In this paper, we study the algebraic structure of additive cyclic codes over the alphabet F2r×F4s=F2rF4s,\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {F}}_{2}^{r}\times {\mathbb {F}}_{4}^{s}={ \mathbb {F}}_{2}^{r}{\mathbb {F}}_{4}^{s},$$\end{document} where r and s are non-negative integers, F2=GF(2)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{2}={\mathbb {GF}}(2)$$\end{document} and F4=GF(4)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{4}={\mathbb {GF}} (4)$$\end{document} are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for F2F4\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{2}\mathbb {F}_{4}$$\end{document}-additive cyclic codes. We also introduce a linear map W that depends on the trace map T to relate these codes to binary linear codes over F2.\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F} _{2}.$$\end{document} Further, we investigate the duals of F2F4\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{2}\mathbb {F}_{4}$$\end{document}-additive cyclic codes. We show that the dual of any F2F4\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{2}\mathbb {F }_{4}$$\end{document}-additive cyclic code is another F2F4\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{2}\mathbb {F}_{4}$$\end{document}-additive cyclic code. Using the mapping W, we provide examples of F2F4\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{2}\mathbb {F}_{4}$$\end{document}-additive cyclic codes whose binary images have optimal parameters. We also consider additive cyclic codes over F4\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{4}$$\end{document} and give some examples of optimal parameter quantum codes over F4\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{4}$$\end{document}.
