Unextendible maximally entangled bases in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$

We investigate the unextendible maximally entangled bases (UMEBs) in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$ and present a 30-number UMEB construction in ${\mathbb{C}}^6\otimes {\mathbb{C}}^6$. For the higher-dimensional case, we show that for a given $N$-number UMEB in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$, there is an $\tilde{N}$-number, $\tilde{N}={\left(qd\right)}^2-(d^2-N)$, UMEB in ${\mathbb{C}}^{qd}\otimes {\mathbb{C}}^{qd}$ for any $q\in \mathbb{N}$. As an example, for ${\mathbb{C}}^{12n}\otimes {\mathbb{C}}^{12n}$ systems, we show that there are at least two sets of UMEBs which are not equivalent.