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In this paper, we study the structure and properties of additive right and left polycyclic codes induced by a binary vector \begin{document}$ a $\end{document} in \begin{document}$ \mathbb{F}_{2}^{n}. $\end{document} We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if \begin{document}$ C $\end{document} is a right polycyclic code induced by a vector \begin{document}$ a\in \mathbb{F}_{2}^{n} $\end{document}, then the Hermitian dual of \begin{document}$ C $\end{document} is a sequential code induced by \begin{document}$ a. $\end{document} As an application of these codes, we present examples of additive right polycyclic codes over \begin{document}$ \mathbb{F}_{4} $\end{document} with more codewords than comparable optimal linear codes as well as optimal binary linear codes and optimal quantum codes obtained from additive right polycyclic codes over \begin{document}$ \mathbb{F}_{4}. $\end{document}