  We provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs)
in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show
that these incomplete sets cannot be extended further using the operators of
the Pauli group. However, specific examples of sets of MUBs obtained using our
construction are shown to be strongly unextendible; that is, there does not
exist another vector that is unbiased with respect to the elements in the set.
We conjecture the existence of such unextendible sets in higher dimensions
(d=2^{n}, n>3) as well. Furthermore, we note an interesting connection between
these unextendible sets and state-independent proofs of the Kochen-Specker
Theorem for two-qubit systems. Our construction also leads to a proof of the
tightness of a H_{2} entropic uncertainty relation for any set of three MUBs
constructed from Pauli classes in d=4.
