A binary linear code C is a Z2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_2$$\end{document}-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the Z2[x]\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_2[x]$$\end{document}-module Z2[x]/(xr-1)×Z2[x]/(xs-1).\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_2[x]/(x^r-1)\times {\mathbb {Z}}_2[x]/(x^s-1).$$\end{document} We determine the structure of Z2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_2$$\end{document}-double cyclic codes giving the generator polynomials of these codes. We give the polynomial representation of Z2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_2$$\end{document}-double cyclic codes and its duals, and the relations between the generator polynomials of these codes. Finally, we study the relations between Z2\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {Z}}}_2$$\end{document}-double cyclic and other families of cyclic codes, and show some examples of distance optimal Z2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_2$$\end{document}-double cyclic codes.
