Advertisement
Published in:
10-10-2023 | Research
Factorization of invariant polynomials under actions of projective linear groups and its applications in coding theory
Published in: Cryptography and Communications | Issue 1/2024
Log inActivate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
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 \([A]\in PGL_2(\mathbb {F}_{q})\).