The Computation Rate Theorem

Fix nested lattices $\Lambda_s \subset \Lambda_c \subset \mathbb{R}^n$ with $V(\Lambda_s) = P^{n/2}$ (so the second-moment power of $\mathcal{V}(\Lambda_s)$ is $P$) and coding rate $R_{\rm cb} = \tfrac{1}{n} \log_2(V(\Lambda_s)/V(\Lambda_c))$. Let $\Lambda_c$ achieve the Poltyrev exponent (from Ch. 16 — AWGN-good). All $K$ users share this codebook. User $k$ picks its message as a lattice point $\mathbf{u}_k \in \Lambda_c \cap \mathcal{V} (\Lambda_s)$, dithers with a uniform dither $\mathbf{d}_k \sim \mathrm{Unif}(\mathcal{V}(\Lambda_s))$ (known to the relay), and transmits $\mathbf{x}_k = (\mathbf{u}_k - \mathbf{d}_k) \bmod \Lambda_s$. The crypto-lemma (Erez-Zamir) makes $\mathbf{x}_k$ uniform over $\mathcal{V}(\Lambda_s)$, hence power $P$ per dimension, hence satisfying the power constraint.

The relay scales $\mathbf{y}$ by $\alpha \in \mathbb{R}$ and adds back the $\mathbf{a}$-weighted dithers: $$\mathbf{y}' = \alpha \mathbf{y} + \sum_{k=1}^K a_k \mathbf{d}_k = \sum_{k=1}^K a_k (\mathbf{x}_k + \mathbf{d}_k) + (\alpha \mathbf{h} - \mathbf{a})^T \text{(signals)} + \alpha \mathbf{w}.$$ Writing $\mathbf{x}_k + \mathbf{d}_k \equiv \mathbf{u}_k \pmod {\Lambda_s}$, we get $\mathbf{y}' \equiv \sum_k a_k \mathbf{u}_k + \mathbf{w}_{\rm eff} \pmod{\Lambda_s}$ where the effective noise is $$\mathbf{w}_{\rm eff} \;=\; (\alpha \mathbf{h} - \mathbf{a})^T \begin{pmatrix} \mathbf{x}_1 \\ \vdots \\ \mathbf{x}_K \end{pmatrix} + \alpha \mathbf{w}.$$

Under the crypto-lemma, $\mathbf{x}_k$ is uniform over $\mathcal{V}(\Lambda_s)$ — in particular, the $\mathbf{x}_k$'s are independent and have covariance $P \mathbf{I}_n /$ (approximately, by the Erez-Zamir second-moment result — sphere-like shaping makes $\mathcal{V}(\Lambda_s)$ look like an isotropic ball). The per-dimension variance of $\mathbf{w}_{\rm eff}$ is $$N(\alpha, \mathbf{h}, \mathbf{a}) \;=\; P \|\alpha \mathbf{h} - \mathbf{a}\|^2 + \alpha^2.$$ Minimise over $\alpha$: $\partial N / \partial \alpha = 2 P \mathbf{h}^T(\alpha \mathbf{h} - \mathbf{a}) + 2 \alpha = 0$, giving $$\alpha^*(\mathbf{h}, \mathbf{a}) = \frac{P\, \mathbf{h}^T \mathbf{a}}{1 + P \|\mathbf{h}\|^2}.$$ Substituting back, with $\text{SNR} = P$: $$N^* = P \|\mathbf{a}\|^2 - \frac{P^2 (\mathbf{h}^T \mathbf{a})^2} {1 + P \|\mathbf{h}\|^2} = P \cdot N_{\rm eff}(\mathbf{h}, \mathbf{a})$$ where we defined $N_{\rm eff}(\mathbf{h}, \mathbf{a}) = \|\mathbf{a}\|^2 - \text{SNR}(\mathbf{h}^T \mathbf{a})^2 / (1 + \text{SNR} \|\mathbf{h}\|^2)$.

The relay applies $\mathbf{y}' \bmod \Lambda_s$ and then rounds to the nearest $\Lambda_c$ point within $\mathcal{V}(\Lambda_s)$. The sum $\sum_k a_k \mathbf{u}_k \in \Lambda_c$ by closure of $\Lambda_c$ under integer linear combinations; its residue mod $\Lambda_s$ lies in $\mathcal{V}(\Lambda_s)$. The decoder succeeds iff $\mathbf{w}_{\rm eff}$ falls inside $\mathcal{V}(\Lambda_c)$ — the fine-lattice Voronoi region. By the Poltyrev exponent (Erez-Zamir), this happens w.p. $\to 1$ whenever the coding rate satisfies $$R_{\rm cb} < \tfrac{1}{2} \log_2 (P / N^*) = \tfrac{1}{2} \log_2(1 / N_{\rm eff}(\mathbf{h}, \mathbf{a})).$$ Taking the truncation $\log_2^+$ (the rate is $0$ when $N_{\rm eff} > 1$, i.e., when the integer-coefficient mismatch dominates) yields the theorem. $\blacksquare$

Every step above uses $\mathbf{a}$ only to define which lattice combination the relay targets. The codebook $\Lambda_c \supset \Lambda_s$ is $\mathbf{a}$-independent; the transmitters do not even know $\mathbf{a}$. The relay's scalar scaling $\alpha^*$ and the effective noise both depend on $\mathbf{a}$, but neither depends on the $K$ individual messages. This is the deep property that makes CF useful: the same codebook works for every $\mathbf{a}$, and the relay can choose $\mathbf{a}$ post-hoc, after observing $\mathbf{h}$.