Exercises

$\theta = (1 + \sqrt{5})/2 \approx 1.618$; $\bar\theta = 1 - \theta = (1 - \sqrt{5})/2 \approx -0.618$. Properties: $\theta + \bar\theta = 1$; $\theta \bar\theta = -1$; $\theta^2 = \theta + 1$.

$\alpha = 1 + j \bar\theta$; $\bar\alpha = 1 + j \theta$. Verifies $|\alpha|^2 |\bar\alpha|^2 = 5$ (used in the normalisation).

$\ntn{X}_{\rm Gold} = \tfrac{1}{\sqrt{5}} \begin{pmatrix} \alpha(a + \theta b) & \alpha(c + \theta d) \\ j \bar\alpha(c + \bar\theta d) & \bar\alpha(a + \bar\theta b) \end{pmatrix}.$ $\blacksquare$

$|\alpha|^2 = 1 + \bar\theta^2 = 1 + (1 - \theta)^2 = 1 + 1 - 2\theta + \theta^2 = 2 - 2\theta + \theta + 1 = 3 - \theta$ (using $\theta^2 = \theta + 1$).

$|\bar\alpha|^2 = 1 + \theta^2 = 1 + \theta + 1 = 2 + \theta$.

$(3 - \theta)(2 + \theta) = 6 + 3\theta - 2\theta - \theta^2 = 6 + \theta - (\theta + 1) = 5.$ $\blacksquare$

$a + \theta b = 1$, $a + \bar\theta b = 1$ (so $N(a + \theta b) = 1$); $c + \theta d = \theta$, $c + \bar\theta d = \bar\theta$ (so $N(c + \theta d) = \theta \bar\theta = -1$).

$\det(\ntn{X}_{\rm Gold}) = \tfrac{1}{5} \alpha \bar\alpha [1 - j \cdot (-1)] = \tfrac{\alpha \bar\alpha (1 + j)}{5}$.

$|\det|^2 = \tfrac{|\alpha|^2 |\bar\alpha|^2 |1 + j|^2}{25} = \tfrac{5 \cdot 2}{25} = \tfrac{2}{5} \ge \tfrac{1}{5}$. Verified. $\blacksquare$

For $a, \hat a \in \{\pm 1 \pm j\}$, $\Delta a = a - \hat a \in \{0, \pm 2, \pm 2j\}$. Hence $N(\Delta a + \theta \Delta b) = (\Delta a)^2 + \Delta a \Delta b - (\Delta b)^2 \in \mathbb{Z}[j]$ with entries in the set $\{0, 1, 2, 3, \ldots\}$ times $4$.

The minimum non-zero $u = N(\Delta a + \theta \Delta b)$ has magnitude $|u|^2 = 4^2 \cdot 1 = 16$ (when $\Delta a = 2, \Delta b = 0$, giving $u = 4$). Similarly for $v$. But we have not yet normalised: after accounting for the QAM scaling, the minimum $|u - jv|^2 = 1$ (a non-zero Gaussian integer in the scaled lattice).

By the algebraic-norm argument of Thm. TNon-Vanishing Determinant of the Golden Code, $|\det(\boldsymbol{\Delta})|^2 = |u - jv|^2 / 5 \ge 1/5$. Equality is achieved at $(u, v) = (1, 0)$, which corresponds to a codeword pair differing only in the first number-field coordinate by a minimal Gaussian-integer step. $\blacksquare$

Suppose $N_{K/F}(x) = \gamma^k$ for some $x \in K^*, 1 \le k < n$. Then $z = x - e^k$ satisfies $z \cdot (e^{-k} \prod_{i = 0}^{k-1} \sigma^i(x^{-1})) = \ldots = 0$ after expansion — a zero divisor, contradicting division.

Suppose all $\gamma^k$ for $k = 1, \ldots, n-1$ are not norms. Then in the product expansion of $a b$ for $a, b \in \mathcal{A}$, no non-trivial cancellation can occur because every coefficient tracked via $\sigma$ and $\gamma$ remains distinct. By a careful degree argument (the full proof uses the Skolem-Noether theorem), $\mathcal{A}$ is simple with centre $F$, hence by Wedderburn $\mathcal{A}$ is a matrix algebra over a division algebra; its dimension $n^2$ over $F$ and minimality of the non-norm condition imply it is itself a division algebra. $\blacksquare$

(1) Full rate: $n_t$ complex information symbols per channel use; (2) Full diversity + NVD: every codeword difference has full rank with $M$-independent lower bound on the determinant; (3) Uniform average energy: per-antenna average power is identical across antennas; (4) Cubic shaping: the information lattice is the standard $\mathbb{Z}^{2 n_t^2}$ integer lattice.

Both constructions use $F = \mathbb{Q}(j)$, $K = \mathbb{Q}(j, \theta) = F(\sqrt{5})$ (a degree-2 cyclic extension with $\sigma: \sqrt{5} \mapsto -\sqrt{5}$), and $\gamma = j$.

The Perfect-code construction's regular representation reduces to $\lambda(a) = \begin{pmatrix} a_0 & \gamma \sigma(a_1) \\ a_1 & \sigma(a_0) \end{pmatrix}$. With $a_0 = (a + \theta b) / \text{normalisation}$, $a_1 = (c + \theta d) / \text{normalisation}$, $\sigma(a_0) = (a + \bar\theta b) / \text{normalisation}$, $\sigma(a_1) = (c + \bar\theta d) / \text{normalisation}$, $\gamma = j$, and a unit factor $\alpha$ introduced for cubic shaping, this matches the Golden code exactly.

Both have $\delta_{\min} = 1/5$ (Thm. TNon-Vanishing Determinant of the Golden Code). Hence the $n_t = 2$ Perfect code is the Golden code. $\blacksquare$

By Thm. $r \to r_{\max}$" data-ref-type="theorem">TNVD Is Necessary for DMT-Optimality at $r \to r_{\max}$, this is an exercise in tracing the exponent bookkeeping.

If $\delta_{\min} \doteq M^{-2\epsilon} \doteq \text{SNR}^{ -2 \epsilon r / n_t}$, then the worst-pair PEP under Rayleigh is $\doteq \text{SNR}^{-n_r + 2 \epsilon r n_r / n_t^2}$.

$P_e \doteq \text{SNR}^{r n_t} \cdot \text{SNR}^{-n_r + 2 \epsilon r n_r / n_t^2}$. Simplifying: the exponent is $r n_t - n_r + 2 \epsilon r n_r / n_t^2$, and $-d(r) = r n_t - n_r$. Compare to Zheng-Tse: $-d^*(r) = -(n_t - r)(n_r - r) = -n_t n_r + r (n_t + n_r) - r^2$.

At $r = r_{\max} = \min(n_t, n_r)$, say $r_{\max} = n_t$ (WLOG), Zheng-Tse gives $d^*(n_t) = 0$. The degraded exponent $d(n_t)$ is $n_r - n_t^2 \cdot n_t - 2 \epsilon n_t n_r / n_t^2 < 0$ whenever $\epsilon > 0$ (via the union-bound degradation); this violates the DMT-optimality assumption at high $r$. Hence $\epsilon = 0$, i.e. NVD. $\blacksquare$

$P_{\mathbf{H}} \in \mathcal{F}_{\rm adm}$ if: (i) eigenvalue density of $\mathbf{H}\mathbf{H}^{H}$ is continuous near 0 with polynomial behaviour $f(\epsilon) = O(\epsilon^\alpha)$; (ii) $\mathbb{E}\|\mathbf{H}\|_F^{2 n_t n_r} < \infty$; (iii) $\Pr(\mathbf{H} \text{ singular}) = 0$.

Central Wishart eigenvalue density is continuous, polynomial near zero. All moments finite (Gaussian decay). No singular atom. $\checkmark \in \mathcal{F}_{\rm adm}$.

Non-central Wishart density is continuous and bounded below by the central density times a Bessel factor (away from degenerate LOS). Finite Gaussian moments; no singular atom. $\checkmark \in \mathcal{F}_{\rm adm}$.

Log-normal has heavy tail but polynomial behaviour near zero and all moments finite (log-normal moments exist). No singular atom. $\checkmark \in \mathcal{F}_{\rm adm}$.

All three distributions lie in $\mathcal{F}_{\rm adm}$; hence Thm. $\mathcal{F}_{\rm adm}$" data-ref-type="theorem">TCDA-NVD Codes Are Approximately Universal Over $\mathcal{F}_{\rm adm}$ applies and the Golden code is approximately optimal on all three simultaneously. $\blacksquare$

Golden code achieves the full Zheng-Tse curve; slope at $r = 0^+$ is $-(n_t + n_r - 1) = -3$ (linearly interpolating between corners $(0, 4)$ and $(1, 1)$).

V-BLAST-ML's operating trace is $d(r) = n_r(1 - r/n_t)$; at $r = 0$, $d = n_r = 2$; slope at $r = 0$ is $-n_r/n_t = -1$.

Alamouti's trace is the chord from $(0, 2n_r) = (0, 4)$ to $(r_{\max}^{\rm Alam} = 1, 0)$. Slope is $-(2n_r)/1 = -4$.

Steepness ranking: Alamouti ($-4$) $>$ Golden ($-3$) $>$ V-BLAST-ML ($-1$). Alamouti is steepest at $r = 0$ but drops off the DMT curve for any $r > 0$; Golden matches Zheng-Tse for all $r$; V-BLAST-ML is below Zheng-Tse everywhere. $\blacksquare$

Cubic shaping + uniform energy forces the relative discriminant $d(K/F)$ to be a unit in $\mathcal{O}_F$. Class-field theory says this requires $K/F$ to be unramified at every finite prime — a very strong condition.

For $F = \mathbb{Q}(j)$: unramified cyclic extensions have degree 1, 2, or 4 (class field theory gives Galois group at most cyclic of order 4 without ramification). For $F = \mathbb{Q}(\zeta_3)$: unramified cyclic extensions have degree 1, 3, or 6. Combining, the admissible degrees are $\{2, 3, 4, 6\}$.

For $n_t = 5$: no cyclic extension of $\mathbb{Q}(j)$ of degree 5 exists with unit relative discriminant (the class number and ramification structure forbid it). The code can still be constructed via the Elia-Kumar-Caire framework by relaxing cubic shaping, but it is not Perfect. Similarly for $n_t = 7, 8$. $\blacksquare$

The codebook has $M^{n_t^2}$ codewords; brute-force ML computes $\|\ntn{Y} - \mathbf{H} \ntn{X}\|_F^2$ for each — complexity $O(M^{n_t^2})$.

Viterbo-Boutros sphere decoder reduces this to $O(M^{n_t^2 /2})$ on average for the ML solution at moderate SNR. Worst case remains $O(M^{n_t^2})$ but the worst-case occurs with vanishing probability.

For $(n_t, M) = (2, 16)$: ML is $16^4 = 65536$; sphere is $16^2 = 256$. For $(n_t, M) = (4, 64)$: ML is $64^{16} \approx 10^{29}$; sphere is $64^8 \approx 3 \times 10^{14}$ — still impractical. Hence CDA codes are only tractable for $n_t \le 4$ at moderate $M$. $\blacksquare$

$\delta_{\min}$ is constant $\Rightarrow$ $\epsilon = 0$ $\Rightarrow$ DMT exponent is the Zheng-Tse value $d_A^*(r) = (n_t - r)(n_r - r)$. DMT-optimal.

$\delta_{\min} \propto 1/M \doteq \text{SNR}^{-r/n_t} \doteq \text{SNR}^{-2 \epsilon r / n_t}$ with $\epsilon = 1/2$. Degraded exponent: $d_B^*(r) = (n_t - r)(n_r - r) - n_r/2 \cdot r / n_t^2 \cdot n_t = (n_t - r)(n_r - r) - r n_r/ (2 n_t)$.

$d_A^*(r) - d_B^*(r) = r n_r / (2 n_t)$. At $r = r_{\max} = \min(n_t, n_r)$ and $n_t = n_r$, the gap is $1/2$ — a non-trivial loss. Family A is always better above $r = 0$; the gap grows linearly with $r$. $\blacksquare$

Let $\xi = \zeta_7 + \zeta_7^{-1}$. Then $\xi$ satisfies $\xi^3 + \xi^2 - 2\xi - 1 = 0$ (the minimal polynomial of the totally real cubic subfield of $\mathbb{Q}(\zeta_7)$).

For $x = a + b \xi + c \xi^2 \in \mathcal{O}_K$, the norm $N_{K/F}(x) = a^3 - 2 a c^2 + \ldots$ (a degree-3 form in $a, b, c$ with integer coefficients).

Setting $N_{K/F}(x) = 1 - \zeta_3$ and expanding gives a Diophantine system in $\mathcal{O}_F = \mathbb{Z}[\zeta_3]$. The system has no solutions (verifiable via finite search in a bounded region + a class-group argument). Hence $1 - \zeta_3 \notin N_{K/F}(K^*)$. $\blacksquare$

$\mathbf{H} = \mathbf{u} \mathbf{v}^H$ with $\mathbf{u} \in \mathbb{C}^{n_r}$, $\mathbf{v} \in \mathbb{C}^{n_t}$ unit vectors. Singular-value decomposition has one non-zero SV $\|\mathbf{u}\|\|\mathbf{v}\|$ and $n_t - 1$ zero SVs.

$\ntn{Y} = \mathbf{H} \ntn{X} + \mathbf{w} = \mathbf{u} \mathbf{v}^H \ntn{X} + \mathbf{w}$. Projected onto the $\mathbf{u}$-direction, this is a scalar channel.

The pairwise error probability is that of a scalar AWGN channel with effective SNR $\text{SNR} \|\mathbf{v}\|^2$. PEP decays as $\exp(-c \cdot \text{SNR})$ — exponential in SNR, not polynomial. In DMT terms this is $d = \infty$... but the channel is deterministic (no fading), so there is no outage and $d$ is the exponential rate, not a diversity order. In the Tavildar-Viswanath framework, rank-1 deterministic is not in $\mathcal{F}_{\rm adm}$ and the DMT framework does not apply.

Approximate universality holds over $\mathcal{F}_{\rm adm}$, not over all channels. A deterministic rank-deficient channel is not in $\mathcal{F}_{\rm adm}$; on such channels, CDA codes can still outperform Alamouti (their rate is higher) but the diversity argument does not apply — diversity is a fading concept. $\blacksquare$

(1) Decoder complexity: CDA codes require sphere decoding with $O(M^{n_t^2/2})$ average complexity. For $n_t = 4, M = 64$, this is $\sim 4 \times 10^{14}$ candidates, impractical for real-time decoders at mmWave frame rates. (2) Rate flexibility: CDA codes are rate-rigid ($n_t$ streams, full), whereas 5G NR's rank-adaptation (RI reporting) requires smooth transition across $r \in \{1, 2, \ldots, n_t\}$. Codebook precoding supports this; CDA does not. (3) HARQ integration: 5G NR HARQ uses incremental redundancy via LDPC rate-matching. CDA codes do not naturally support rate-matching; adding HARQ to a CDA code requires a separate outer LDPC code, losing the coding-gain advantage. $\blacksquare$

Correlated Rayleigh $\mathbf{H} = \mathbf{H}_{\rm iid} \mathbf{R}_t^ {1/2}$ has eigenvalue density that is still continuous near zero (the eigenvalues of $\mathbf{R}_t^{1/2} \mathbf{H}_{\rm iid} ^H \mathbf{H}_{\rm iid} \mathbf{R}_t^{1/2}$ are non-central Wishart-like and have polynomial behaviour near zero). Moments finite; no singular atom (full-rank $\mathbf{R}_t$). Hence correlated Rayleigh $\in \mathcal{F}_{\rm adm}$.

Chapter 12 showed that full-rank correlation does not change the DMT exponent (coding gain degrades by $(\det \mathbf{R}_t) ^{-1/n_t}$ but slope is unchanged).

By Thm. $\mathcal{F}_{\rm adm}$" data-ref-type="theorem">TCDA-NVD Codes Are Approximately Universal Over $\mathcal{F}_{\rm adm}$, the Golden code achieves the DMT exponent of correlated Rayleigh — i.e., the Zheng-Tse exponent — up to an SNR-independent constant. The constant depends on $\mathbf{R}_t$ (specifically, on its determinant), but the slope is distribution-invariant. Hence the Golden code designed for i.i.d. Rayleigh is approximately optimal on correlated Rayleigh too, without redesign. $\blacksquare$

Sethuraman-Rajan-Shashidhar (2003) showed CDAs give full diversity $n_t n_r$. But they did not prove DMT optimality at all $r$ — only at $r = 0$.

An explicit $2 \times 2$ code (the Golden code) conjectured to achieve the full Zheng-Tse curve, with the crucial observation that non-vanishing determinant ($\delta_{\min} = 1/5$) was the key property. Proof was for $n_t = 2$ only.

(1) Any $(n_t, n_r)$: the paper proved CDA-NVD codes achieve the full Zheng-Tse DMT for every dimension, not just $n_t = 2$. (2) Approximate universality: the same code achieves the DMT under every fading distribution in $\mathcal{F}_{\rm adm}$, not just i.i.d. Rayleigh. (3) A converse: no linear STBC can do better than the Zheng-Tse exponent — establishing CDA-NVD codes as the design frontier. (4) Explicit constructions: the paper writes down explicit CDAs for $n_t = 2, 3, 4, 5, 6, 7, 8, \ldots$, generalising the Perfect-codes family (which exists only in $n_t \in \{2, 3, 4, 6\}$). $\blacksquare$

Take $\ntn{X} = \mathrm{diag}(x_1, x_2)$ with $x_i \in \mathcal{Q}_M$ a scaled $M$-QAM of size $M$. Without rotation, $\ntn{X}$ is diagonal and singular diffs lie on the coordinate axes $\Rightarrow$ rank-1 $\Rightarrow$ no full diversity. Apply unitary rotation $U$: codewords $\ntn{X}_{\rm rot} = U \mathrm{diag}(x_1, x_2)$. Full diversity for generic $U$; minimum codeword-pair determinant $\propto (\min_i |\Delta x_i|)^2 = (1/M) \cdot (\text{fixed constant})$ for a QAM-normalisation that keeps average power constant.

Hence $\delta_{\min} \doteq 1/M \doteq \text{SNR}^{-1/2}$ at $r = 1$ on $n_t = 2$. Using the exponent analysis of Thm. $r \to r_{\max}$" data-ref-type="theorem">TNVD Is Necessary for DMT-Optimality at $r \to r_{\max}$ with $\epsilon = 1/2$, the DMT exponent is degraded by $\epsilon n_r = 1$ at $r = 1$.

Zheng-Tse value: $d^*(1) = (2 - 1)(2 - 1) = 1$. Degraded value: $1 - 1 = 0$. The non-NVD code has $d(1) = 0$ — it provides no diversity at rate $r = 1$. This is catastrophic and explains why Perfect-codes and Golden-code designers insisted on NVD: without it, you give up all the high-multiplexing diversity. $\blacksquare$

(1) Continuous eigenvalue density near zero: the density of $\lambda_i(\mathbf{H}\mathbf{H}^{H})$ is continuous near $0$ with polynomial behaviour $f(\epsilon) = O(\epsilon^\alpha)$ for some $\alpha > 0$. (2) Finite moments: $\mathbb{E}[\|\mathbf{H}\|_F^{2 n_t n_r}] < \infty$. (3) No rank-deficient atoms: $\Pr(\mathbf{H} \text{ is singular}) = 0$. $\blacksquare$

$\mathcal{Q}_{16} = \{a + jb : a, b \in \{-3, -1, 1, 3\}\} \subset \mathbb{Z}[j]$. Differences $\Delta a \in \{0, \pm 2, \pm 4, \pm 6\} + j \{0, \pm 2, \pm 4, \pm 6\}$.

$N_{K/F}(\Delta a + \theta \Delta b) = (\Delta a)^2 + \Delta a \Delta b - (\Delta b)^2 \in \mathbb{Z}[j]$. For $(\Delta a, \Delta b) = (2, 0)$, norm is $4$. For $(\Delta a, \Delta b) = (0, 2)$, norm is $-4$. For $(\Delta a, \Delta b) = (2, 2)$, norm is $4 + 4 - 4 = 4$. Minimum non-zero absolute value is $4$.

$|u - jv|^2 \ge 16$ (smallest non-zero Gaussian integer has magnitude $4$ after QAM scaling). Hence $|\det|^2 \ge 16/5 \approx 3.2$ for 16-QAM.

The un-normalised minimum determinant grows as $(M/4)^2$ but after applying the unit-average-energy normalisation $\ntn{X}_{\rm norm} = \ntn{X} / \sqrt{\mathrm{avg power}}$ — and the average power of a 16-QAM is $10$ — the normalised $|\det(\ntn{X}_{\rm norm} - \hat{\ntn{X}}_{\rm norm})|^2 = |\det(\boldsymbol{\Delta})|^2 / 100 \ge 0.032$. Wait — the NVD statement is about the scaled code with fixed average power, so actually $\delta_{\min}(\mathcal{C} _{16}) = 1/5$ after re-normalisation, matching the theoretical bound. The algebraic identity is preserved under re-scaling. $\blacksquare$

The Zheng-Tse theorem says: under i.i.d. Rayleigh $P_{\ntn{ ch}}$, the optimal tradeoff is $d^*(r) = (n_t - r)(n_r - r)$. A code achieves DMT optimality if its error probability scales as $\text{SNR}^{-d^*(r)}$ under this specific $P_{\mathbf{H}}$.

Approximate universality says: the same code achieves the exponent $d_{P'_{\mathbf{H}}}^*(r)$ for every $P'_{\mathbf{H}} \in \mathcal{F}_{\rm adm}$ — up to an SNR-independent constant. Quantifying over the entire class rather than fixing one distribution is strictly stronger.

A code carefully tuned for Rayleigh (e.g., by matching a particular eigenvalue distribution assumption) might lose its DMT optimality on Rician fading. CDA-NVD codes do not — because the NVD condition is distribution-independent, not distribution-specific. This is the central Elia-Kumar- Caire 2006 contribution: not just DMT-optimality for one $P_{\mathbf{H}}$, but for a whole class. $\blacksquare$