Exercises

$|\mathcal{C}| = |\Lambda_c / \Lambda_s| = V(4 \mathbb{Z}^4)/V(\mathbb{Z}^4) = 4^4 = 256$.

$R = T^{-1} \log_2 256 = (1/2) \cdot 8 = 4$ bits/ch.use. $\blacksquare$

The dither makes the transmit signal $\mathbf{x} = [\mathbf{G} \mathbf{u} + \mathbf{d}] \bmod \Lambda_s$ uniform on $\mathcal{V}(\Lambda_s)$ regardless of the message $\mathbf{u}$ (Erez-Zamir crypto lemma). This uniformity is the lattice analog of i.i.d. Gaussian random coding: it lets us compute ensemble- averaged error probabilities.

If $\mathbf{d} = \mathbf{0}$, the transmit signal is $\mathbf{G} \mathbf{u} \bmod \Lambda_s$, which depends deterministically on $\mathbf{u}$ and does not average to uniform. The DMT proof via Minkowski-Hlawka averaging then breaks: the error probability cannot be bounded by $\gamma_c^{-n_r}$ without the uniformity. Practically, undithered LAST codes have larger finite-SNR BER and worse asymptotic slope. $\blacksquare$

$\alpha = 1/10 = 0.1$, $\sqrt{\alpha} \approx 0.316$.

$\bar{\mathbf{H}} = \begin{pmatrix} 1 & 0.5 \\ 0.5 & 1 \\ 0.316 & 0 \\ 0 & 0.316 \end{pmatrix}$, a $4 \times 2$ real matrix. $\blacksquare$

From $\mathbf{Q}^H \mathbf{Q} = \mathbf{I}$ and the block partition: $\mathbf{Q}_1^H \mathbf{Q}_1 + \mathbf{Q}_2^H \mathbf{Q}_2 = \mathbf{I}$.

The bottom block of $\bar{\mathbf{H}} = \mathbf{Q} \mathbf{R}$ is $\sqrt{\alpha} \mathbf{I} = \mathbf{Q}_2 \mathbf{R}$, so $\mathbf{Q}_2 = \sqrt{\alpha} \mathbf{R}^{-1}$ and $\mathbf{Q}_2^H \mathbf{Q}_2 = \alpha (\mathbf{R}^H \mathbf{R})^{-1} = \alpha (\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I})^{-1}$.

$\mathbf{F}^H \mathbf{F} = \mathbf{Q}_1^H \mathbf{Q}_1 = \mathbf{I} - \alpha (\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I})^{-1} = (\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I} - \alpha \mathbf{I})(\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I})^{-1} = \mathbf{H}^{H} \mathbf{H} (\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I})^{-1}$, i.e., a non-unitary positive-definite operator. $\blacksquare$

(1) MMSE-GDFE preserves mutual information (this chapter, §2 Thm. TMMSE-GDFE Preserves Mutual Information). (2) Erez-Zamir lattice coding achieves the AWGN capacity on the effective channel. (3) Zheng-Tse Wishart-Laplace analysis giving $\Pr(\text{outage}) \doteq \text{SNR}^{-d^*(r)}$.

(1) is this chapter's §2. (2) is Ch. 16 (Erez-Zamir 2004) plus Ch. 15 (Minkowski-Hlawka random-lattice averaging). (3) is Ch. 12 (Zheng-Tse 2003). Composition of three prior results gives the LAST theorem. $\blacksquare$

$d^*(0) = 4$, $d^*(0.5) = 4 + (1-4) \cdot 0.5 = 2.5$, $d^*(1) = 1$, $d^*(1.5) = 1 + (0-1) \cdot 0.5 = 0.5$, $d^*(2) = 0$.

Diversity $d^*(1.5) = 0.5$. BER-vs-SNR slope is $-0.5$, i.e., BER $\propto \text{SNR}^{-0.5}$ at high SNR. This is a very shallow slope — at $r = 1.5$ on a $2 \times 2$ channel, the diversity advantage of the LAST code is minimal; we are essentially multiplexing at the full MIMO rate. $\blacksquare$

Per layer $i \in \{1, \ldots, n_t T\}$, the signal is $R_{ii} x_i$, the effective noise has unit variance, and the effective SNR is $R_{ii}^2$.

$\prod_i R_{ii}^2 = \det(\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I})^T$ (by Thm. TMMSE-GDFE Triangularises the MIMO Channel). Taking log: $\sum_i \log_2 R_{ii}^2 = T \log_2 \det(\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I})$, which equals the MIMO mutual information (up to the MMSE bias offset). The equivalent channel is a product of $n_t T$ scalar AWGN channels with varying per-layer SNR, whose aggregate capacity equals the full MIMO capacity. $\blacksquare$

$\gamma_c(D_4) = 2 / (1/2)^{2/4} = 2 / (1/\sqrt{2}) = 2\sqrt{2} \approx 2.83$. $\gamma_c(\mathbb{Z}^4) = 1 / 1^{1/2} = 1$. Ratio $\gamma_c(D_4) / \gamma_c(\mathbb{Z}^4) = 2\sqrt{2}$, or $\approx 4.5$ dB.

$\gamma_c^{-n_r} = (2\sqrt{2})^{-2} = 1/8$, i.e., $9$ dB of SNR shift at the same BER target. At the BER $10^{-4}$ operating point, structured-$D_4$-LAST operates $9$ dB below $\mathbb{Z}^4$-LAST. $\blacksquare$

The MMSE-GDFE filter $\mathbf{F} = \mathbf{Q}_1^H$ applied to $\mathbf{y}$ gives $\mathbf{z} = \mathbf{F} \mathbf{y}$ that is a sufficient statistic for $\mathbf{x}$, i.e., $I(\mathbf{x}; \mathbf{z}) = I(\mathbf{x}; \mathbf{y})$ and $\mathbf{x} \to \mathbf{z} \to \mathbf{y}$ form a Markov chain (trivially, as $\mathbf{z}$ is a function of $\mathbf{y}$).

From Exercise 4, $\mathbf{F}^H \mathbf{F} = \mathbf{H}^{H} \mathbf{H} (\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I})^{-1}$, which is positive-definite (hence full column rank) whenever $\mathbf{H}$ has full rank. A full-column-rank linear transformation of $\mathbf{y}$ preserves mutual information with $\mathbf{x}$ (data-processing equality for invertible linear maps). Hence $\mathbf{z}$ is a sufficient statistic for $\mathbf{x}$. $\blacksquare$

$M^{n_t T / 2} = (n_t T)^3 \implies M^4 = 512 \implies M = 512^{1/4} = 2^{9/4} \approx 4.76$.

At $M \lesssim 5$ (e.g., BPSK/QPSK), sphere decoding is cheaper than MMSE-GDFE. At $M = 16$ (16-QAM), MMSE-GDFE wins: sphere decoder cost is $16^4 = 65536$, much larger than MMSE-GDFE's $\sim 512$. For standard MIMO rates (16-QAM and above), MMSE-GDFE is the clear winner. $\blacksquare$

$r = R / \log_2(\text{SNR}) = 2 / \log_2(100) = 2 / 6.64 \approx 0.30$.

$d^*(0.30) = (2 - 0.30)(2 - 0.30) = 1.70 \cdot 1.70 = 2.89$. BER decays as $\text{SNR}^{-2.89}$.

$E_8$ lives in $\mathbb{R}^8$, so we need $2 n_t T = 8$, i.e., $n_t T = 4$. With $n_t = 2$, we need $T = 2$. Yes, structured- $E_8$-LAST with $T = 2$ fits the $(2, 2)$ setting.

From Ex. 8's pattern: $\gamma_c(E_8) = 2$ (i.e., $3$ dB); BER shift factor $\gamma_c^{-n_r} = 2^{-2}$, i.e., $6$ dB of SNR shift. Structured-$E_8$-LAST operates $6$ dB below random LAST at BER $10^{-3}$. $\blacksquare$

$R_{ii}^2$ are the squared diagonals of the QR factor of $[\mathbf{H}^{T}, \sqrt{\alpha}\mathbf{I}]^T$. By QR properties, $\prod_i R_{ii}^2 = \det(\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I})$. For $(2,2)$: $\det = \lambda_1 \lambda_2 + \alpha(\lambda_1 + \lambda_2) + \alpha^2$ where $\lambda_i$ are eigenvalues of $\mathbf{H}^{H} \mathbf{H}$.

Under i.i.d. Rayleigh, $\mathbb{E}[\lambda_1 \lambda_2] = \mathbb{E}[\det(\mathbf{H}^{H} \mathbf{H})] = 1$ (for $n_t = n_r = 2$ normalised), $\mathbb{E}[\lambda_1 + \lambda_2] = \mathbb{E}[\mathrm{tr}(\mathbf{H}^{H} \mathbf{H})] = n_t \cdot n_r \cdot 1 = 4$. With $\alpha = 0.1$: $\mathbb{E}[\det(\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I})] = 1 + 0.1 \cdot 4 + 0.01 = 1.41$.

$\mathbb{E}[\sum_i \log_2 R_{ii}^2] = \mathbb{E}[\log_2 \det] \approx \log_2 1.41 = 0.50$ (per block, $T = 1$). The lattice decoder sees an ensemble-averaged aggregate SNR in this range — specific block realisations may be higher or lower depending on channel eigenvalues. $\blacksquare$

After ZF, each stream sees a Gaussian channel with effective noise variance $[(\mathbf{H}^{H} \mathbf{H})^{-1}]_{ii}$. The per-stream SNR is $\text{SNR} \cdot [(\mathbf{H}^{H} \mathbf{H})^{-1}]_{ii}^{-1}$.

$(\mathbf{H}^{H} \mathbf{H})^{-1}$ is inverse-Wishart; its diagonals are distributed as $1/\chi^2_{n_r - n_t + 1}$. Hence the per-stream SNR is distributed as $\text{SNR} \cdot \chi^2_{n_r - n_t + 1}$, which has diversity order $n_r - n_t + 1$.

At $r = n_t$, each of the $n_t$ streams carries rate $\log_2 \text{SNR}$, and the probability of per-stream outage is $\text{SNR}^{-(n_r - n_t + 1)}$. This is strictly smaller than the Zheng-Tse $d^*(n_t) = 0$ (degenerate case, where we should have no outage asymptotically); but at intermediate $r < n_t$ the ZF diversity is stuck at $n_r - n_t + 1$, below the Zheng-Tse $(n_t - r)(n_r - r)$. Hence ZF-V-BLAST is not DMT-optimal. This is precisely the pitfall that MMSE-GDFE fixes. $\blacksquare$

Per layer $i$ of the triangular system, the achievable rate with a Gaussian random code is $\log_2(1 + R_{ii}^2 / \sigma_{\text{eff}}^2)$; with a lattice code, the achievable rate is $\log_2(R_{ii}^2 / \sigma_{\text{eff}}^2)$ (Erez-Zamir gives $\tfrac12 \log_2(1 + \text{SNR})$ per real dimension, or $\log_2 \text{SNR}$ at high SNR per complex dimension).

$\sum_i \log_2 R_{ii}^2 = T \log_2 \det(\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I}) = T \log_2 \det(\mathbf{I} + \text{SNR} \mathbf{H}^{H} \mathbf{H}) - n_t T \log_2 \alpha + \text{(offset)} = T \cdot I_{\text{MIMO}}(\mathbf{H}) - O(1)$.

Both MMSE-GDFE (lattice) and MMSE-SIC (Gaussian) achieve the same aggregate capacity $I_{\text{MIMO}}(\mathbf{H})$ on the triangularised channel. The receivers differ only in which codebook they assume — Gaussian random code versus lattice code. This is the precise sense in which MMSE-GDFE is the lattice analog of MMSE-SIC. $\blacksquare$

$2 n_t T = 2 \cdot 3 \cdot 4 = 24$. Matches Leech dim.

$|\mathcal{C}| = k^{24} = 8^{24} = 2^{72}$ codewords, i.e., a rate of $(1/T) \log_2 |\mathcal{C}| = 72/4 = 18$ bits per channel use across 3 transmit antennas ($= 6$ bits/ch.use per antenna, equivalent to $64$-QAM per layer).

$\gamma_c(\Lambda_{24}) = 4$ ($6$ dB), so BER shift factor is $\gamma_c^{-n_r} = 4^{-3} = 1/64 = -18$ dB. Structured-Leech-LAST operates $18$ dB below random LAST at the same BER target — an enormous finite-SNR advantage from the Leech lattice's extreme density. $\blacksquare$

Yes — each layer achieves its own AWGN capacity with a sufficiently dense lattice code; no — the per-layer rates are not chosen to match the per-layer capacities (they are chosen to match the overall MIMO rate). What MMSE-GDFE actually achieves is aggregate capacity: $\sum_i \log_2(1 + \text{per-layer SNR}_i) = I(\mathbf{x}; \mathbf{y})$. If the code rate is fixed at $R$ bits/ch.use, the per-layer rate is $R / (n_t T)$ per scalar channel use, which may be below or above the per-layer AWGN capacity depending on the channel realisation.

The DMT-achievability of LAST does not rely on per-layer rate-matching; it relies on the aggregate capacity being above the aggregate rate (the non-outage event). On non-outage channels the code succeeds; on outage channels it fails. This is the fundamental trade-off being optimised in Thm. TLAST + MMSE-GDFE Achieves the Zheng-Tse DMT (El Gamal-Caire-Damen 2004). $\blacksquare$

CDA sphere decoder at $n_t = 5$: $O(M^{12.5}) = 16^{12.5}$ per block — $\approx 10^{15}$, intractable at $M = 16$. MMSE-GDFE

• structured LAST: $O((n_t T)^3) = 125 \cdot T^3$. For $T = 2$ (gives $n_t T = 10$, matching $K_{12}$ after padding), complexity $\approx 1000$. MMSE-GDFE wins by 12 orders of magnitude.

$K_{12}$ (Coxeter-Todd lattice in dim 12) has $\gamma_c \approx 4/3 \cdot \sqrt[12]{3} \approx 1.5$ — modest. Still, it gives $\sim 1.7$ dB over $\mathbb{Z}^{12}$. Combined with polynomial decoding, this gives structured LAST the edge at moderate SNR.

For $n_t \ge 5$, CDA codes are impractical (exponential decoding); structured LAST is practical with some coding-gain penalty. At $n_t = 5$, structured LAST is the clear winner. This is the scaling-robustness argument for lattice codes over algebraic codes. $\blacksquare$

Breaks DMT. Without the dither, the crypto-lemma (uniform codeword distribution) fails, and the Minkowski-Hlawka averaging argument cannot be applied. The Erez-Zamir step of the proof breaks. Undithered LAST has strictly worse DMT than $d^*(r)$.

Preserves DMT. As long as $\Lambda_c$ has density $\ge 2^{-c n_t T}$ for some $c < \log_2 e$ (Thm. TStructured LAST with Dense Inner Lattice Achieves the DMT (Kumar-Caire 2008)), DMT is preserved. Half-density is still asymptotically adequate; only the finite-SNR coding gain degrades.

Breaks DMT. Plain MMSE achieves only diversity $n_r - n_t + 1$ (ZF baseline), not the full $(n_t - r)(n_r - r)$. The backsubstitution is what extracts the transmit diversity. See pitfall ⚠MMSE-GDFE Is Not Plain MMSE — The Feedback Structure Is Essential.

Preserves DMT and gains coding gain. Thm. TStructured LAST with Dense Inner Lattice Achieves the DMT (Kumar-Caire 2008) (Kumar-Caire 2008). DMT preserved; $+3$ dB coding gain over $\mathbb{Z}^8$. $\blacksquare$

import numpy as np

def last_decode(y_vec, H, snr_lin, dither, E8_lattice_points):
    # 1. Augment channel
    alpha = 1.0 / snr_lin
    H_tilde = np.kron(np.eye(T), H)        # n_r T x n_t T
    sqrt_alpha_I = np.sqrt(alpha) * np.eye(H_tilde.shape[1])
    H_aug = np.vstack([H_tilde, sqrt_alpha_I])
    # 2. QR
    Q, R = np.linalg.qr(H_aug)
    F = Q[:H_tilde.shape[0], :].conj().T   # upper block
    # 3. Filter and remove dither
    z = F @ y_vec - F @ H_tilde @ dither
    # 4. Layer-by-layer lattice decode
    x_hat = np.zeros_like(z)
    for i in range(len(z) - 1, -1, -1):
        z_i_tilde = (z[i] - R[i, i+1:] @ x_hat[i+1:]) / R[i, i]
        # E8 nearest-neighbour via inner-lattice table
        x_hat[i] = nearest_in_E8_coord(z_i_tilde, E8_lattice_points)
    # 5. Undo modulo shaping
    u_hat = inverse_G @ ((x_hat + dither) % Lambda_s)
    return u_hat

def nearest_in_E8_coord(z_i, E8_points):
    # Return the nearest coordinate from precomputed E8 table
    return E8_points[np.argmin(np.abs(E8_points - z_i))]

In production one precomputes the 240 nearest-neighbour offsets of $E_8$ for fast per-layer lookup. For Leech, the Conway-Sloane efficient decoder uses a tree-based algorithm that avoids the full $196560$-point search. The overall structure — augment, QR, filter, backsubstitute — is identical across inner lattices. $\blacksquare$

$d^*(0) = 6, d^*(1) = 2, d^*(2) = 0$. Piecewise linear between integer points.

$(2 - r)(3 - r) = 2 \implies 6 - 5 r + r^2 = 2 \implies r^2 - 5 r + 4 = 0 \implies r = 1$ or $r = 4$. Only $r = 1$ is in range $[0, 2]$. So $r^* = 1$.

Zheng-Tse requires $T \ge n_t$ for the DMT to be achievable. At $n_t = 2$, $T \ge 2$ suffices. Standard choice: $T = 2$ ($n_t T = 4$, so $E_8$ or $D_4$ inner lattice if complex-to- real). At $T = 2$, LAST achieves the full DMT $d^* = 2$ at $r = 1$. $\blacksquare$

The most defensible reading is (a), two steps of a single programme. The 2004 paper established the existence of DMT-optimal LAST codes via a random-lattice argument; its limitation (non-constructive) was explicitly noted in the paper's Section IV. The 2008 paper addressed the constructive question left open in 2004: it replaced the random inner lattice by dense structured lattices and showed that (i) DMT is preserved, (ii) finite-SNR coding gain is gained. This is the natural follow-up.

(1) The 2008 paper cites the 2004 paper as its starting point and uses the same DMT framework (Zheng-Tse 2003) and the same decoder (MMSE-GDFE). (2) The authors overlap (G. Caire on both). (3) The 2008 paper proves its theorem by reducing to the 2004 theorem — an internal dependency (Thm. TStructured LAST with Dense Inner Lattice Achieves the DMT (Kumar-Caire 2008), Step 1). (4) The chapter's §1-5 structure is built around the two papers as complementary pieces of the LAST story, with §4 explicitly bridging theory and practice.

A reading favoring (b) or (c) would note that the two papers have four years between them and use different technical tools (Minkowski-Hlawka vs. Conway-Sloane catalogue). But given Caire's consistent authorship and the explicit forward/ backward references between the papers, (a) is the most accurate summary. $\blacksquare$

At high SNR, the joint density of $\{\alpha_i\}$ (sorted so $\alpha_1 \le \ldots \le \alpha_{n_t}$) is $\doteq \text{SNR}^{-\sum_i (n_t + n_r - 2 i + 1) \alpha_i / 2}$ under i.i.d. Rayleigh.

Outage: $\sum_i (1 - \alpha_i)^+ < r$. Feasible region: $\alpha_i \ge 0$, sorted. For integer $r = k$, the most probable point in the outage region has $\alpha_i = 0$ for $i \le k$ (no outage from the first $k$ eigenvalues) and $\alpha_i > 0$ for $i > k$.

Minimise $\sum_{i > k} (n_t + n_r - 2 i + 1) \alpha_i / 2$ subject to $\alpha_i \ge 0$. The linear minimum is $\alpha_i = 1$ for $i > k$, giving exponent $\sum_{i > k} (n_t + n_r - 2 i + 1) / 2 = (n_t - k)(n_r - k)$ (after summation algebra).

At $r = 0$: $(n_t - 0)(n_r - 0) = n_t n_r$ — the full-diversity order, correct. At $r = 1$: $(n_t - 1)(n_r - 1)$ — consistent with the $(2, 2)$ case's $d^*(1) = 1$ and $(3, 3)$'s $d^*(1) = 4$. At $r = \min(n_t, n_r)$: $0$ — the full-multiplexing degenerate case. All consistent with Zheng-Tse. $\blacksquare$

Both MMSE-SIC and MMSE-GDFE triangularise the MIMO channel: they convert the $n_t \to n_r$ channel into $n_t$ scalar channels, decoded sequentially, with earlier decisions feeding back into later ones. In both cases the aggregate rate equals the full MIMO mutual information (no capacity lost). In both cases the order of decoding matters (strongest first for MMSE-SIC; the QR structure determines order for MMSE-GDFE).

MMSE-SIC is iterative: decode stream 1, make a hard decision, subtract its reconstructed contribution from the channel output, redo the MMSE filter on the residual, decode stream 2, and so on. MMSE-GDFE is non-iterative: form the augmented matrix, do one QR decomposition, filter through it once, and then backsubstitute through the triangular system. The feedback is implicit in the triangularisation rather than explicit in an iterative loop.

For Gaussian random codes, iterative MMSE-SIC makes sense because each stream is independently coded — soft information flows cleanly between the demodulator and the outer-code decoder. For lattice codes, the codewords span multiple layers simultaneously (they are structured jointly on the lattice), so iterative per-stream decoding with hard decisions would lose the lattice structure. The non-iterative QR-backsubstitution form of MMSE-GDFE preserves the lattice structure throughout. $\blacksquare$

For $n_t = n_r = 64, T = 1$: $n_t T = 64$, Barnes-Wall BW_128 has dim $128 = 2 n_t T$ (after complex-to-real), $\gamma_c \approx 2.9$ — strong coding gain. MMSE-GDFE complexity $O(64^3) = 2.6 \times 10^5$ flops per block. Single-block latency $\sim 10 \mu s$ at 100 GFLOPS. Feasible.

For $n_t T = 128, 256$ (to get $E_{8n}$ compatibility): $n_t T$ cubed is $2 \times 10^6$ or $1.6 \times 10^7$ — pushing against real-time limits.

(1) Finite-SNR performance: structured-BW-LAST must beat Type-II codebook precoding at typical 5G operating points (SNR 10-30 dB, rate $\sim 10$ bits/ch.use/user). (2) Real-time implementation: the MMSE-GDFE at $n_t = 64$ must fit in the receiver latency budget ($\sim 1$ ms) across realistic user configurations. (3) Robustness: the code must be robust to CSI estimation errors, which are significant in massive-MIMO regimes. (4) Standards inertia: 3GPP has significant investment in the codebook paradigm; switching would require a compelling gain ($\ge 3$ dB at standard operating points).

None of these have been demonstrated at the scale required for a 6G standard. Research in this direction is active but has not crossed the threshold for standardisation adoption. This is an open problem with significant practical implications. $\blacksquare$