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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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
Keywords
- Irreducible polynomials
- Projective linear group
- Extended Goppa codes
- Parity-check subcodes of Goppa codes
- Quasi-cyclic codes