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Non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

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Abstract

In this paper, we construct a family of non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes. Moreover, we present a sufficient and necessary condition for irreducible Goppa polynomial g(x) of degree \(r=2\) or 3 over \({\mathbb {F}}_{q}\). In certain conditions, we give a determination and enumeration of irreducible Goppa polynomials g(x) of degree 2s or 3s as above, where s is a positive integer.

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Authors and Affiliations

  1. College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, People’s Republic of China

    Xia Li

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, People’s Republic of China

    Qin Yue

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  1. Xia Li
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  2. Qin Yue
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Correspondence to Xia Li.

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Communicated by T. Feng.

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The paper was supported by National Natural Science Foundation of China (Nos. 62172219 and 12171420), the Foundation of State Key Laboratory of Integrated Services Networks under Grant ISN23-22, Natural Science Foundation of Shandong Province under Grant ZR2021MA046, Natural Science Foundation of Jiangsu Province under Grant BK20200268.

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Li, X., Yue, Q. Non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes. Des. Codes Cryptogr. 90, 1629–1647 (2022). https://doi.org/10.1007/s10623-022-01062-y

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  • DOI: https://doi.org/10.1007/s10623-022-01062-y

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