Additive and linear conjucyclic codes over $ {\mathbb{F}}_4 $

Abstract

<p style='text-indent:20px;'>Conjucyclic codes were first introduced by Calderbank, Rains, Shor and Sloane in [<xref ref-type="bibr" rid="b1">1</xref>] because of their applications in quantum error-correction. In this paper, we study linear and additive conjucyclic codes over the finite field <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb{F}}_{4} $\end{document}</tex-math></inline-formula> and produce a duality for which the orthogonal, with respect to that duality, of conjucyclic codes is conjucyclic. Moreover, we show that this is not the case for other standard dualities. We show that additive conjucyclic codes are the only non-trivial conjucyclic codes over <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb{F}}_{4} $\end{document}</tex-math></inline-formula> and we use a linear algebraic approach to classify these codes. We will also show that additive conjucyclic codes of length <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M5">\begin{document}$ {\mathbb{F}}_{4} $\end{document}</tex-math></inline-formula> are isomorphic to binary cyclic codes of length <inline-formula><tex-math id="M6">\begin{document}$ 2n. $\end{document}</tex-math></inline-formula></p>