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