Ferkans — Interactive Telecom Tutor

Full Rate, Full Diversity, NVD — and What Else?

The CDA framework of §2 gives a full-diversity code with non- vanishing determinant in any dimension $n_t$ . But for a practical code we want more. We want the average energy per antenna to be uniform — no single antenna transmitting twice the power of another, since per-antenna power amplifiers are the real-world cost. We want the code to be a cubic shaping lattice — one where the $2 n_t^2$ -dimensional real signal space is a cube, not a parallelepiped, so QAM transmission is efficient. And we want the coding gain — the actual $\delta_{\min}$ — to be as large as possible within the NVD constraint.

Oggier, Rekaya, Belfiore, and Viterbo (2006) bundled all of these desiderata into the Perfect code definition. A Perfect code is a CDA-NVD code with (i) full rate $n_t$ , (ii) full diversity $n_t n_r$ , (iii) uniform average energy, and (iv) cubic shaping. The remarkable fact — which the next theorem proves — is that Perfect codes exist in only four dimensions: $n_t \in \{2, 3, 4, 6\}$ . All four constructions are explicit and the paper tabulates their cyclic algebras, units, and non-norm elements.

This is the family that the Elia-Kumar-Caire paper generalises to arbitrary $n_t$ at the cost of dropping the cubic-shaping and uniform-energy constraints.

Definition:
Perfect Code (Oggier-Rekaya-Belfiore-Viterbo 2006)

A Perfect code is a CDA space-time code $\mathcal{C} = \{\lambda(a) : a \in \mathcal{I}\}$ satisfying all four properties:

Full rate. The code transmits $n_t$ complex information symbols per channel use.
Full diversity + NVD. Every non-zero codeword difference has full rank $n_t$ , and the minimum $|\det(\boldsymbol{ \Delta})|^2$ is bounded below by a positive constant independent of the input QAM size $M$ .
Uniform average energy. Each antenna transmits the same average power: $\mathbb{E}[|(\lambda(a))_{i j}|^2]$ is independent of $(i, j)$ .
Cubic shaping. The real-vector representation of $\mathcal{I}$ is an integer lattice $\mathbb{Z}^{2 n_t^2}$ — i.e., the information symbols can be drawn from a standard integer QAM without any rotation of the lattice basis.

Property (4) ensures that Perfect codes are efficient to use in practice: no complex shaping transform is needed to match the code to a QAM constellation.

Notice how each of the four properties rules out a different class of "lesser" codes. Properties 1+2 are what the Zheng-Tse DMT theorem requires. Properties 3+4 are what practical implementation requires. Prior to Oggier et al., one could achieve any subset of three properties but not all four. The Golden code of §1 is the $n_t = 2$ Perfect code — same construction, rediscovered and re-motivated under the Perfect- code framework.

Theorem: Perfect Codes Exist Only in Dimensions $n_t \in \{2, 3, 4, 6\}$

A Perfect code on $n_t$ transmit antennas exists if and only if $n_t \in \{2, 3, 4, 6\}$ . For each of these four values, explicit cyclic division algebras are constructed over $\mathbb{Q}(j)$ (for $n_t = 2, 4$ ) and over $\mathbb{Q}(\zeta_3)$ where $\zeta_3 = e^{2\pi j/3}$ (for $n_t = 3, 6$ ), together with explicit non-norm elements $\gamma$ that satisfy both the NVD and the cubic-shaping constraints simultaneously.

The constraint boils down to a compatibility condition between the cyclotomic field needed for cubic shaping ( $\mathbb{Q}(j)$ or $\mathbb{Q}(\zeta_3)$ ) and the degree of the cyclic extension $K/F$ . The lattice $\mathbb{Z}[j]^{n_t}$ or $\mathbb{Z}[\zeta_3] ^{n_t}$ must be cubic (not just an integer lattice) when pulled back through the CDA regular representation. This is a number- theoretic tightness condition on the discriminant of $K/F$ that is satisfied for the four special dimensions and fails elsewhere. The result is a cousin of the classical Wedderburn theorem on finite division rings, where only certain dimensions support certain algebraic structures.

Show Hint

Cubic shaping requires $\mathcal{O}_K$ to be a free rank- $n_t$ module over $\mathbb{Z}[j]$ or $\mathbb{Z}[\zeta_3]$ — a strong discriminant condition.

Full diversity + cubic shaping together require $K/F$ to be totally real with prescribed conductor; only $n_t \in \{2, 3, 4, 6\}$ admit such $K$ .

Verify each case by explicit construction: $K = \mathbb{Q}(j, \theta)$ for $n_t = 2, 4$ ; $K = \mathbb{Q}(\zeta_3, \zeta_7)$ for $n_t = 3, 6$ .

Proof

Step 1 — Reduce to a number-field existence problem

Uniform average energy plus cubic shaping forces $\mathcal{O}_K$ to be a free rank- $n_t$ module over $\mathcal{O}_F$ where $F = \mathbb{Q}(j)$ or $\mathbb{Q}(\zeta_3)$ . This is equivalent to the relative discriminant $d(K/F)$ being a unit in $\mathcal{O}_F$ , an extremely restrictive condition.

Step 2 — Enumerate admissible $K$

A theorem of class-field theory (Hasse, Chevalley, Noether) classifies the cyclic extensions of $\mathbb{Q}(j)$ and $\mathbb{Q}(\zeta_3)$ with unit relative discriminant. Only degrees $n_t = 2, 4$ have such $K$ over $\mathbb{Q}(j)$ ; only degrees $n_t = 3, 6$ over $\mathbb{Q}(\zeta_3)$ . Together these give $n_t \in \{2, 3, 4, 6\}$ .

Step 3 — Verify non-norm $\gamma$ exists for each case

For each admissible $(F, K, n_t)$ one must exhibit a non-norm $\gamma \in F^*$ so that $\mathcal{A}(F, K, \sigma, \gamma)$ is a division algebra. The paper exhibits: $n_t = 2$ : $F = \mathbb{Q}(j), \gamma = j$ (Golden code); $n_t = 3$ : $F = \mathbb{Q}(\zeta_3), \gamma = 1 - \zeta_3$ ; $n_t = 4$ : $F = \mathbb{Q}(j), \gamma = j$ ; $n_t = 6$ : $F = \mathbb{Q}(\zeta_3), \gamma = -\zeta_3$ .

Step 4 — No Perfect code outside $\{2, 3, 4, 6\}$

For $n_t \notin \{2, 3, 4, 6\}$ , the unit-discriminant requirement forces $\mathcal{O}_K$ to have a non-cubic embedding as a lattice in $\mathbb{R}^{2 n_t^2}$ . Equivalently, the Gram matrix of the lattice is not the identity. Hence cubic shaping fails. The code can still be constructed as a general CDA (e.g., the Elia-Kumar-Caire construction), but it is not Perfect in the strict sense. $\blacksquare$

Example: The $n_t = 3$ Perfect Code

State the cyclic algebra underlying the $n_t = 3$ Perfect code and write down the regular-representation codeword matrix.

Solution

Number-field setup

Base field $F = \mathbb{Q}(\zeta_3)$ where $\zeta_3 = e^{2\pi j /3}$ is a primitive cube root of unity. Cyclic extension $K = \mathbb{Q}(\zeta_7 + \zeta_7^{-1}) \cdot F$ of degree $3$ over $F$ (the totally real cubic subfield of $\mathbb{Q}(\zeta_7)$ , adjoined to $F$ ). Galois generator $\sigma$ cyclically permutes the three real embeddings of $\mathbb{Q}(\zeta_7 + \zeta_7^{-1})$ .

Non-norm element

$\gamma = 1 - \zeta_3 \in F$ . One verifies that $\gamma$ is not the image of any element of $K$ under the norm $N_{K/F}$ (this is a finite computation in the unit group of $K$ ). Hence $\mathcal{A}(F, K, \sigma, \gamma)$ is a division algebra.

Codeword matrix

For information triple $(a_0, a_1, a_2) \in K^3$ embedded via $\mathbb{Z}[\zeta_3]$ -QAM, the Perfect codeword is $\lambda(a) = \begin{pmatrix} a_0 & \gamma \sigma(a_2) & \gamma \sigma^2(a_1) \\ a_1 & \sigma(a_0) & \gamma \sigma^2(a_2) \\ a_2 & \sigma(a_1) & \sigma^2(a_0) \end{pmatrix}.$ Each row is a $\gamma$ -twisted Galois orbit; the diagonal $(a_0, \sigma(a_0), \sigma^2(a_0))$ is the three real embeddings of $a_0$ . A finite verification gives $\delta_{\min} = 1/49$ (independent of QAM size $M$ ). $\blacksquare$

Golden Code vs Alamouti vs V-BLAST at Fixed Rate

BER versus SNR comparison on a $2 \times 2$ i.i.d. Rayleigh channel for three $n_t = 2$ schemes — Alamouti (rate 1, full diversity, not full rate), uncoded V-BLAST-ML (full rate, low diversity), and the Golden code (full rate, full diversity, NVD) — at a common target spectral efficiency. Observe how Alamouti's slope is steep but saturates at rate 1; V-BLAST is flat in slope; the Golden code combines a steep slope with full rate. The Golden code's advantage grows with target rate.

Parameters

Target rate [bits/ch.use]4

Classical Space-Time Codes on the $2 \times 2$ DMT Curve

Code	Rate	Diversity at $r = 0$	DMT slope at $r = 0$	DMT-optimal?	Decoder
Alamouti	$1$ (not full)	$4$ (full)	$-4/r_{\max}^{\mathrm{Alam}} = -4$	No (off curve for $r > 0$ )	Linear
V-BLAST-ZF	$2$ (full)	$1$	$-1/n_t = -0.5$	No (shallow chord)	Linear / MMSE-SIC
V-BLAST-ML	$2$ (full)	$2$	$-n_r/n_t = -1$	No (above VBLAST-ZF but below DMT)	Exponential
Golden code	$2$ (full)	$4$ (full)	$-(n_t + n_r - 1) = -3$	Yes (matches Zheng-Tse)	Sphere decoder $O(M^2)$
Perfect code ( $n_t=2$ )	$2$ (full)	$4$ (full)	$-3$	Yes (identical to Golden)	Sphere decoder $O(M^2)$
CDA-NVD general	$n_t$ (full)	$n_t n_r$ (full)	$-(n_t + n_r - 1)$	Yes for all $(n_t, n_r)$	Sphere decoder $O(M^{n_t^2/2})$

Why This Matters: Perfect Codes in Practice: DVB-NGH and Beyond

The Golden code was adopted (in its scaled integer form) by the DVB-NGH (Next-Generation Handheld) mobile TV standard in 2012 as the optional $2 \times 2$ MIMO mode for high-mobility broadcast. DVB-NGH receivers implement a sphere decoder for Golden-code decoding, taking advantage of the full diversity at moderate target rates. The Perfect codes for $n_t = 3, 4$ have been proposed for WLAN 802.11n/ac $3 \times 3$ and $4 \times 4$ modes but were not adopted in favour of simpler Alamouti-based space-time block codes (higher rates via single-user MIMO + BICM instead).

5G NR (Release 15, 2018) does not use CDA codes. Instead it relies on low-dimensional transmit precoding (codebook-based beamforming) combined with BICM outer codes (LDPC or polar), trading DMT optimality for lower decoder complexity, rate flexibility, and seamless integration with HARQ. The CDA family lives on in academic research and as a theoretical upper bound against which standardised schemes are benchmarked.

Quick Check

For which values of $n_t$ do Perfect space-time codes exist?

All $n_t \ge 2$

$n_t \in \{2, 3, 4, 6\}$

$n_t \in \{2, 4, 8\}$

Only $n_t = 2$ (the Golden code)

Correction:

n_t \in \{2, 3, 4, 6\}

Correct. The Oggier-Rekaya-Belfiore-Viterbo classification identifies exactly these four dimensions, determined by the number-theoretic condition that $\mathcal{O}_K$ be a free rank- $n_t$ module over $\mathbb{Z}[j]$ or $\mathbb{Z}[\zeta_3]$ with unit relative discriminant.

Perfect Code

A cyclic-division-algebra space-time code with four properties: full rate ( $n_t$ complex symbols per channel use), full diversity and non-vanishing determinant, uniform average energy across antennas, and cubic shaping of the information lattice. Exists in dimensions $n_t \in \{2, 3, 4, 6\}$ only.

The Perfect-Codes Family