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The nonsquare unextendible maximally entangled basis (nUMEB) is a set of orthonormal maximally entangled states in \(C^{d}\otimes C^{d^{\prime }}~(d^{\prime }> d)\) which have no additional maximally entangled vectors orthogonal to all of them. We study nUMEBs in arbitrary bipartite spaces and present a constructive proof of the existence of nUMEBs in \(C^{d}\otimes C^{d^{\prime }}~(d^{\prime }\geq 2d)\) . Furthermore, a bound condition for the existence of nUMEBs for \(C^{d}\otimes C^{d^{\prime }}~(d^{\prime }\geq 2d)\) is obtained.
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