Mutually unbiased unextendible maximally entangled bases in some systems of higher dimension

We study the construction of mutually unbiased bases such that all the bases are unextendible maximally entangled ones. By using some results from the theory of finite fields, we construct mutually unbiased unextendible maximally entangled bases in some bipartite systems of higher dimension: $${\mathbb {C}}^{4} \otimes {\mathbb {C}}^{5}$$ , $${\mathbb {C}}^{6} \otimes {\mathbb {C}}^{7}$$ , $${\mathbb {C}}^{10} \otimes {\mathbb {C}}^{11}$$ and $${\mathbb {C}}^{12} \otimes {\mathbb {C}}^{13}$$ , which extend the known result of $${\mathbb {C}}^{2} \otimes {\mathbb {C}}^{3}$$ . We also generalize these results to more bipartie systems of specific dimension.