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\begin{document}$$\mathrm {R}={\mathbb {F}}_4+v{\mathbb {F}}_4, v^2=v$$\end{document}. A linear code over R\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {R}$$\end{document} is a double cyclic code of length (r, s), if the set of its coordinates can be partitioned into two parts of sizes r and s, so that any cyclic shift of coordinates of both parts leave the code invariant. In polynomial representation, these codes can be viewed as R[x]\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {R}[x]$$\end{document}-submodules of R[x]⟨xr-1⟩×R[x]⟨xs-1⟩\documentclass[12pt]{minimal}
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\begin{document}$$\frac{\mathrm {R}[x]}{\langle x^r-1\rangle }\times \frac{\mathrm {R}[x]}{\langle x^s-1\rangle }$$\end{document}. In this paper, we determine generator polynomials of R\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {R}$$\end{document}-double cyclic codes and their duals for arbitrary values of r and s. We enumerate R\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {R}$$\end{document}-double cyclic codes of length (2e1,2e2)\documentclass[12pt]{minimal}
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\begin{document}$$(2^{e_1},2^{e_2})$$\end{document} by giving a mass formula, where e1\documentclass[12pt]{minimal}
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\begin{document}$$e_1$$\end{document} and e2\documentclass[12pt]{minimal}
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\begin{document}$$e_2$$\end{document} are positive integers. Some structural properties of double constacyclic codes over R\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {R}$$\end{document} are also studied. These results are illustrated with some good examples.
