  Many deep, mysterious connections have been observed between collections of
mutually unbiased bases (MUBs) and combinatorial designs called $k$-nets (and
in particular, between complete collections of MUBs and finite affine - or
equivalently: finite projective - planes). Here we introduce the notion of a
$k$-net over an algebra $\mathfrak{A}$ and thus provide a common framework for
both objects. In the commutative case, we recover (classical) $k$-nets, while
choosing $\mathfrak{A} := M_d(\mathbb C)$ leads to collections of MUBs.
  A common framework allows one to find shared properties and proofs that
"inherently work" for both objects. As a first example, we derive a certain
rigidity property which was previously shown to hold for $k$-nets that can be
completed to affine planes using a completely different, combinatorial
argument. For $k$-nets that cannot be completed and for MUBs, this result is
new, and, in particular, it implies that the only vectors unbiased to all but
$k \leq \sqrt{d}$ bases of a complete collection of MUBs in $\mathbb C^d$ are
the elements of the remaining $k$ bases (up to phase factors). In general, this
is false when $k$ is just the next integer after $\sqrt{d}$; we present an
example of this in every prime-square dimension, demonstrating that the derived
bound is tight.
  As an application of the rigidity result, we prove that if a large enough
collection of MUBs constructed from a certain type of group representation
(e.g. a construction relying on discrete Weyl operators or generalized Pauli
matrices) can be extended to a complete system, then in fact every basis of the
completion must come from the same representation. In turn, we use this to show
that certain large systems of MUBs cannot be completed.
