  We compute the escape of ionizing radiation from galaxies in the redshift
interval z=4-10, i.e., during and after the epoch of reionization, using a
high-resolution set of galaxies, formed in fully cosmological simulations. The
simulations invoke early, energetic feedback, and the galaxies evolve into a
realistic population at z=0. Our galaxies cover nearly four orders of magnitude
in masses (10^{7.8}-10^{11.5}\msun) and more than five orders in star formation
rates (10^{-3.5}-10^{1.7}\msun\yr^{-1}), and we include an approximate
treatment of dust absorption. We show that the source-averaged Lyman-limit
escape fraction at z=10.4 is close to 80% declining monotonically with time as
more massive objects build up at lower redshifts. Although the amount of dust
absorption is uncertain to 1-1.5 dex, it is tightly correlated with
metallicity; we find that dust is unlikely to significantly impact the observed
UV output. These results support reionization by stellar radiation from
low-luminosity dwarf galaxies and are also compatible with Lyman continuum
observations and theoretical predictions at z\sim3-4.
