Ferkans — Interactive Telecom Tutor

From Random Existence to Structured Construction

The 2004 LAST paper of §1-3 answered a foundational question: lattice codes, in composition with MMSE-GDFE, achieve the Zheng-Tse DMT on MIMO fading channels. But the proof used random lattices drawn from the Minkowski-Hlawka ensemble — not an explicit, deployable code. A practitioner reading the 2004 paper would rightly ask: can we replace the random lattice with a known dense lattice — $E_8$ , Leech, Barnes-Wall — and keep DMT-optimality while also getting the finite-SNR coding gain of the chosen lattice?

The 2008 paper of K. R. Kumar and G. Caire answered this question with a clean yes. Their structured LAST construction replaces the random fine lattice $\Lambda_c$ with an explicit dense lattice — $E_8$ for $n_t T = 8$ , Leech for $n_t T = 24$ , Barnes-Wall for dyadic dimensions — and shows that (i) DMT-optimality is preserved, and (ii) at moderate SNR ( $\text{BER} \approx 10^{-3}$ ) the structured code gains $3-6$ dB of SNR over random LAST. This closed the theory-practice gap: one can now deploy a concrete, fully-specified LAST code whose performance matches or exceeds the best CDA-NVD construction in the finite-SNR regime.

The section is organised around a single theorem (structured LAST is DMT-optimal, with quantified coding gain) and three worked constructions: $E_8$ -LAST in dimension $n_t T = 8$ , Leech-LAST in dimension $n_t T = 24$ , and BW_16-LAST in dimension $n_t T = 16$ . A simulated BER comparison against CDA-NVD and random LAST makes the case graphically.

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Definition:
Structured LAST Code

A structured LAST code is a LAST codebook $\mathcal{C} = \Lambda_c \cap \mathcal{V}(\Lambda_s)$ in which the fine lattice $\Lambda_c$ is an explicit dense lattice taken from the Conway-Sloane catalogue: $\Lambda_c \in \{E_8, D_4, K_{12}, \Lambda_{16}^{BW}, \Lambda_{24}, \ldots\}$ , where the choice matches the dimension $n_t T$ (or a multiple, if the block length $T$ can be adjusted) of the real-vectorised code space.

The shaping lattice $\Lambda_s \subset \Lambda_c$ is typically a scaled copy of $\Lambda_c$ itself — $\Lambda_s = k \cdot \Lambda_c$ for some positive integer $k$ — so that the codebook $\mathcal{C} = \Lambda_c / (k \Lambda_c)$ has cardinality $k^{n_t T}$ (the index of $k \Lambda_c$ in $\Lambda_c$ ). The common random dither $\mathbf{d} \sim \mathrm{Unif}(\mathcal{V}(\Lambda_s))$ and the modulo-lattice transmission rule $\mathbf{x} = [\mathbf{G} \mathbf{u} + \mathbf{d}] \bmod \Lambda_s$ are unchanged from the generic LAST construction (§1).

The design freedoms collapse to three clean choices: (i) the inner lattice $\Lambda_c$ , selected from the catalogue for the target dimension $n_t T$ ; (ii) the integer shaping factor $k$ , which sets the rate $R = n_t \log_2 k$ bits per channel use; (iii) the seeded random dither $\mathbf{d}$ . All other LAST-code parameters (block length $T$ , antenna configuration $n_t \times n_r$ ) are externally specified. This is as close to a deterministic construction as a lattice space-time code gets.

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🎓CommIT Contribution(2008)

Structured LAST Codes from Dense Lattice Packings

K. R. Kumar, G. Caire — IEEE Trans. Inf. Theory, vol. 55, no. 12, pp. 5556–5578

K. Raj Kumar (USC) and Giuseppe Caire (USC at the time, now TU Berlin) addressed the constructive question left open by the 2004 LAST paper: how do we deploy a LAST code in practice, when the existence proof uses random lattices that nobody can write down? Their 2008 paper proposes an explicit recipe — take a dense known lattice ( $E_8$ , Leech $\Lambda_{24}$ , Barnes-Wall $\Lambda_{16}^{BW}$ , $K_{12}$ ) as the inner lattice of the LAST code — and proves that this preserves DMT-optimality while delivering the finite-SNR coding gain of the chosen dense lattice.

The question. The 2004 LAST paper proved DMT-optimality using Minkowski-Hlawka random lattices (Ch. 15). This is a clean existence proof, but non-constructive: random-ensemble lattices have no closed-form generator matrix and require high-precision sampling to implement. Moreover, the typical random lattice is not optimally dense — Minkowski-Hlawka saturates at density $\sim 2^{-n}$ , while the best-known dense lattices in specific dimensions ( $E_8$ , Leech) achieve considerably higher density. This gap translates directly into coding-gain at finite SNR.

The insight. Kumar and Caire observed that the LAST DMT proof uses only two properties of the inner lattice: (i) adequate normalised density (enough to support Erez-Zamir capacity achievement), and (ii) sufficient shaping via the Voronoi region (the crypto-lemma dither argument). Every lattice in the Conway-Sloane catalogue — $E_8$ , Leech, Barnes-Wall, checkerboards — satisfies both. Therefore any dense lattice inner to a Voronoi-shaped shell lattice gives a DMT-optimal LAST code. The random-lattice machinery of the 2004 paper was mathematical convenience, not essential structure.

The main result. The paper's central theorem (reproduced in TStructured LAST with Dense Inner Lattice Achieves the DMT (Kumar-Caire 2008)) is that structured LAST with inner lattice $E_8$ achieves the Zheng-Tse DMT for every $(n_t, n_r)$ such that $n_t T = 8$ , with coding gain $\gamma_c(E_8) \cdot \gamma_c(\text{LAST setup}) \approx 6 \cdot \gamma_c(\text{LAST setup})$ . Translated into BER at moderate SNR: structured- $E_8$ -LAST is $\sim 3$ dB better than random LAST at $\text{BER} = 10^{-3}$ , and $\sim 6$ dB better when using Leech $\Lambda_{24}$ at the appropriate $n_t T = 24$ .

Why it redefined the field. Before 2008 the reading of LAST was "beautiful theorem, but the code is a random-lattice abstraction — use CDA codes (Ch. 13) in practice." After 2008 the reading was "take $E_8$ , apply the LAST recipe, win both DMT and finite-SNR coding gain." This collapsed a theory-practice gap that had persisted for four years. Structured LAST codes became the competitive benchmark against which all subsequent MIMO codes — including integer-forcing linear receivers (Zhan-Nazer-Erez 2014) and compute-and-forward (Ch. 18) — compare themselves. The key message is that lattice-theoretic structure (Ch. 15) and information-theoretic receiver design (Ch. 17 §2) are compatible and combinable — a non-trivial composition that collapsed "choose between coding gain and DMT optimality" into "have both."

laststructured-latticecoding-gaine8leechView Paper →

Theorem: Structured LAST with Dense Inner Lattice Achieves the DMT (Kumar-Caire 2008)

Let $\Lambda_c$ be any lattice in dimension $n_t T$ with packing density $\Delta(\Lambda_c) \ge 2^{-c n_t T}$ for some constant $c < \log_2 e$ , and let $\Lambda_s = k \Lambda_c$ for positive integer $k$ . Let the transmit rate scale as $R(\text{SNR}) = r \log_2 \text{SNR}$ , so that $k = k(\text{SNR}) \doteq \text{SNR}^{r / n_t}$ . Then the structured LAST code $\mathcal{C} = \Lambda_c / (k \Lambda_c)$ with MMSE-GDFE lattice decoding achieves $\lim_{\text{SNR} \to \infty} - \frac{\log \bar{P}_e(\text{SNR})} {\log \text{SNR}} \;=\; d^{*}(r) \;=\; (n_t - r)(n_r - r).$ Furthermore, at finite SNR, the BER of the structured code satisfies $\text{BER}(\text{SNR}) \;\le\; C \cdot \gamma_c(\Lambda_c)^{-n_r} \cdot \text{SNR}^{-d^{*}(r)},$ where $\gamma_c(\Lambda_c) = d_{\min}^2(\Lambda_c) / V(\Lambda_c)^{2/n_t T}$ is the normalised coding gain of the inner lattice and $C$ is an $\text{SNR}$ -independent constant. In particular, replacing a random $\Lambda_c$ by $E_8$ (which has $\gamma_c(E_8) = 2^{1/2}$ , i.e., $+3$ dB over $\mathbb{Z}^8$ ) gives $+3 n_r$ dB of coding-gain advantage at moderate SNR.

The theorem has two parts. The first part — asymptotic DMT — follows immediately from the 2004 LAST theorem because the proof only used adequate density, not randomness. The second part — the finite-SNR coding-gain bound — uses the classical union- bound argument: the PEP between a pair of codewords is controlled by $d_{\min}^2(\Lambda_c) / \text{SNR}$ , and the kissing number $K(\Lambda_c)$ gives the number of nearest-neighbour errors. Replacing $\mathbb{Z}^8$ by $E_8$ doubles $d_{\min}^2$ (gaining $3$ dB per receive antenna) and increases the kissing number from $16$ to $240$ (adding a small over-count but not enough to erase the $3$ dB gain). Leech is even better: $d_{\min}^2$ increases by a factor of $4$ over $\mathbb{Z}^{24}$ , giving $6$ dB.

Show Hint

Reduce to the 2004 LAST theorem (Thm. TLAST + MMSE-GDFE Achieves the Zheng-Tse DMT (El Gamal-Caire-Damen 2004)) for the asymptotic DMT part — the proof uses only adequate density.

For the finite-SNR part, combine the PEP bound (Ch. 11 or Ch. 16 Erez-Zamir analysis) with the lattice-specific $d_{\min}$ , $K(\Lambda)$ .

The coding gain is $\gamma_c(\Lambda) = d_{\min}^2 / V^{2/n}$ ; a $+3$ dB increase in $\gamma_c$ translates to $+3 n_r$ dB in received PEP due to the $n_r$ -fold receiver diversity.

Proof

Step 1 — DMT part follows from 2004 LAST theorem

The El Gamal-Caire-Damen 2004 theorem (Thm. TLAST + MMSE-GDFE Achieves the Zheng-Tse DMT (El Gamal-Caire-Damen 2004)) uses only two properties of $\Lambda_c$ : Minkowski-Hlawka adequate density and Voronoi-based shaping. Both are satisfied by $E_8$ , Leech, Barnes-Wall, and any lattice with $\Delta(\Lambda_c) \ge 2^{-c n_t T}$ and a round Voronoi region. Therefore the asymptotic DMT $\lim_{\text{SNR} \to \infty} -\log \bar{P}_e / \log \text{SNR} = d^*(r)$ is inherited.

Step 2 — PEP between lattice codewords

Conditioned on $\mathbf{H}$ , the pairwise error probability between two codewords differing by the minimum-norm lattice vector $\mathbf{e} \in \Lambda_c$ , with $\|\mathbf{e}\|^2 = d_{\min}^2(\Lambda_c)$ , is bounded by $\mathrm{Q}(\sqrt{ \text{SNR} \cdot d_{\min}^2(\Lambda_c) / (4 V(\Lambda_c)^{2/n_t T}) \cdot \sum_{i} R_{ii}^2 / n_t T})$ after MMSE-GDFE triangularisation. The ratio $d_{\min}^2 / V^{2/n_t T} = \gamma_c(\Lambda_c)$ is the normalised coding gain.

Step 3 — Union bound over nearest neighbours

By the lattice structure, the number of codewords at minimum-distance $d_{\min}$ of any given codeword is the kissing number $K(\Lambda_c)$ . Union-bound over these $K(\Lambda_c)$ dominant error events: $\bar{P}_{e|\mathbf{H}} \le K(\Lambda_c) \cdot \mathrm{Q}\bigl( \sqrt{\gamma_c(\Lambda_c) \cdot \text{SNR} \cdot f(\mathbf{H})}\bigr),$ where $f(\mathbf{H})$ is the Wishart-eigenvalue combination that gives rise to the DMT in the channel outage.

Step 4 — Channel averaging with coding-gain prefactor

Averaging over $\mathbf{H}$ with the Zheng-Tse Wishart-Laplace analysis (Ch. 12) gives $\bar{P}_e \le C K(\Lambda_c) \cdot \gamma_c(\Lambda_c)^{-n_r} \cdot \text{SNR}^{-d^*(r)}$ . The kissing number contributes a multiplicative constant; the coding gain contributes a $\gamma_c^{-n_r}$ factor that shifts the BER curve left by $n_r \log_2 \gamma_c$ dB.

Step 5 — Substituting $E_8$ and Leech

For $E_8$ : $\gamma_c(E_8) = 2^{1/2}$ (i.e., $1.5$ dB in $d_{\min}^2$ terms, $3$ dB in terms of the normalised $\text{SNR}$ ), $K(E_8) = 240$ , $n_t T = 8$ . Hence BER shifts by $3 n_r$ dB relative to random- $\Lambda_c$ . For $\Lambda_{24}$ : $\gamma_c(\Lambda_{24}) = 2$ ( $6$ dB), $K(\Lambda_{24}) = 196560$ , $n_t T = 24$ . Shifts by $6 n_r$ dB — a substantial coding-gain boost that reshapes the BER plot at moderate SNR. $\blacksquare$

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Example: Coding Gain of $E_8$ over $\mathbb{Z}^8$ in a Structured LAST Code

Compute the normalised coding gain of $E_8$ relative to $\mathbb{Z}^8$ , and translate it into a BER-vs-SNR shift for a structured LAST code at $(n_t, n_r) = (4, 4)$ and $r = 1$ multiplexing gain. Use the standard Conway-Sloane data: $d_{\min}^2(E_8) = 2$ , $V(E_8) = 1$ (in the conventional scaling where $E_8 \subset \mathbb{R}^8$ ), $d_{\min}^2(\mathbb{Z}^8) = 1$ , $V(\mathbb{Z}^8) = 1$ . Assume the LAST block length $T = 2$ so that $n_t T = 8$ matches $E_8$ .

Solution

Coding gain of $E_8$

$\gamma_c(\Lambda) = d_{\min}^2 / V^{2/n}$ . $\gamma_c(E_8) = 2 / 1^{2/8} = 2$ . $\gamma_c(\mathbb{Z}^8) = 1 / 1^{2/8} = 1$ . Ratio $\gamma_c(E_8) / \gamma_c(\mathbb{Z}^8) = 2$ , or $3$ dB.

BER shift on $4 \times 4$ at $r = 1$

By Thm. TStructured LAST with Dense Inner Lattice Achieves the DMT (Kumar-Caire 2008), the coding gain affects the BER as $\gamma_c^{-n_r}$ . With $n_r = 4$ : $\gamma_c(E_8)^{-n_r} = 2^{-4}$ , i.e., $12$ dB shift. Translated into SNR: at $\text{BER} = 10^{-4}$ , structured- $E_8$ -LAST at $(4,4)$ operates $12$ dB below random LAST in SNR. This is the concrete design payoff. $\blacksquare$

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Example: Structured LAST with Leech Lattice for $n_t T = 24$

A $3 \times n_r$ MIMO system uses a structured LAST code with inner lattice $\Lambda_c = \Lambda_{24}$ (the Leech lattice) and block length $T = 8$ . (a) Compute the total information bits per block for a $4$ -bit-per-dimension design ( $k = 16$ ). (b) State the DMT achieved on $(3, 3)$ Rayleigh fading. (c) Compute the coding-gain advantage over random LAST at the same rate.

Solution

Part (a): Information bits

$\log_2 |\Lambda_c / (k \Lambda_c)| = 24 \log_2 16 = 96$ bits per block of $T = 8$ channel uses, i.e., $12$ bits/ch.use across the $3$ transmit antennas, or $4$ bits/ch.use/antenna.

Part (b): DMT on $(3, 3)$

$d^*(r) = (3 - r)(3 - r)$ for $r \in \{0, 1, 2, 3\}$ . The Kumar-Caire theorem gives full DMT optimality of Leech-based structured LAST. At $r = 1$ : $d^* = 4$ ; at $r = 2$ : $d^* = 1$ .

Part (c): Coding gain

$\gamma_c(\Lambda_{24}) = d_{\min}^2(\Lambda_{24}) / V(\Lambda_{24})^{2/24} = 4 / 1^{1/12} = 4$ , i.e., $6$ dB over $\mathbb{Z}^{24}$ . Translated into the BER shift on a $(3,3)$ channel with $n_r = 3$ : $\gamma_c^{-n_r} = 4^{-3} = 2^{-6}$ , i.e., $18$ dB shift at the same BER target. Leech is a remarkable coding-gain workhorse in dimension $24$ , which is exactly why it is chosen here. $\blacksquare$

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Structured LAST BER vs. SNR for Different Inner Lattices

BER curves for a $2 \times 2$ MIMO (for example) at rate $2$ bits/ch.use/antenna, varying the inner lattice $\Lambda_c$ in the structured LAST construction. Choices: $\mathbb{Z}^n$ (baseline), $D_4$ (2-dim dense), $E_8$ (8-dim best-known and optimal), $\Lambda_{24}$ (Leech, best-known and optimal in 24 dim). The plot shows the progressive coding-gain improvement as the inner lattice becomes denser: $+1.5$ dB for $D_4$ , $+3$ dB for $E_8$ , $+6$ dB for Leech at moderate BER. All curves have the same slope (the DMT is preserved) but different intercepts.

Parameters

Inner lattice

Structured LAST: Wrapping $E_8$ Inside Nested Shaping

An animated construction of the structured LAST codebook. Starting from scalar QAM (a piece of

\mathbb{Z}^2

), we substitute the

E_8

inner lattice, which sits in

\mathbb{R}^8

with

d_{\min}^2 = 2

and kissing number

240

. We then wrap

E_8

inside the nested shaping lattice

\Lambda_s = k E_8

and choose

k

to set the rate. The final codebook is

E_8 / (k E_8)

— a finite number of cosets that, composed with the dither, give the LAST space-time code.

Structured LAST: (1) scalar QAM as a finite patch of

\mathbb{Z}^2

, (2) replace by

E_8

in

\mathbb{R}^8

, (3) wrap

E_8

in

k E_8

for shaping, (4) resulting codebook

|E_8 / k E_8| = k^8

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🔧Engineering Note

Where $E_8$ Actually Appears in Practice: Optical Coherent Systems

While $E_8$ -based space-time codes are not in any 5G NR release, they do appear in modern optical coherent communication systems. Optical coherent receivers use dual-polarisation 16-QAM or 64-QAM, operating on the complex 2D constellation per polarisation. A 4-dimensional $D_4$ -lattice constellation — the densest 4D lattice — has been proposed and partially deployed for coherent 100G/200G systems because it offers $\sim 1.5$ dB of shaping gain over naive 16-QAM.

The path forward — $E_8$ -lattice constellations in 4-D polarisation-multiplexed coherent optical transmission — would double the constellation dimension by using 4 consecutive time slots per polarisation, giving an $8$ -D constellation that can host $E_8$ . The theoretical gain is $3$ dB, matching the LAST-code analysis of §4. Commercial deployment has been slow because of decoder complexity and standardisation inertia, but several research groups (Millar, Koike-Akino, NTT, Nokia Bell Labs) have demonstrated $E_8$ -coded optical coherent transmission in lab experiments.

The lesson: lattice codes are already in deployment, just not in the wireless MIMO channels this book has focused on. The wavelength-division-multiplexed optical fibre channel is where $E_8$ is finding its first commercial home.

Practical Constraints

•
Optical coherent: dual-polarisation 16-QAM/64-QAM at 100G-400G
•
$D_4$ shaped constellations: partial deployment at 100G
•
$E_8$ constellations: experimental lab demonstrations at 200G

📋 Ref: ITU-T G.709 (OTU4, OTU5) — framework for coherent optical with FEC; $E_8$ not yet standardised.

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Random LAST vs. Structured LAST vs. CDA-NVD vs. V-BLAST

Property	Random LAST (2004)	Structured LAST (2008)	CDA-NVD (Ch. 13)	V-BLAST (1998)
Codebook structure	Random lattice (Minkowski-Hlawka)	Explicit dense lattice ( $E_8$ , Leech)	Cyclic division algebra	Scalar QAM per layer
Deterministic code?	No (ensemble)	Yes	Yes	Yes
DMT-optimal?	Yes	Yes	Yes	No (only $r = \min(n_t, n_r)$ )
Coding gain $\gamma_c$	$\sim 1$ (Minkowski-Hlawka baseline)	$\gamma_c(E_8) = 2$ , $\gamma_c(\Lambda_{24}) = 4$	Algebraic constant; $\delta_0^{1/n_t}$	$1$ (no coding)
Decoder	MMSE-GDFE + lattice	MMSE-GDFE + lattice	Sphere + ML	MMSE-SIC
Decoder complexity	Polynomial (MMSE-GDFE)	Polynomial (MMSE-GDFE)	Exponential in $n_t^2$	Polynomial
Dimension constraint	Any $n_t T$	$n_t T$ matches lattice dim	$n_t$ matches CDA degree	Any $n_t$
Practical at $n_t = 5, 6$ ?	Yes	Yes (if matching dim)	Marginal (sphere decoder limit)	Yes (no coding gain)

Common Mistake: Structured LAST Requires Dimension Match: $n_t T \in \{8, 16, 24, \ldots\}$

Mistake:

A reader might assume that structured LAST with inner lattice $E_8$ or Leech works for any $(n_t, n_r, T)$ configuration.

Correction:

Structured LAST requires the dimension $n_t T$ (after complex-to-real conversion, $2 n_t T$ ) to match the dimension of the chosen inner lattice. $E_8$ works for $2 n_t T = 8$ , i.e., $n_t T = 4$ (for example $n_t = 2, T = 2$ , or $n_t = 4, T = 1$ ). Leech works for $2 n_t T = 24$ , i.e., $n_t T = 12$ (for example $n_t = 3, T = 4$ , or $n_t = 4, T = 3$ ). Barnes-Wall $\Lambda_{16}^{BW}$ works for $n_t T = 8$ (matching $16$ -dim real). For $n_t T$ not matching a catalogued dense lattice dimension, one either (a) pads with a scaled $\mathbb{Z}^k$ factor (which dilutes the coding gain), (b) uses a different dense lattice of the right dimension (e.g., $K_{12}$ for $n_t T = 6$ ), or (c) falls back to random LAST. The dimension-matching constraint is the main practical limitation of structured LAST and is why the technique is most often applied in dimensions $8$ , $12$ , $16$ , $24$ , $32$ where the Conway-Sloane catalogue is dense.

Structured LAST Codes from Dense Lattice Packings

From Random Existence to Structured Construction

Definition: Structured LAST Code

Structured LAST Codes from Dense Lattice Packings

Theorem: Structured LAST with Dense Inner Lattice Achieves the DMT (Kumar-Caire 2008)

Step 1 — DMT part follows from 2004 LAST theorem

Step 2 — PEP between lattice codewords

Step 3 — Union bound over nearest neighbours

Step 4 — Channel averaging with coding-gain prefactor

Step 5 — Substituting $E_8$ and Leech

Example: Coding Gain of E8E_8E8​ over Z8\mathbb{Z}^8Z8 in a Structured LAST Code

Coding gain of $E_8$

BER shift on $4 \times 4$ at $r = 1$

Example: Structured LAST with Leech Lattice for ntT=24n_t T = 24nt​T=24

Part (a): Information bits

Part (b): DMT on $(3, 3)$

Part (c): Coding gain

Structured LAST BER vs. SNR for Different Inner Lattices

Parameters

Structured LAST: Wrapping E8E_8E8​ Inside Nested Shaping

Where E8E_8E8​ Actually Appears in Practice: Optical Coherent Systems

Random LAST vs. Structured LAST vs. CDA-NVD vs. V-BLAST

Common Mistake: Structured LAST Requires Dimension Match: ntT∈{8,16,24,…}n_t T \in \{8, 16, 24, \ldots\}nt​T∈{8,16,24,…}

Definition:
Structured LAST Code

Example: Coding Gain of $E_8$ over $\mathbb{Z}^8$ in a Structured LAST Code

Example: Structured LAST with Leech Lattice for $n_t T = 24$

Structured LAST: Wrapping $E_8$ Inside Nested Shaping

Where $E_8$ Actually Appears in Practice: Optical Coherent Systems

Common Mistake: Structured LAST Requires Dimension Match: $n_t T \in \{8, 16, 24, \ldots\}$