# A Decentralized Receiver in Gaussian Interference
## Appendix B Achievability of Bounds
* **Decoding** At the base, receive all all the binned helper encodings \(\mathbf{V}=((V_{n}^{\ell})_{\ell})_{n}\). Take \(A_{\varepsilon}^{T}(\mathbf{X},\mathbf{U}|\mathbf{W}_{Q})\) to be the set of \[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(Y _{A}+W_{A})]\]3 (103) which are jointly \(\varepsilon\)-weakly-typical with respect to the joint distribution of \((\mathbf{X},\mathbf{U})\), conditional on all the dithers \((\tilde{\mathbf{W}}_{Q,n})_{n\in[0:N]}\). Weak- and joint-typicality are defined in Reference [14]. For an ensemble of bin indices from all the receivers, \(\mathbf{v}=((v_{n}^{\ell})_{\ell})_{n}\in\prod_{n=0}^{N}[1:2^{t\cdot r_{n}}]^{t}\), define: \[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(Y _{A}+W_{A})]\]4 (104)Each \(B_{\mathbf{v}}\) represents the set of helper lattice quantizations represented by the ensemble of helper bin indices \(\mathbf{v}\). At the base, find \(\hat{m}\) for which there is some \(\hat{\mathbf{u}}\in B_{\mathbf{V}}\) where \((\phi(\hat{m}),\hat{\mathbf{u}})\in A_{\varepsilon}^{T}(\mathbf{X},\mathbf{U}|\mathbf{W}_{Q})\). Declare the message associated with \(\hat{x}\) to be the broadcast. * **Error analysis** We have the following error events:
* \(\mathcal{E}_{\mathcal{I}T}\): \(\mathbf{X}\) is not typical with anything in \(B_{\mathbf{V},[0:N]}\). * \(\mathcal{E}_{\mathcal{I}n,S}\): For \(S\subseteq[0:N]\), there is some \(\hat{\mathbf{m}}\
eq M\) and \(\hat{\mathbf{u}}=(\hat{\mathbf{u}}_{S},\hat{\mathbf{u}}_{SC})\in B_{\mathbf{V}}\) where \((\phi(\hat{m}),\hat{\mathbf{u}})\in A_{\varepsilon}^{T}(\mathbf{X},\mathbf{U}|\mathbf{W}_{Q})\) and \(\hat{\mathbf{u}}_{SC}=\mathbf{U}_{SC}\), but any \(n\in S\) has \(\hat{\mathbf{u}}_{n}\
eq\mathbf{U}_{n}\). \(\mathcal{E}_{\mathcal{I}n,S}\) denotes the situation where the base identifies the wrong quantizations for all the receivers in \(S\). By the law of large numbers and properties of typical sets, as \(t\) becomes large, eventually \(P(\mathcal{E}_{\mathcal{I}T})<\varepsilon\). For each \(S\subseteq[0:N]\), take \(A_{\varepsilon}^{T}(\mathbf{U}|\mathbf{W}_{Q})\) to be the collection of jointly-typical sequences in \((\prod_{n}\mathcal{L}_{n}^{t})^{t}\) up to the joint distribution on \((\mathbf{U}_{n}^{t})_{n\in S}\) (any \(\ell\in[1:t]\)), conditioned on \((\tilde{\mathbf{W}}_{Q,n})_{n}\). Now compute:  
\[\sum_{\hat{m}\
eq M}P(\mathcal{E}_{\mathcal{I}n,S})\leq 2^{TR}\cdot\sum_{ \begin{subarray}{c}\mathbf{\hat{u}}\in A_{\varepsilon}^{T}(\mathbf{U}|\mathbf{W}_{Q}):\\ \hat{\mathbf{u}}_{SC}=\mathbf{U}_{SC}\end{subarray} }\mathbf{1}_{\tilde{\mathbf{W}}_{\mathbf{V}}}( \tilde{\mathbf{u}})\cdot P\left((\tilde{\mathbf{X}},\tilde{\mathbf{u}})\in A_{\varepsilon }^{T}(\mathbf{X},\mathbf{U}|\mathbf{W}_{Q})\right) \tag{101}\]  
where \(\tilde{\mathbf{X}}\) is a random variable independent of and distributed identically to \(\mathbf{X}\). Then, with high probability for large enough \(t\),  
\[\sum_{\hat{m}\
eq M}P(\mathcal{E}_{\mathcal{I}n,S}) \leq 2^{TR}\cdot\sum_{ \begin{subarray}{c}\mathbf{\hat{u}}\in A_{ \varepsilon}^{T}(\mathbf{U}|\mathbf{W}_{Q}):\\ \tilde{\mathbf{u}}_{SC}=\mathbf{U}_{SC}\end{subarray} }\mathbf{1}_{\tilde{\mathbf{W}}_{\mathbf{V}}}( \tilde{\mathbf{u}})\cdot 2^{-t\cdot(I(\mathbf{X};\mathbf{U}|(\mathbf{W}_{Q,n})_{n})-3\varepsilon)} \tag{102}\] \[\leq 2^{TR}\cdot\left|\left\{\tilde{\mathbf{u}}\in A_{\varepsilon}^{T}( \mathbf{U}|\mathbf{W}_{Q})|\hat{\mathbf{u}}_{SC}=\mathbf{U}_{SC}\right\}\right|\cdot(1+ \varepsilon)\cdot P(\mathbf{U}\in B_{\mathbf{V}})\cdot\] (103) \[2^{-t\cdot(I(\mathbf{X};\mathbf{U}|(\mathbf{W}_{Q,n})_{n})-3\varepsilon)}\] \[\leq 2^{TR}\cdot 2^{t\cdot(H(\mathbf{u}_{S}|\mathbf{u}_{SC},(\mathbf{W}_{Q,n})_{n })+\varepsilon)}\cdot 2^{-t\cdot(t\cdot\sum_{n\in S}r_{n}-\varepsilon)}\cdot 2^{-t \cdot(I(\mathbf{X};\mathbf{U}|(\mathbf{W}_{Q,n}^{t})_{n})-3\varepsilon)} \tag{104}\]  
where Equation (104) follows by construction of \(B_{\mathbf{V}}\).