\(\sigma \)-LCD codes over finite chain rings

Abstract

In this work, we first generalize the \(\sigma \)-LCD codes over finite fields to \(\sigma \)-LCD codes over finite chain rings. Under suitable conditions, linear codes over finite chain rings that are \(\sigma \)-LCD codes are characterized. Then we provide a necessary and sufficient condition for free constacyclic codes over finite chain rings to be \(\sigma \)-LCD. We also get some new binary LCD codes of different lengths which come from Gray images of constacyclic \(\sigma \)-LCD codes over \(\mathbb {F}_{2}+\gamma \mathbb {F}_{2}+\gamma ^2\mathbb {F}_{2}\). Finally, for special finite chain rings \(\mathbb {F}_{q}+\gamma \mathbb {F}_{q}\), we define a new Gray map \(\Phi \) from \((\mathbb {F}_{q}+\gamma \mathbb {F}_{q})^n\) to \(\mathbb {F}_{q}^{2n}\), and by using \(\sigma \)-LCD codes over finite chain rings \(\mathbb {F}_{q}+\gamma \mathbb {F}_{q}\), we construct new entanglement-assisted quantum error-correcting (abbreviated to EAQEC) codes with maximal entanglement and parts of them are MDS EAQEC codes.

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Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for their valuable comments and suggestions, which improved the quality of the manuscript. This research was supported by Research Funds of Hubei Province (Grant No. Q20174503), and Research Project of Hubei Polytechnic University (Grant No. 17xjz03A).

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Correspondence to Xiusheng Liu.

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Liu, X., Liu, H. \(\sigma \)-LCD codes over finite chain rings. Des. Codes Cryptogr. 88, 727–746 (2020). https://doi.org/10.1007/s10623-019-00706-w

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Keywords

  • Finite chain ring
  • Generator matrix
  • \(\sigma \)-LCD code
  • Entanglement-assisted quantum error-correcting code

Mathematics Subject Classification

  • 94B15
  • 94B60
  • 11T71