  In this paper we study the structure and properties of additive right and
left 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 also
study different duals for these codes. In particular, we show that if $C$ is a
right polycyclic code induced by a vector $a\in \mathbb{F}_{2}^{n}$, then the
Hermitian dual of $C$ is a sequential code induced by $a.$ As an application of
these codes, we present examples of additive right polycyclic codes over
$\mathbb{F}_{4}$ 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 $\mathbb{F}_{4}.$
