factor of 2 from , since $$\begin{aligned}
 & \mathbf{E}\left[\left(1+P_{Z}(\mathcal{B}_{\mathsf{D}}(W))2^{\iota_{X;Y}(X;Y)}\right)^{-1}\right]\\
 & \le\mathbf{E}\left[\left(1+(2J)^{-1}2^{\iota_{X;Y}(X;Y)}\right)^{-1}\right]+\mathbf{P}\left\{ (2J)^{-1}\ge P_{Z}(\mathcal{B}_{\mathsf{D}}(W))\right\} \\
 & \le\mathbf{E}\left[\min\left\{ 2J2^{-\iota_{X;Y}(X;Y)},\,1\right\} \right]+2\mathbf{E}\left[\max\{1-JP_{Z}(\mathcal{B}_{\mathsf{D}}(W)),\,0\}\right]\\
 & \le2\mathbf{E}\left[\min\left\{ J2^{-\iota_{X;Y}(X;Y)},\,1\right\} \right]+2\mathbf{E}\left[(1-P_{Z}(\mathcal{B}_{\mathsf{D}}(W)))^{J}\right].\end{aligned}$$ However, does not imply a bound that is within a constant factor from our bound. Theorem 8 in [@kostina2013joint] is obtained by substituting $J=\lfloor\gamma/P_{Z}(\mathcal{B}_{\mathsf{D}}(W))\rfloor$ in : $$P_{e}\le\mathbf{E}\left[\min\left\{ \gamma P_{Z}(\mathcal{B}_{\mathsf{D}}(W))^{-1}2^{-\iota_{X;Y}(X;Y)},\,1\right\} \right]+e^{1-\gamma},$$ which is strictly weaker than our bound with an unbounded multiplicative gap $\gamma$ (that tends to $\infty$ when the bound tends to 0). Hence our bound is stronger than Theorem 7 and 8 in [@kostina2013joint] (ignoring constant multiplicative gaps). Also our proof is significantly shorter than that of Theorem 7 in [@kostina2013joint].

Please refer to Appendix \[subsec:second\_jscc\] for the proof that Theorem \[thm:jscc\] achieves the optimal dispersion.

Poisson Matching Lemma Beyond the First Index\[sec:gen\_pml\]
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The Poisson functional representation concerns the point with the smallest $T_{i}((dP/d\mu)(\bar{U}_{i}))^{-1}$. We can generalize it to obtain a sequence ordered in ascending order of $T_{i}((dP/d\mu)(\bar{U}_{i}))^{-1}$.

\[Mapped Poisson process\] Let $\{\bar{U}_{i},T_{i}\}_{i\in\mathbb{N}}$ be the points of a Poisson process with intensity measure $\mu\times\lambda_{\mathbb{R}_{\ge0}}$ on $\mathcal{U}\times\mathbb{R}_{\ge0}$ (where $\mathcal{U}$ is a Polish space with its Borel $\sigma$-algebra, and $\mu$ is $\sigma$-finite). For $P\ll\mu$ a probability measure over $\mathcal{U}$, let $i_{P,1},i_{P,2},\ldots\in\mathbb{N}$ be a sequence of distinct integers such 