Seeding plasma-based soft-x-ray lasers (PBSXRL) with high order harmonics (HOH) has been demonstrated in plasmas created from gas targets (Zeitoun et al. in Nature 431:426, 2004) and solid targets (Wang et al. in Nat. Photonics 2:94, 2008), obtaining 1 mu J, 1 ps pulses. Reaching multi-microJoule, hundreds of fs regime is the ultimate goal. Recent papers (Oliva et al. in Opt. Lett. 34(17):26402642, 2009; Phys. Rev. E 82(5):056408, 2010) showed that increasing the width (up to 1 mm) of the plasma increases the amplification surface and improves the gain zone properties. Up to 20 mu J could be extracted when seeding but the temporal duration and profile was not studied. Simulations show that the HOH is weakly amplified whereas most of the energy is within a long (several picoseconds) wake induced by the HOH (Al'miev et al. in Phys. Rev. Lett. 99(12):123902, 2007; Kim et al. in Phys. Rev. Lett. 104:053901, 2010). Amplified Spontaneous Emission (ASE) is also present in the output beam. Using the 1D Maxwell-Bloch code DeepOne (Oliva et al. in Phys. Rev. A 84(1):013811, 2011) we will show that fully coherent, wake and ASE-suppressed, 21.6 mu J, 80 fs pulse can be obtained when optimizing at the same time both the seed and the plasma conditions