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On the structure of 1-generator quasi-polycyclic codes over finite chain rings

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Abstract

Quasi-polycyclic (QP for short) codes over a finite chain ring R are a generalization of quasi-cyclic codes, and these codes can be viewed as an R[x]-submodule of \({\mathcal {R}}_m^{\ell }\), where \({\mathcal {R}}_m:= R[x]/\langle f\rangle \), and f is a monic polynomial of degree m over R. If f factors uniquely into monic and coprime basic irreducibles, then their algebraic structure allow us to characterize the generator polynomials and the minimal generating sets of 1-generator QP codes as R-modules. In addition, we also determine the parity check polynomials for these codes by using the strong Gröbner bases. In particular, via Magma system, some quaternary codes with new parameters are derived from these 1-generator QP codes.

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Acknowledgements

The authors would like to thank the anonymous referees for their careful checking and helpful suggestions which have improved the manuscript. The authors are also grateful to the Assoc. Prof. Edgar Martínez-Moro for helpful discussions.

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Authors and Affiliations

  1. Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, School of Mathematical Sciences, Anhui University, Hefei, 230601, China

    Rongsheng Wu & Minjia Shi

  2. Lab I2M (CNRS, Aix-Marseille University, Centrale Marseille), Marseilles, 13009, France

    Patrick Solé

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  1. Rongsheng Wu
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  2. Minjia Shi
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Correspondence to Minjia Shi.

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This research is supported by the National Natural Science Foundation of China (12071001), the Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20).

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Wu, R., Shi, M. & Solé, P. On the structure of 1-generator quasi-polycyclic codes over finite chain rings. J. Appl. Math. Comput. 68, 3491–3503 (2022). https://doi.org/10.1007/s12190-021-01669-9

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