New Nonasymptotic Channel Coding Theorems for Structured Codes

New nonasymptotic random coding theorems (with error probability E and finite block length n) based on Gallager parity check ensemble and general parity check ensembles are derived in this paper. The resulting nonasymptotic achievability bounds, when combined with nonasymptotic equipartition properties developed in this paper, can be easily computed. Analytically, these nonasymptotic 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 subexponentially decreasing E in the case of Gallager parity check ensemble, and to imply that low density parity check (LDPC) codes be capacity-achieving in the case of LDPC ensembles. Numerically, they are also compared favorably, for finite n and E of practical interest, with existing nonasymptotic achievability bounds in the literature.