The mod-$\Lambda$ Scheme: Erez–Zamir Capacity

Fix any $\mathbf{x} = \mathbf{x}_0$. Then $\mathbf{x}_0 + \mathbf{d}$ has the distribution of $\mathbf{d}$ shifted by $\mathbf{x}_0$: uniform on the set $\mathbf{x}_0 + \mathcal{V}(\Lambda)$. This is another fundamental region of $\Lambda$, because any translate of a fundamental region is again a fundamental region.

The map $\mathbf{y} \mapsto [\mathbf{y}] \bmod \Lambda$ is a bijection from any fundamental region of $\Lambda$ onto the canonical Voronoi cell $\mathcal{V}(\Lambda)$ (it pushes each coset representative to its canonical representative). Moreover the map is Lebesgue-measure-preserving because translation by a lattice vector preserves Lebesgue measure.

Therefore $[\mathbf{x}_0 + \mathbf{d}] \bmod \Lambda$ is uniform on $\mathcal{V}(\Lambda)$ — and this holds for every $\mathbf{x}_0$. Since the conditional distribution of $[\mathbf{x} + \mathbf{d}] \bmod \Lambda$ given $\mathbf{x}$ is the same uniform distribution for every value of $\mathbf{x}$, the two random variables are independent. $\blacksquare$

By the crypto-lemma (Theorem above), the transmit signal $\mathbf{x} = [\mathbf{c}(u) - \mathbf{d}] \bmod \Lambda_s$ is uniform on $\mathcal{V}(\Lambda_s)$ and statistically independent of the message $u$. Its per-dimension second moment is $P =$ second moment of $\mathcal{V}(\Lambda_s)$, satisfying the power constraint with equality.

At the decoder, $\mathbf{r} = [\alpha \mathbf{y} + \mathbf{d}] \bmod \Lambda_s = [\alpha(\mathbf{x} + \mathbf{w}) + \mathbf{d}] \bmod \Lambda_s = [\mathbf{c}(u) + (1 - \alpha)(-\mathbf{d}) + \alpha \mathbf{w}] \bmod \Lambda_s$, where we used $\alpha \mathbf{x} + \mathbf{d} = \alpha[\mathbf{c}(u) - \mathbf{d}] + \mathbf{d} = \alpha \mathbf{c}(u) + (1 - \alpha) \mathbf{d} \equiv \mathbf{c}(u) + (1-\alpha)(-\mathbf{d}) \pmod{\Lambda_s}$ by $\mathbf{c}(u) \in \Lambda_c \supset \Lambda_s$ and a $(1 - \alpha) \alpha \mathbf{c}(u)$ absorption into the mod. Define the effective noise $\mathbf{z} := (1 - \alpha)( -\mathbf{d}) + \alpha \mathbf{w}$. Its per-dimension second moment is $$\mathbb{E}\bigl[\tfrac{\|\mathbf{z}\|^2}{n}\bigr] \;=\; (1 - \alpha)^2 P + \alpha^2 \sigma^2 \;=\; \alpha \sigma^2.$$ The last equality uses $\alpha = P/(P + \sigma^2)$: $(1 - \alpha)^2 P + \alpha^2 \sigma^2 = (\sigma^2/(P + \sigma^2))^2 P + (P/(P + \sigma^2))^2 \sigma^2 = P \sigma^2 / (P + \sigma^2) = \alpha \sigma^2$.

The decoder sees $\mathbf{r} = [\mathbf{c}(u) + \mathbf{z}] \bmod \Lambda_s$ with $\mathbb{E}[\|\mathbf{z}\|^2]/n = \alpha \sigma^2$. If $\mathbf{z}$ were purely Gaussian, this would be the unconstrained lattice channel (Poltyrev channel) at effective noise power $\alpha \sigma^2$, whose capacity is $\tfrac12 \log_2 (1 / (2 \pi e G(\Lambda_s)) \cdot V(\Lambda_s)^{2/n} / (\alpha \sigma^2))$ by Poltyrev's analysis. The dither component $(1 - \alpha)(-\mathbf{d})$ is not Gaussian — it is uniform on a scaled Voronoi region — but a key lemma (Erez–Zamir's MMSE-inflation lemma, also known as the "Linder–Zamir" / Zamir–Feder moment inequality) shows that for any lattice $\Lambda_s$ with finite second moment, Gaussian decoding of $\mathbf{c}(u)$ through the $\mathbf{z}$- noise achieves the same asymptotic error performance as through a pure Gaussian of the same variance. The key step is Lemma 6 of Erez–Zamir: the cumulant-generating function of $\mathbf{z}$ is dominated by that of a Gaussian of variance $\alpha \sigma^2$.

By the Loeliger random-lattice averaging argument (Ch. 15 Minkowski–Hlawka, Theorem 15.x), for any rate $R' < \tfrac12 \log_2(1/(2 \pi e G(\Lambda_s)) \cdot V(\Lambda_s)^{2/n} / (\alpha \sigma^2))$ there exists a fine lattice $\Lambda_c \supset \Lambda_s$ with $\log_2 |\Lambda_c / \Lambda_s|/n \ge R'$ and decoding error probability (at effective noise variance $\alpha \sigma^2$) vanishing exponentially in $n$. The Poltyrev exponent is strictly positive whenever we are below the Poltyrev capacity, so ensemble averaging gives us what we need.

Rogers (1959) and Poltyrev (1994) proved that there exist lattices with normalised second moment $G(\Lambda_s) \to 1/(2 \pi e)$ as $n \to \infty$. (The cube has $G(\mathbb{Z}^n) = 1/12$, so the shaping gap is $1/12 \div 1/(2 \pi e) = \pi e / 6 \approx 1.53$ dB — Ch. 4 and s03 of this chapter.) Substituting $G(\Lambda_s) \to 1/(2 \pi e)$ and $V(\Lambda_s)^{2/n} = P / G(\Lambda_s) \to 2 \pi e P$ (the second-moment identity $V^{2/n} G = P$), the rate bound in Step 4 becomes $$R' \;<\; \tfrac12 \log_2 \frac{2 \pi e P}{2 \pi e \cdot \alpha \sigma^2} \;=\; \tfrac12 \log_2 \frac{P}{\alpha \sigma^2} \;=\; \tfrac12 \log_2 \frac{P + \sigma^2}{\sigma^2} \;=\; \tfrac12 \log_2(1 + \text{SNR}).$$ The last equality used $\alpha = P/(P + \sigma^2)$ to cancel. Hence every $R < C$ is achievable by the mod-$\Lambda_s$ scheme with a suitable $(\Lambda_c, \Lambda_s)$. $\blacksquare$