# A Decentralized Receiver in Gaussian Interference
## Appendix B Achievability of Bounds
If some \(n\in[0:N]\) has \(r_{n}=0\) or \(\rho_{n}=0\), the system is equivalent to the case where the \(n\)th helper is not present, so, without loss of generality, we assert that each \(r_{n},\rho_{n}>\varepsilon\). Fix some rate \(R<R_{G}(\boldsymbol{\rho}-\boldsymbol{1}\cdot\varepsilon)-\lambda\) and a block length \(T=t^{2}\in\mathbb{N}\). Form a message \(M\) uniform on \([1:2^{tR}]\). * **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 \(r_{n}\). Randomly bin the encodings down to rate \(\rho_{n}\). At the base find the codeword jointly typical with the binned encodings. Observe that probability of error is low by recognizing that the unbinned \(t\)-length encodings from helpers approximately contain the joint-information content of the helper's observations plus Gaussian distortion. * **Transmitter setup** Generate a codebook mapping \(\phi:[1:2^{TR}]\to\mathbb{C}^{T\times 1}\) where \[\geq\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+ th(W_{A})+tI(Y_{A};Y_{A}+W_{A})]\]9 (49) 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: \[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(Y _{A}+W_{A})]\]0 (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 \(\rho_{0}\) chosen large enough for Lemma 3 to hold. For each receiver \(n\in[0:N]\), form a random mapping \(\mathsf{Index}_{n}:\mathcal{L}_{n}^{t}\to[1:2^{t_{n}}]\) which takes on iid values at each \(u_{n}\in\mathcal{L}_{n}^{t}\):  
\[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(Y _{A}+W_{A})]\]1  
\(\mathsf{Index}_{n}\) is the binning scheme used by receiver \(n\). Distribute each \(\mathsf{Index}_{n}\) to helper \(n\) and the base. * **Transmission** To send a message \(M\in[1:2^{TR}]\), have the transmitter broadcast \(\mathbf{X}=(X_{1},\ldots,X_{T})=\phi(M)\). * **Helper encoding** Helper \(n\) (\(n\in[0:N]\)) observes a sequence of length \(T\), \(\mathbf{Y}_{n}=(Y_{n}^{\ell})_{\ell}\in(\mathbb{C}^{t\times 1})^{t}\). Now \(\mathbf{Y}_{n}-\tilde{\mathbf{W}}_{Q,n}=(Y_{n}^{\ell}-\tilde{\mathbf{W}}_{Q,n}^{\ell})_{ \ell}\in\mathbb{C}^{T}=(\mathbb{C}^{t})^{t}\) 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]. Applying Lemma 2, then for large enough \(t\), any \(S\subseteq[0:N]\) has  
\[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(Y _{A}+W_{A})]\]2  
where the right-hand inequality comes from choice of \(\mathbf{\rho}\). * **Helper joint-compression and forwarding** At receiver \(n\in[0:N]\), form binned encodings \((V_{n}^{\ell})_{\ell\in[1:t]}\) where for each \(\ell\), \(V_{n}^{\ell}=\mathsf{Index}_{n}(U_{n}^{\ell})\). Note that \(|\mathsf{Range}(\mathsf{Index}_{n})|\leq 2^{t\cdot r_{n}}\) so \(H(V_{n}^{\ell})\leq t\cdot r_{n}\) and \(H((V_{n}^{\ell})_{\ell})\leq T\cdot r_{n}\) so that the LAN constraint is satisfied. Each receiver forwards \((V_{n}^{\ell})_{\ell}\) to the base.