Mutually unbiased bases plays a central role in quantum mechanics and quantum information processing. As an important class of mutually unbiased bases, mutually unbiased maximally entangled bases (MUMEBs) in bipartite systems have attracted much attention in recent years. In the paper, we try to construct MUMEBs in C2s⊗C2s\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}^{2^s}\otimes {\mathbb {C}}^{2^s}$$\end{document} by using Galois rings, which is different from the work in [17], where finite fields are used. As applications, we obtain several new types of MUMEBs in C2s⊗C2s\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}^{2^s}\otimes {\mathbb {C}}^{2^s}$$\end{document} and prove that M(2s,2s)≥3(2s-1)\documentclass[12pt]{minimal}
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\begin{document}$$M(2^s,2^s)\ge 3(2^s-1)$$\end{document}, which raises the lower bound of M(2s,2s)\documentclass[12pt]{minimal}
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\begin{document}$$M(2^s,2^s)$$\end{document} given in [16].
