Unextendible entangled bases with fixed Schmidt number

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