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Erschienen in:
10.10.2023 | Research
Factorization of invariant polynomials under actions of projective linear groups and its applications in coding theory
Erschienen in: Cryptography and Communications | Ausgabe 1/2024
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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 \([A]\in PGL_2(\mathbb {F}_{q})\).