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 ${\mathbb{C}}^d\otimes {\mathbb{C}}^{d^{\prime }}$ $\left(2\le k<d\le d^{\prime }\right)$, 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 $d^{\prime }$ is proposed. Consequently, it is shown that there are at least $k-r$ (here $r=d\text{mod}k$ or $r=d^{\prime }\text{mod}k)$ sets of UEBks when $d$ or $d^{\prime }$ is not a multiple of $k$, while there are at least $2(k-1)$ sets of UEBks when both $d$ and $d^{\prime }$ are multiples of $k$.