  In this article we focus on the parameterized complexity of the
Multidimensional Binary Vector Assignment problem (called \BVA). An input of
this problem is defined by $m$ disjoint sets $V^1, V^2, \dots, V^m$, each
composed of $n$ binary vectors of size $p$. An output is a set of $n$ disjoint
$m$-tuples of vectors, where each $m$-tuple is obtained by picking one vector
from each set $V^i$. To each $m$-tuple we associate a $p$ dimensional vector by
applying the bit-wise AND operation on the $m$ vectors of the tuple. The
objective is to minimize the total number of zeros in these $n$ vectors. mBVA
can be seen as a variant of multidimensional matching where hyperedges are
implicitly locally encoded via labels attached to vertices, but was originally
introduced in the context of integrated circuit manufacturing.
  We provide for this problem FPT algorithms and negative results ($ETH$-based
results, $W$[2]-hardness and a kernel lower bound) according to several
parameters: the standard parameter $k$ i.e. the total number of zeros), as well
as two parameters above some guaranteed values.
