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Recently, a new decoding rule called jar decoding was proposed; under jardecoding, a non-asymptotic achievable tradeoff between the coding rate and worderror probability was also established for any discrete input memorylesschannel with discrete or continuous output (DIMC). Along the path ofnon-asymptotic analysis, in this paper, it is further shown that jar decodingis actually optimal up to the second order coding performance by establishingnew non-asymptotic converse coding theorems, and determining the Taylorexpansion of the (best) coding rate $R_n (\epsilon)$ of finite block length forany block length $n$ and word error probability $\epsilon$ up to the secondorder. Finally, based on the Taylor-type expansion and the new converses, twoapproximation formulas for $R_n (\epsilon)$ (dubbed "SO" and "NEP") areprovided; they are further evaluated and compared against some of the bestbounds known so far, as well as the normal approximation of $R_n (\epsilon)$revisited recently in the literature. It turns out that while the normalapproximation is all over the map, i.e. sometime below achievable bounds andsometime above converse bounds, the SO approximation is much more reliable asit is always below converses; in the meantime, the NEP approximation is thebest among the three and always provides an accurate estimation for $R_n(\epsilon)$. An important implication arising from the Taylor-type expansion of$R_n (\epsilon)$ is that in the practical non-asymptotic regime, the optimalmarginal codeword symbol distribution is not necessarily a capacity achievingdistribution.

Jar Decoding: Non-Asymptotic Converse Coding Theorems, Taylor-Type Expansion, and Optimality