  Collisionless shock acceleration of carbon ions (C$^{6+}$) is investigated in
the ultra-relativistic regime of laser-plasma interaction by accounting for the
radiation reaction force and the pair production in particle-in-cell
simulations. Both radiation reaction force and pair plasma formation tend to
slow down the shock velocity, reducing the energy of the accelerated ions,
albeit extending the time scales of the acceleration process. Slab plasma
target achieves lower energy spread while target with a tailored density
profile yields higher ion acceleration energies.
