  In the recent years, intensive research work has been dedicated to prove
conditional lower bounds in order to reveal the inner structure of the class P.
These conditional lower bounds are based on many popular conjectures on
well-studied problems. One of the most heavily used conjectures is the
celebrated Strong Exponential Time Hypothesis (SETH). It turns out that
conditional hardness proved based on SETH goes, in many cases, through an
intermediate problem - the Orthogonal Vectors (OV) problem.
  Almost all research work regarding conditional lower bound was concentrated
on time complexity. Very little attention was directed toward space complexity.
In a recent work, Goldstein et al.[WADS 2017] set the stage for proving
conditional lower bounds regarding space and its interplay with time. In this
spirit, it is tempting to investigate the space complexity of a data structure
variant of OV which is called \emph{OV indexing}. In this problem $n$ boolean
vectors of size $c\log{n}$ are given for preprocessing. As a query, a vector
$v$ is given and we are required to verify if there is an input vector that is
orthogonal to it or not.
  This OV indexing problem is interesting in its own, but it also likely to
have strong implications on problems known to be conditionally hard, in terms
of time complexity, based on OV. Having this in mind, we study OV indexing in
this paper from many aspects. We give some space-efficient algorithms for the
problem, show a tradeoff between space and query time, describe how to solve
its reporting variant, shed light on an interesting connection between this
problem and the well-studied SetDisjointness problem and demonstrate how it can
be solved more efficiently on random input.
