  Let $G$ be a finite permutation group on $\Omega$. An ordered sequence
$(\omega_1,\ldots,\omega_\ell)$ of elements of $\Omega$ is an irredundant base
for $G$ if the pointwise stabilizer is trivial and no point is fixed by the
stabilizer of its predecessors. If all irredundant bases of $G$ have the same
cardinality, $G$ is said to be an IBIS group.
  Lucchini, Morigi and Moscatiello have proved a theorem reducing the problem
of classifying finite primitive IBIS groups $G$ to the case that the socle of
$G$ is either abelian or non-abelian simple.
  In this paper, we classify the finite primitive IBIS groups having socle an
alternating group. Moreover, we propose a conjecture aiming to give a
classification of all almost simple primitive IBIS groups.
