[@yassaee2013oneshot].

Using the generalized Poisson matching lemma that will be introduced in Section \[sec:gen\_pml\], we can prove the following bound. The proof is in Appendix \[subsec:channel2\].

\[thm:channel2\]Fix any $P_{X}$. There exists a code for the channel $P_{Y|X}$, with message $M\sim\mathrm{Unif}[1:\mathsf{L}]$, with average error probability $$P_{e}\le\mathbf{E}\left[1-\left(1-\min\left\{ 2^{-\iota_{X;Y}(X;Y)},\,1\right\} \right)^{(\mathsf{L}+1)/2}\right]$$ if $P_{XY}\ll P_{X}\times P_{Y}$.

Compare this to the dependence testing bound [@polyanskiy2010channel]: $$P_{e}\le\mathbf{E}\left[\min\left\{ \frac{\mathsf{L}-1}{2}\cdot2^{-\iota_{X;Y}(X;Y)},\,1\right\} \right].$$ Theorem \[thm:channel2\] is at least as strong (with a slight penalty of having $(\mathsf{L}+1)/2$ instead of $(\mathsf{L}-1)/2$) since $$\begin{aligned}
 & \mathbf{E}\left[1-\left(1-\min\left\{ 2^{-\iota_{X;Y}(X;Y)},\,1\right\} \right)^{(\mathsf{L}+1)/2}\right]\\
 & \le\mathbf{E}\left[\min\left\{ \frac{\mathsf{L}+1}{2}\cdot2^{-\iota_{X;Y}(X;Y)},\,1\right\} \right].\end{aligned}$$

Apart from the dependence testing bound [@polyanskiy2010channel], there are other one-shot bounds for channel coding such as the random-coding union (RCU) bound and the $\kappa\beta$ bound in [@polyanskiy2010channel], which are tighter in certain situations (e.g. the RCU bound is suitable for error exponent analysis). The technique introduced in this paper is suitable for first and second order analysis, but does not seem to give tight error exponent bounds.

One-shot Coding for Channels with State Information at the Encoder\[sec:channel\_state\]
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The one-shot coding setting for a channel with state information at the encoder is described as follows. Upon observing $M\sim\mathrm{Unif}[1:\mathsf{L}]$ and $S\sim P_{S}$, the encoder produces $X$, which is sent through the 