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Linear complementary pairs of codes over rings

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Abstract

In this work, we first prove a necessary and sufficient condition for a pairs of linear codes over finite rings to be linear complementary pairs (abbreviated to LCPs). In particular, a judging criterion of free LCP of codes over finite commutative rings is obtained. Using the criterion of free LCP of codes, we construct a maximum-distance-separable (MDS) LCP of codes over ring \(\mathbb {Z}_4\). Then, all the possible LCP of codes over chain rings are determined. We also generalize the criterions for constacyclic and quasi-cyclic LCP of codes over finite fields to constacyclic and quasi-cyclic LCP of codes over chain rings. Finally, we give a characterization of LCP of codes over principal ideal rings. Under suitable conditions, we also obtain the judging criterion for a pairs of cyclic codes over principal ideal rings \(\mathbb {Z}_{k}\) to be LCP, and illustrate a MDS LCP of cyclic codes over the principal ideal ring \(\mathbb {Z}_{15}\).

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Acknowledgements

X. Liu was supported by Research Funds of Hubei Province (Grant No. Q20174503). P. Hu was supported by Research Project of Hubei Polytechnic University (Grant No. numbers 17xjz03A ).

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Authors and Affiliations

  1. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, 435003, Hubei, China

    Peng Hu & Xiusheng Liu

  2. School of Science and Technology, College of Arts and Science of Hubei Normal University, Huangshi, 435109, Hubei, China

    Xiusheng Liu

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  1. Peng Hu
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  2. Xiusheng Liu
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Correspondence to Xiusheng Liu.

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Communicated by J.-L. Kim.

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Hu, P., Liu, X. Linear complementary pairs of codes over rings. Des. Codes Cryptogr. 89, 2495–2509 (2021). https://doi.org/10.1007/s10623-021-00933-0

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  • DOI: https://doi.org/10.1007/s10623-021-00933-0

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