  We consider the quantum Ising chain with uniformly distributed random
antiferromagnetic couplings $(1 \le J_i \le 2)$ and uniformly distributed
random transverse fields ($\Gamma_0 \le \Gamma_i \le 2\Gamma_0$) in the
presence of a homogeneous longitudinal field, $h$. Using different numerical
techniques (DMRG, combinatorial optimisation and strong disorder RG methods) we
explore the phase diagram, which consists of an ordered and a disordered phase.
At one end of the transition line ($h=0,\Gamma_0=1$) there is an infinite
disorder quantum fixed point, while at the other end ($h=2,\Gamma_0=0$) there
is a classical random first-order transition point. Close to this fixed point,
for $h>2$ and $\Gamma_0>0$ there is a reentrant ordered phase, which is the
result of quantum fluctuations by means of an order through disorder
phenomenon.
