Additive polycyclic codes over $ \mathbb{F}_{4} $ induced by binary vectors and some optimal codes

<p style='text-indent:20px;'>In this paper, we study the structure and properties of additive right left polycyclic codes induced by a binary vector <inline-formula><tex-math id="M2">\begin{document}$ $\end{document}</tex-math></inline-formula> in id="M3">\begin{document}$ \mathbb{F}_{2}^{n}. We find generator polynomials cardinality these codes. also different duals for In particular, show that if id="M4">\begin{document}$ C is code id="M5">\begin{document}$ a\in \mathbb{F}_{2}^{n} $\end{document}</tex-math></inline-formula>, then Hermitian dual id="M6">\begin{document}$ sequential id="M7">\begin{document}$ a. As an application codes, present examples over id="M8">\begin{document}$ \mathbb{F}_{4} with more codewords than comparable optimal linear as well quantum obtained from id="M9">\begin{document}$ \mathbb{F}_{4}. $\end{document}</tex-math></inline-formula></p>