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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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