We present the results of a survey of damped (DLA, log N(H I) > 20.3) and sub-damped Lyman-α systems (19.5 2.55 along the lines-of-sight to 77 quasars with emission redshifts in the range 4 19.5 were detected of which 40 systems are damped Lyman-α systems for an absorption length of ΔX = 378. About half of the lines of sight of this homogeneous survey have never been investigated for DLAs. We study the evolution with redshift of the cosmological density of the neutral gas and find, consistent with previous studies at similar resolution, that Ω_(DLA,HI) decreases at z > 3.5. The overall cosmological evolution of Ω_(HI) shows a peak around this redshift. The H I column density distribution for log N(H I) ≥ 20.3 is fitted, consistent with previous surveys, with a single power-law of index α ~ −1.8 ± 0.25. This power-law overpredicts data at the high-end and a second, much steeper, power-law (or a gamma function) is needed. There is a flattening of the function at lower H I column densities with an index of α ~ −1.4 for the column density range log N(H I) = 19.5−21. The fraction of H I mass in sub-DLAs is of the order of 30%. The H I column density distribution does not evolve strongly from z ~ 2.5 to z ~ 4.5
