  New non-asymptotic random coding theorems (with error probability $\epsilon$
and finite block length $n$) based on Gallager parity check ensemble and
Shannon random code ensemble with a fixed codeword type are established for
discrete input arbitrary output channels. The resulting non-asymptotic
achievability bounds, when combined with non-asymptotic equipartition
properties developed in the paper, 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 $\epsilon$. Numerically, they are also
compared favourably, for finite $n$ and $\epsilon$ of practical interest, with
existing non-asymptotic achievability bounds in the literature in general.
