# A Decentralized Receiver in Gaussian Interference
## Appendix B Achievability of Bounds
Note that, for any \(S\subseteq[1:N]\),  
\[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(W_ {A})+I(\mathbf{Y}_{A};\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})]\]3 \[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(W_ {A})+I(\mathbf{Y}_{A};\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})]\]4 (113) \[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(W_ {A})+I(\mathbf{Y}_{A};\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})]\]5 (114) \[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(W_ {A})+I(\mathbf{Y}_{A};\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})]\]6  
where Equation (114) follows because conditioning reduces mutual information, and Equation (115) follows from what was shown at the start. For a distributed receive system as described in Section 2 with noise covariance matrix \(\mathbf{\Sigma}\), LAN constraint \(L\) and fixed average helper quantization rates \(\mathbf{r}\in\mathcal{R}_{\text{LAN}}(L)\), a rate \(R_{G}(\mathbf{r})\) is achievable. Proof. : If some \(n\in[0:N]\) has \(r_{n}=0\), the system is equivalent to the case where the \(n\)th receiver's observation is not present, so, without loss of generality, we assert that \(r_{n}>\varepsilon\). Fix some rate \(R<R_{G}(\mathbf{r})\) and a block length \(T=t^{2}\in\mathbb{N}\). Form a message \(M\) uniform on \([1:2^{tR}]\). \(\bullet\) **Outline**  
Form a rate-\(R\) codebook with length-\(T\) codewords. At receiver \(n\in[0:N]\), form a length-\(t\) regular lattice encoder with lattice chosen coarse enough that its encodings have rate \(r_{n}\). Combine encodings at base and find the typical codeword. Observe that probability of error is low by recognizing that each \(t\)-length encoding from a helper approximately contains the information content of the helper's observation plus Gaussian distortion. \(\bullet\) **Transmitter setup**  
Generate a codebook mapping \(\phi:[1:2^{TR}]\to\mathbb{C}^{T\times 1}\) where:  
\[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(W_ {A})+I(\mathbf{Y}_{A};\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})]\]7  
with each component \(x_{m,\ell}\) drawn iid from \(\mathcal{CN}(0,1)\). Reveal \(\phi\) to the transmitter and base. * **Helper encoder setup** For each helper \(n,\ n\in[1:N]\), generate a dither vector \(\tilde{\mathbf{W}}_{Q,n}=(\tilde{\mathbf{W}}_{Q,n}^{\ell})_{\ell}\in(\mathbb{C}^{t\times 1 })^{t}\), where each successive \(t\)-segment is uniform in the base region of the Voroni partition \(\mathcal{P}_{n}^{t}\) of a regular, white \(t\)-dimensional lattice \(\mathcal{L}_{n}^{t}\) which has the following normalized-second-moment: \[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(W_ {A})+I(\mathbf{Y}_{A};\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})]\]8 (100)  
These terms are detailed at the beginning of Reference [20]. Reveal \(\tilde{\mathbf{W}}_{Q,n}\) to the base and helper \(n\). Similarly, at the base, generate a dither vector \(\tilde{\mathbf{W}}_{Q,0}=(\tilde{\mathbf{W}}_{Q,0}^{\ell})_{\ell}\in(\mathbb{C}^{t \times 1})^{t}\) with \(r_{0}\) chosen large enough for Lemma 3 to hold. * **Transmission** To send message \(M\in[1:2^{TR}]\), have the transmitter broadcast \(\mathbf{X}=(X_{1},\ldots,X_{T})=\phi(M)\). * **Helper encoding and forwarding** Higher \(n\) (\(n\in[0:N]\)) observes a sequence of length \(T\), \(\mathbf{Y}_{n}=(\mathbf{Y}_{n}^{\ell})_{\ell}\in(\mathbb{C}^{t\times 1})^{t}\). Now \(\mathbf{Y}_{n}-\tilde{\mathbf{W}}_{Q,n}=(\mathbf{Y}_{n}^{\ell}-\tilde{\mathbf{W}}_{Q,n}^{\ell} )_{\ell}\in\mathbb{C}^{T}=(\mathbb{C}^{t})^{t}\) is composed of \(t\) consecutive length-\(t\) segments. For the \(\ell\)-th length-\(t\) segment (\(\ell\in[1:t]\)), form a quantization \(U_{n}^{\ell}\in\mathcal{L}_{n}^{t}\) by finding the point in \(\mathcal{L}_{n}^{t}\) associated with the region in \(\mathcal{P}_{n}^{t}\) in which that segment resides. The properties of such \(U_{n}\) are the subject of References [20; 29].