Unextendible entangled bases with fixed Schmidt number

The unextendible product basis is generalized to the unextendible entangled basis with any arbitrarily given Schmidt number $k$ (UEBk) for any bipartite system ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}$ $(2\ensuremath{\le}k<d\ensuremath{\le}{d}^{\ensuremath{'}})$, which can also be regarded as a generalization of the unextendible maximally entangled basis. A general way of constructing such a basis with arbitrary $d$ and ${d}^{\ensuremath{'}}$ is proposed. Consequently, it is shown that there are at least $k\ensuremath{-}r$ (here $r=d\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k$ or $r={d}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k)$ sets of UEBks when $d$ or ${d}^{\ensuremath{'}}$ is not a multiple of $k$, while there are at least $2(k\ensuremath{-}1)$ sets of UEBks when both $d$ and ${d}^{\ensuremath{'}}$ are multiples of $k$.