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Factorization of invariant polynomials under actions of projective linear groups and its applications in coding theory

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Abstract

In this paper, let \(\mathbb {F}_q\) be a finite field with \(q=2^n\) elements and let \([A] \in PGL_2(\mathbb {F}_q)\) be of order 2 or 3, where \(A= \left( \begin{array}{cc} a&{}b\\ 1&{}d\end{array}\right) \). We determine all invariant irreducible (monic) polynomials by the action of \([A]\in PGL_2(\mathbb {F}_{q})\) and have irreducible factorizations of polynomials \(F_s(x)=x^{q^s+1}+dx^{q^s}+ax+b\) over \(\mathbb {F}_{q}\) in two cases: (1) \(s=t^e\) and t is an odd prime; (2) \(s=t_1t_2\) and \(t_1, t_2\) are two distinct odd primes. Moreover, we construct some binary irreducible quasi-cyclic expurgated Goppa codes and extended Goppa codes by invariant irreducible polynomials under the action of \([A]\in PGL_2(\mathbb {F}_{q})\).

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Funding

The paper is supported by National Natural Science Foundation of China(No.62172219 and No.12171420), Innovation Program for Quantum Science and Technology under grant 2021ZD0302902 and Talent Research Initiation Fund Project under grant XK0070922088.

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Authors and Affiliations

  1. College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 211100, People’s Republic of China

    Xia Li

  2. State Key Laboratory of Cryptology, P.O. Box 5159, Beijing, 100878, People’s Republic of China

    Xia Li & Qin Yue

  3. College of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, People’s Republic of China

    Qin Yue

  4. College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, People’s Republic of China

    Daitao Huang

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  1. Xia Li
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  2. Qin Yue
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  3. Daitao Huang
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Contributions

Xia Li and Qin Yue wrote the main manuscript text and Daitao Huang prepared examples. All authors reviewed the manuscript.

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Correspondence to Qin Yue.

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Li, X., Yue, Q. & Huang, D. Factorization of invariant polynomials under actions of projective linear groups and its applications in coding theory. Cryptogr. Commun. 16, 185–207 (2024). https://doi.org/10.1007/s12095-023-00675-x

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