(σ, δ)-Skew quasi-cyclic codes over the ring \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\)

Abstract

Let \(R=\mathbb {Z}_{4}+u\mathbb {Z}_{4}\) be a finite non-chain ring, where u2 = 1. In this paper, we consider (σ, δ)-skew quasi-cyclic codes over the ring R, where σ is an automorphism of R and δ is an inner σ-derivation of R. We determine the structure of 1-generator (σ, δ)-skew quasi-cyclic codes over R and give a sufficient condition for 1-generator (σ, δ)-skew quasi-cyclic codes over R to be free. We also determine a distance bound for free 1-generator (σ, δ)-skew quasi-cyclic codes. Moreover, using the residue codes of these codes over R we obtain some good \(\mathbb {Z}_{4}\)-linear codes. Finally, we give the characterization of Euclidean dual codes of (σ, δ)-skew quasi-cyclic codes.

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Acknowledgements

This research is supported by the 973 Program of China (Grant No. 2013CB834204), the National Natural Science Foundation of China (Grant No. A010206, 61571243, 11701336, 11626144 and 11671235, 12071264).

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Correspondence to Jian Gao.

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Ma, F., Gao, J., Li, J. et al. (σ, δ)-Skew quasi-cyclic codes over the ring \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\). Cryptogr. Commun. (2021). https://doi.org/10.1007/s12095-020-00467-7

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Keywords

  • -skew quasi-cyclic codes
  • residue codes
  • good -linear codes
  • Euclidean dual codes