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

Additive Polycyclic Codes over $\mathbb{F}_{4}$ Induced by Binary Vectors and Some Optimal Codes