a one-shot version of the inner bound in [@liang2007broadcast Theorem 5] (which is shown to be equivalent to [@gelfand1980capacity Theorem 1] in [@liang2011equivalence]). The proof is given in Appendix \[subsec:pf\_bc\_cm\].

\[thm:bc\_cm\]Fix any $P_{U_{0},U_{1},U_{2}}$ and function $x:\mathcal{U}_{0}\times\mathcal{U}_{1}\times\mathcal{U}_{2}\to\mathcal{X}$. For any $\mathsf{J},\mathsf{K}_{1},\mathsf{K}_{2}\in\mathbb{N}$, there exists a code for the broadcast channel $P_{Y_{1},Y_{2}|X}$ for independent messages $M_{j}\sim\mathrm{Unif}[1:\mathsf{L}_{j}]$, $j=0,1,2$, with the error probability bounded by $$\begin{aligned}
P_{e} & \le\mathbf{E}\biggl[\min\biggl\{\tilde{\mathsf{L}}_{0}\tilde{\mathsf{L}}_{1}\mathsf{J}A2^{-\iota_{U_{0},U_{1};Y_{1}}(U_{0},U_{1};Y_{1})}+\tilde{\mathsf{L}}_{1}\mathsf{J}A2^{-\iota_{U_{1};Y_{1}|U_{0}}(U_{1};Y_{1}|U_{0})}\\
 & \;\;\;\;\;\;+\tilde{\mathsf{L}}_{0}\tilde{\mathsf{L}}_{2}\mathsf{J}^{-1}B2^{\iota_{U_{1};U_{2}|U_{0}}(U_{1};U_{2}|U_{0})-\iota_{U_{0},U_{2};Y_{2}}(U_{0},U_{2};Y_{2})}+\tilde{\mathsf{L}}_{0}\tilde{\mathsf{L}}_{2}(1-\mathsf{J}^{-1})B2^{-\iota_{U_{0},U_{2};Y_{2}}(U_{0},U_{2};Y_{2})}\\
 & \;\;\;\;\;\;+\tilde{\mathsf{L}}_{2}\mathsf{J}^{-1}B2^{\iota_{U_{1};U_{2}|U_{0}}(U_{1};U_{2}|U_{0})-\iota_{U_{2},Y_{2}|U_{0}}(U_{2};Y_{2}|U_{0})}+\tilde{\mathsf{L}}_{2}(1-\mathsf{J}^{-1})B2^{-\iota_{U_{2},Y_{2}|U_{0}}(U_{2};Y_{2}|U_{0})},\,1\biggr\}\biggr]\end{aligned}$$ if all the information density terms are defined almost surely, where $$\begin{aligned}
\tilde{\mathsf{L}}_{0} & :=\mathsf{L}_{0}\mathsf{K}_{1}\mathsf{K}_{2},\\
\tilde{\mathsf{L}}_{a} & :=\lceil\mathsf{L}_{a}/\mathsf{K}_{a}\rceil\;\mathrm{for}\,a=1,2,\\
A & :=(\log(\tilde{\mathsf{L}}_{1}^{-1}\mathsf{J}^{-1}2^{\iota_{U_{1};Y_{1}|U_{0}}(U_{1};Y_{1}|U_{0})}+1)+1)^{2},\\
B & :=\bigl(\log((\tilde{\mathsf{L}}_{2}\mathsf{J}^{-1}2^{\iota_{U_{1};U_{2}|U_{0}}(U_{1};U_{2}|U_{0})-\iota_{U_{2},Y_{2}|U_{0}}(U_{2};Y_{2}|U_{0})}\\
 & \;\;\;\;+\tilde{\mathsf{L}}_{2}(1-\mathsf{J}^{-1})2^{-\iota_{U_{2},Y_{2}|U_{0}}(U_{2};Y_{2}|U_{0})})^{-1}+1)+1\bigr)^{2}.\end{aligned}$$ As a result, for $\gamma>0$, $$\begin{aligned}
P_{e} & \le\mathbf{P}\biggl\{\log\tilde{\mathsf{L}}_{1}\mathsf{J}>\iota_{U_{1};Y_{1}|U_{0}}(U_{1};Y_{1}|U_{0})-\gamma\;\;\mathrm{or}\;\;\log\tilde{\mathsf{L}}_{2}>\iota_{U_{2},Y_{2}|U_{0}}(U_{2};Y_{2}|U_{0})-\gamma\nonumber \\
 & \;\;\;\;\mathrm{or}\;\;\log\tilde{\mathsf{L}}_{2}\mathsf{J}^{-1}>\iota_{U_{2},Y_{2}|U_{0}}(U_{2};Y_{2}|U_{0})-\iota_{U_{1};U_{2}|U_{0}}(U_{1};U_{2}|U_{0})-\gamma\nonumber \\
 & \;\;\;\;\mathrm{or}\;\;\log\tilde{\mathsf{L}}_{0}\tilde{\mathsf{L}}_{1}\mathsf{J}>\iota_{U_{0},U_{1};Y_{1}}(U_{0},U_{1};Y_{1})-\gamma\;\;\mathrm{or}\;\;\log\tilde{\mathsf{L}}_{0}\tilde{\mathsf{L}}_{2}>\iota_{U_{0},U_{2};Y_{2}}(U_{0},U_{2};Y_{2})-\gamma\nonumber \\
 & \;\;\;\;\mathrm{or}\;\;\log\tilde{\mathsf{L}}_{0}\tilde{\mathsf{L}}_{2}\mathsf{J}^{-1}>\iota_{U_{0},U_{2};Y_{2}}(U_{0},U_{2};Y_{2})-\iota_{U_{1};U_{2}|U_{0}}(U_{1};U_{2}|U_{0})-\gamma\biggr\}\nonumber \\
 & +2^{-\gamma}\left(8\mathbf{E}\left[(\iota_{U_{1};Y_{1}|U_{0}}(U_{1};Y_{1}|U_{0}))^{2}+(\iota_{U_{2},Y_{2}|U_{0}}(U_{2};Y_{2}|U_{0}))^{2}\right]+12\gamma^{2}+84\right).\label{eq:bc_pe2}\end{aligned}$$

The logarithmic terms $A$ and $B$ (or the last term in ) result in an $O(n^{-1}\log n)$ penalty on the rate in the finite blocklength regime, and do not affect the second order result. Ignoring the last term in , the error event in is a strict subset of those in [@yassaee2013oneshot eqn (32)] and [@liu2015oneshotmutual eqn (49)]. This is because the error event in [@yassaee2013oneshot] is a superset of by Fourier-Motzkin elimination on $\mathsf{J}_{2}$ in the error event in [@yassaee2013oneshot], but the reverse is not 