We study the construction of unextendible maximally entangled bases in bipartite systems [Formula: see text] ([Formula: see text], [Formula: see text], [Formula: see text]). We provide a class of maximally entangled states in [Formula: see text] by constructing some unitary matrices in [Formula: see text]. In particular, we present two complete unextendible maximally entangled bases in [Formula: see text] which are mutually unbiased.