  Starting from its original definition in module categories with respect to
projective modules, the index has played an important role in various aspects
of homological algebra, categorification of cluster algebras and $K$-theory. In
the last few years, the notion of index has been generalised to several
different contexts in (higher) homological algebra, typically with respect to a
(higher) cluster-tilting subcategory $\mathcal{X}$ of the relevant ambient
category $\mathcal{C}$. The recent tools of extriangulated and
higher-exangulated categories have permitted some conditions on the subcategory
$\mathcal{X}$ to be relaxed. In this paper, we introduce the index with respect
to a generating, contravariantly finite subcategory of a $d$-exact category
that has $d$-kernels. We show that our index has the important property of
being additive on $d$-exact sequences up to an error term.
