence to Raj Kumar.

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Kumar, R., Bhaintwal, M. A class of constacyclic codes and skew constacyclic codes over \(\pmb {\mathbb {Z}}_{2^s}+u\pmb {\mathbb {Z}}_{2^s}\) and their gray images. J. Appl. Math. Comput. 66, 111–128 (2021). https://doi.org/10.1007/s12190-020-01425-5

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  • DOI: https://doi.org/10.1007/s12190-020-01425-5

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