  We study mutually unbiased bases (MUBs) in which all the bases are
unextendible maximally entangled ones. We first present a necessary and
sufficient condition of constructing a pair of MUBs in $C^2 \otimes C^4$. Based
on this condition, an analytical and necessary condition for constructing MUBs
is given. Moreover we illustrate our approach by some detailed examples in $C^2
\otimes C^4$. The results are generalized to $C^2 \otimes C^d$ $(d\geq 3)$ and
a concrete example in $C^2 \otimes C^8$ is given.
