New non-asymptotic random coding theorems (with error probability 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

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... stant or sub-exponentially decreasing . Numerically, they are also compared favourably, for finite n and of practical interest, with existing non-asymptotic achievability bounds in the literature in general. j4meng@uwaterloo.ca * This result on Gallager parity check ensemble was later enhanced by Shulman and Feder [5], who showed that the non-exponential term could be further eliminated. § Replacing M in (3.2) by (M −1)/2 yields exactly the Dependence Testing Bound [1, Theorem 34].