New non-asymptotic random channel coding theorems

New non-asymptotic random coding theorems (with error probability ϵ and finite block length n) based on Gallager parity check ensemble are established for binary input arbitrary output channels. The resulting non-asymptotic achievability bounds, when combined with non-asymptotic equipartition properties, can be easily computed. Analytically, these non-asymptotic achievability bounds are shown to be asymptotically tight up to the second order of the coding rate as n goes to infinity with either constant or sub-exponentially decreasing ϵ. Numerically, they are also compared favourably, for finite n and ϵ of practical interest, with existing non-asymptotic achievability bounds in the literature in general.