  The study of distribution testing has become ubiquitous in the area of
property testing, both for its theoretical appeal, as well as for its
applications in other fields of Computer Science. The original distribution
testing model relies on samples drawn independently from the distribution to be
tested. However, when testing distributions over the $n$-dimensional Hamming
cube $\left\{0,1\right\}^{n}$ for a large $n$, even reading a few samples is
infeasible. To address this, Goldreich and Ron [ITCS 2022] have defined a model
called the huge object model, in which the samples may only be queried in a few
places.
  In this work, we initiate a study of a general class of properties in the
huge object model, those that are invariant under a permutation of the indices
of the vectors in $\left\{0,1\right\}^{n}$, while still not being necessarily
fully symmetric as per the definition used in traditional distribution testing.
  We prove that every index-invariant property satisfying a bounded
VC-dimension restriction admits a property tester with a number of queries
independent of n. To complement this result, we argue that satisfying only
index-invariance or only a VC-dimension bound is insufficient to guarantee a
tester whose query complexity is independent of n. Moreover, we prove that the
dependency of sample and query complexities of our tester on the VC-dimension
is tight. As a second part of this work, we address the question of the number
of queries required for non-adaptive testing. We show that it can be at most
quadratic in the number of queries required for an adaptive tester of
index-invariant properties. This is in contrast with the tight exponential gap
for general non-index-invariant properties. Finally, we provide an
index-invariant property for which the quadratic gap between adaptive and
non-adaptive query complexities for testing is almost tight.
