  We present the results of a survey for damped (DLA, log N(H I) > 20.3) and
sub-damped Lyman-? systems (19.5 < log N(H I) < 20.3) at z > 2.55 along the
lines-of-sight to 77 quasars with emission redshifts in the range 4 < zem <
6.3. Intermediate resolution (R ? 4300) spectra have been obtained with the
Echellette Spectrograph and Imager (ESI) mounted on the Keck telescope. A total
of 100 systems with log N(H I) > 19.5 are detected of which 40 systems are
damped Lyman-? systems for an absorption length of ?X = 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, consis- tently with previous studies at similar
resolution, that ?DLA,H I decreases at z > 3.5. The overall cosmological
evolution of ?HI shows a peak around this redshift. The H I column density
distribution for log N(H I) ? 20.3 is ?tted, consistently with previous
surveys, with a single power-law of index ? ? -1.8$\pm$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 ? ? ?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 column density distribution does not evolve strongly from z ? 2.5 to z ? 4.5.
