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We derive the optimum second-order coding rates, known as second-ordercapacities, for erasure and list decoding. For erasure decoding for discretememoryless channels, we show that second-order capacity is$\sqrt{V}\Phi^{-1}(\epsilon_t)$ where $V$ is the channel dispersion and$\epsilon_t$ is the total error probability, i.e., the sum of the erasure andundetected errors. We show numerically that the expected rate at finiteblocklength for erasures decoding can exceed the finite blocklength channelcoding rate. We also show that the analogous result also holds for losslesssource coding with decoder side information, i.e., Slepian-Wolf coding. Forlist decoding, we consider list codes of deterministic size that scales as$\exp(\sqrt{n}l)$ and show that the second-order capacity is$l+\sqrt{V}\Phi^{-1}(\epsilon)$ where $\epsilon$ is the permissible errorprobability. We also consider lists of polynomial size $n^\alpha$ and derivebounds on the third-order coding rate in terms of the order of the polynomial$\alpha$. These bounds are tight for symmetric and singular channels. Thedirect parts of the coding theorems leverage on the simple threshold decoderand converses are proved using variants of the hypothesis testing converse.

Fixed Error Asymptotics For Erasure and List Decoding