$U_{1},U_{2}$, and does not appear to yield a one-shot bound due to the use of typical sequences.

In the conventional mutual covering approach in [@yassaee2013oneshot; @liu2015oneshotmutual], sub-codebooks for both $U_{1}$ and $U_{2}$ are generated, whereas in our approach we generate a sub-codebook only for $U_{1}$, and the codebook of $U_{2}$ adapts to the sub-codebook automatically, eliminating the need for a sub-codebook for $U_{2}$.

\[thm:bc\_ncm\]Fix any $P_{U_{1},U_{2}}$ and function $x:\mathcal{U}_{1}\times\mathcal{U}_{2}\to\mathcal{X}$. For any $\mathsf{J}\in\mathbb{N}$, there exists a code for the broadcast channel $P_{Y_{1},Y_{2}|X}$ for independent private messages $M_{j}\sim\mathrm{Unif}[1:\mathsf{L}_{j}]$, $j=1,2$, with the error probability bounded by $$\begin{aligned}
P_{e} & \le\mathbf{E}\biggl[\min\biggl\{\mathsf{L}_{1}\mathsf{J}2^{-\iota_{U_{1};Y_{1}}(U_{1};Y_{1})}+\mathsf{L}_{2}(1-\mathsf{J}^{-1})2^{-\iota_{U_{2};Y_{2}}(U_{2};Y_{2})}+\mathsf{L}_{2}\mathsf{J}^{-1}2^{\iota_{U_{1};U_{2}}(U_{1};U_{2})-\iota_{U_{2};Y_{2}}(U_{2};Y_{2})},\,1\biggr\}\biggr]\end{aligned}$$ if all the information density terms are defined, where $(U_{1},U_{2},X,Y_{1},Y_{2})\sim P_{U_{1}U_{2}}\delta_{x(U_{1},U_{2})}P_{Y_{1},Y_{2}|X}$.

Let $\{(\bar{U}_{1,i},\bar{M}_{1,i}),T_{1,i}\}_{i\in\mathbb{N}}$, $\{(\bar{U}_{2,i},\bar{M}_{2,i}),T_{2,i}\}_{i\in\mathbb{N}}$ be two independent Poisson processes with intensity measures $P_{U_{1}}\times P_{M_{1}}\times\lambda_{\mathbb{R}_{\ge0}}$ and $P_{U_{2}}\times P_{M_{2}}\times\lambda_{\mathbb{R}_{\ge0}}$ respectively, independent of $M_{1},M_{2}$.

The encoder would generate $X$ such that $$(M_{1},M_{2},K,\{\check{U}_{1j}\}_{j\in[1:\mathsf{J}]},U_{1},U_{2},X)\sim P_{M_{1}}\times P_{M_{2}}\times P_{K}P_{U_{1}}^{\otimes\mathsf{J}}\delta_{\check{U}_{1K}}P_{U_{2}|U_{1}}\delta_{x(U_{1},U_{2})},\label{eq:bc_dist}$$ where $P_{K}=\mathrm{Unif}[1:\mathsf{J}]$, and $\{\check{U}_{1j}\}_{j\in[1:\mathsf{J}]}\in\mathcal{U}_{1}^{\mathsf{J}}$ is an intermediate list (which can be regarded as a sub-codebook). The term $P_{U_{1}}^{\otimes\mathsf{J}}\delta_{\check{U}_{1K}}$ in means that $\{\check{U}_{1j}\}_{j}$ are i.i.d. $P_{U_{1}}$, and $U_{1}=\check{U}_{1K}$. To accomplish this, the encoder computes $\check{U}_{1j}=(\tilde{U}_{1})_{P_{U_{1}}\times\delta_{M_{1}}}(j)$ for $j=1,\ldots,\mathsf{J}$ (which Poisson process we are referring to can be deduced from whether we are discussing $U_{1}$ or $U_{2}$), $U_{2}=(\tilde{U}_{2})_{\mathsf{J}^{-1}\sum_{j=1}^{\mathsf{J}}P_{U_{2}|U_{1}}(\cdot|\check{U}_{1j})\times\delta_{M_{2}}}$, and $(K,U_{1})|(\{\check{U}_{1j}\}_{j},U_{2})\sim P_{K,U_{1}|\{\check{U}_{1j}\}_{j},U_{2}}$ (where $P_{K,U_{1}|\{\check{U}_{1j}\}_{j},U_{2}}$ is 