Channel capacity in the non-asymptotic regime: Taylor-type expansion and computable benchmarks

In this paper, the non-asymptotic counterpart of Shannon capacity is investigated for any discrete input memory-less channel with discrete or continuous output (DIMC). Given any block length n and word error probability ϵ, let R_n(ϵ) be the best channel coding rate achievable with the block length n subject to the error probability e. Based on the non-asymptotic equipartition properties (NEP) established recently by Yang and Meng, a quantity δ_{t, n}(ϵ) is first defined to measure the relative magnitude of error probability ϵ and block length n with respect to a given DIMC and an input distribution t. Then, by combining the non-asymptotic achievability and converse established recently by Yang and Meng via jar decoding, it is shown that, given n and ϵ <; 1/2, R_n(ϵ) has a "Taylor-type expansion" with respect to δ_{t, n}(ϵ), with the first two terms of the expansion being max_t[I (t; P)-δ_{t, n}(ϵ)] = I(t*;P)-δ_{t*, n}(ϵ) for some optimal distribution t*, and the third order term being O(δ²_{t*, n}) + O(ln n/n). Finally, based on the Taylor-type expansion and the non-asymptotic converse, two easy to compute approximation formulas for R_n(ϵ) (dubbed “SO” and “NEP”) are provided. Numerical results show that both the SO and NEP approximation formulas provide reliable and accurate estimation, in contrast with the normal approximation which sometimes falls below achievable bounds and sometimes rises above converses. An important implication arising from the Taylor-type expansion of R_n(ϵ) is that in the practical non-asymptotic regime, the optimal marginal codeword symbol distribution is not necessarily a Shannon capacity achieving distribution.