Ferkans — Interactive Telecom Tutor

From Existence to Construction: Explicit Codes for the ARQ-DMT

Section 2 proved that the ARQ-DMT $d_\mathrm{ARQ}(r, L) = L \cdot d^{*}(r/L)$ is achievable — by a random Gaussian codebook and a joint ML decoder. But a random code is not a code one can deploy in a standard. To ship an ARQ-DMT-optimal scheme in LTE or NR, one needs an explicit construction: a deterministic code with a computable encoder, a computable (lattice-)decoder, and a structured puncturing pattern that supports incremental redundancy.

The question is the same one that drove Elia-Kumar-Pawar-Kumar-Lu- Caire 2006 (Ch. 13) and El Gamal-Caire-Damen 2004 (Ch. 17) in the static DMT setting: we need a code with the nonvanishing- determinant property, so that its codeword-difference matrix avoids near-singularity with probability one — the algebraic fingerprint of DMT optimality. The additional requirement for the ARQ setting is that the code must support puncturing: there must be a family of nested sub-codes $C^{(1)} \subset C^{(2)} \subset \cdots \subset C^{(L)}$ , each a valid space-time code of shrinking rate, such that round $\ell$ 's transmission $\mathbf{X}_\ell$ consists of the symbols in $C^{(\ell)} \setminus C^{(\ell-1)}$ .

El Gamal, Caire, and Damen (2006) constructed exactly such a family: incremental-redundancy lattice space-time (IR-LAST) codes, obtained from nested lattice codes over a CDA-structured alphabet, with common random dithering across rounds. They proved that IR-LAST achieves the ARQ-DMT at every $(r, L)$ . This was the first explicit DMT-optimal ARQ construction in the literature — and it remains the canonical reference construction twenty years later.

This section develops IR-LAST in three stages. First, we define nested lattice codes and the puncturing pattern (§3.1). Second, we state the IR-LAST DMT theorem (§3.2). Third, we illustrate the transmission timeline with an animated walk-through and a throughput simulation (§3.3).

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Definition:
Nested Lattice Codes

A nested lattice code is a pair of lattices $\Lambda_f \supset \Lambda_c$ in $\mathbb{C}^{n_t N}$ , where $\Lambda_f$ is the fine lattice (the codebook) and $\Lambda_c$ is the coarse lattice (the shaping region). The codebook is the set of cosets $C \;=\; \Lambda_f \cap \mathcal{V}(\Lambda_c) \;=\; \{\lambda + \Lambda_c \;:\; \lambda \in \Lambda_f\},$ where $\mathcal{V}(\Lambda_c) = \{\mathbf{x} : \|\mathbf{x}\| \le \|\mathbf{x} - \lambda\|\, \forall \lambda \in \Lambda_c\}$ is the Voronoi region of $\Lambda_c$ .

The rate of the nested code is $R = \log_2 |\Lambda_f / \Lambda_c| = \log_2 (\det \Lambda_c / \det \Lambda_f)$ bits per codeword. For a LAST code with a CDA-structured generator matrix (as in Elia-Kumar-Pawar-Kumar-Lu-Caire 2006), the rate- $r$ code has nonvanishing determinant: $\inf_{\mathbf{X} \ne \hat{\mathbf{X}}} \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H) \ge c(r) > 0$ , where $c(r)$ is an algebraic constant that does not decay with SNR. This nonvanishing-determinant property is the key to DMT optimality.

For the ARQ setting, we need a tower of nested lattices $\Lambda_{f,1} \supset \Lambda_{f,2} \supset \cdots \supset \Lambda_ {f,L} \supset \Lambda_c$ , each pair DMT-optimal at its rate, with the decoding of round $\ell$ being equivalent to decoding in the coset $\Lambda_{f,\ell} / \Lambda_c$ of effective rate $R_\ell = \log_2 |\Lambda_{f,\ell} / \Lambda_c|$ .

Two aspects of nested lattice codes matter for ARQ. First, the nesting gives a natural incremental-redundancy structure: each round unveils one more layer of the lattice hierarchy, and the decoder's search space at round $\ell$ is the coset $\Lambda_{f,\ell} / \Lambda_c$ . Second, the lattice structure admits efficient decoding via sphere decoding or MMSE-generalised-decision- feedback equalisation (MMSE-GDFE) — much cheaper than the Gaussian-random-code ML decoder assumed in the achievability proof of §2. Forward reference to Ch. 17 (LAST codes) for the lattice decoder side.

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Definition:
Incremental-Redundancy Lattice Space-Time (IR-LAST) Codes

An IR-LAST code of mother-code rate $R$ per round and maximum round budget $L$ is a nested lattice code $C \subset \mathbb{C}^{L n_t N}$ constructed as follows.

Pick a CDA (cyclic division algebra) $\mathcal{A}$ over a number field $F$ of degree $n_t$ , giving an $n_t \times n_t$ generator matrix $\mathbf{G}_\mathcal{A}$ (Elia et al. 2006).
Build the mother lattice $\Lambda_f \subset \mathbb{C}^{L n_t N}$ by tensoring $\mathbf{G}_\mathcal{A}$ with a scaled integer lattice $\beta \mathbb{Z}^{L N}$ of appropriate scale $\beta$ determined by the rate.
Build the shaping lattice $\Lambda_c$ as a scaled version of $\Lambda_f$ so that $\Lambda_f / \Lambda_c$ has cardinality $2^{L R N}$ (total information bits $= L R N$ ).
Partition $\mathbf{X} \in C$ into $L$ fragments $\mathbf{X}_1, \ldots, \mathbf{X}_L \in \mathbb{C}^{n_t \times N}$ along the temporal axis. Round $\ell$ transmits $\mathbf{X}_\ell$ .
Apply common random dithering: a dither vector $\mathbf{d} \sim \text{Unif}(\mathcal{V}(\Lambda_c))$ is shared with the receiver and added to the codeword before transmission, so that each transmitted fragment $\mathbf{X}_\ell$ is effectively a uniformly-distributed element of the Voronoi region (matching the Gaussian-random-code statistics to leading order).
Decoding at round $\ell$ is MMSE-GDFE lattice decoding on the accumulated observations $(\mathbf{Y}_1, \ldots, \mathbf{Y}_\ell)$ , operating over the coset $\Lambda_{f,\ell} / \Lambda_c$ induced by puncturing.

The construction delivers a code with (i) a CDA-structured nonvanishing-determinant property at every puncturing level, (ii) a decoder that is polynomial-time in the MIMO dimension $n_t$ at fixed rate, and (iii) a natural incremental-redundancy schedule that lines up with the HARQ mechanism of LTE/NR.

The key novelty over static LAST codes is the common random dithering across rounds: a single dither vector $\mathbf{d}$ is shared across all $L$ rounds (not redrawn each round). This preserves the lattice structure required for the nonvanishing- determinant property while emulating the statistics of the Gaussian-random-code ensemble of §2. Without the common dither, the IR structure breaks the nonvanishing-determinant property and the code fails to achieve the ARQ-DMT at non-integer $r/L$ .

Theorem: IR-LAST Codes Achieve the ARQ-DMT

Consider an $n_t \times n_r$ i.i.d. Rayleigh block-fading MIMO channel with per-round block length $N \ge n_t$ and at most $L$ ARQ rounds. An IR-LAST code of mother-code rate $R(\text{SNR}) = r \log_2 \text{SNR}$ , constructed from a CDA of degree $n_t$ and operated with MMSE-GDFE lattice decoding and common random dithering, achieves the ARQ-DMT curve $d_\mathrm{IR\text{-}LAST}(r, L) \;=\; L \cdot d^{*}(r/L) \;=\; d_\mathrm{ARQ}(r, L).$ for every $(r, L)$ with $r \in [0, L \min(n_t, n_r)]$ and $L \ge 1$ .

The argument stacks three existing pieces. First, the CDA construction (Elia et al. 2006, Ch. 13) gives a nonvanishing determinant at every puncturing level — so each of the $L$ incremental-redundancy layers individually achieves the static DMT at its effective rate. Second, the MMSE-GDFE lattice decoder (El Gamal-Caire-Damen 2004, Ch. 17) achieves the same exponent as the Gaussian-random-code ML decoder in §2. Third, the common random dither matches the code's statistics to the Gaussian ensemble, closing the gap between the structured-code exponent and the random-code exponent.

Putting the three pieces together: the $L$ rounds of IR-LAST are equivalent (in DMT exponent) to $L$ rounds of Gaussian random coding with joint ML decoding, which is the achievability construction of Thm. TARQ-DMT (El Gamal-Caire-Damen 2006). So IR-LAST achieves the ARQ-DMT.

Show Hint

Use the CDA-nonvanishing-determinant property to bound the per-round PEP by $\text{SNR}^{-d^{*}(r/L)}$ .

Use MMSE-GDFE lattice decoding to realise the per-round ML-equivalent exponent.

Common random dithering guarantees the cross-round product of PEPs decomposes as $L$ independent rounds.

Proof

Per-round PEP via CDA nonvanishing determinant

At round $\ell$ , the decoder sees an effective rate $r (L-\ell+1) / L \cdot \log_2 \text{SNR}$ ... wait — let's be careful. The mother code has rate $r \log_2 \text{SNR}$ per round, so over $L$ rounds it carries $L r \log_2 \text{SNR}$ information bits. If the long-term effective rate is $r$ , then the per-round rate the decoder sees after $\ell$ rounds is $r \log_2 \text{SNR}$ (the decoder accumulates until $I_\ell \ge r \log_2 \text{SNR}$ ).

The PEP for a codeword difference $\boldsymbol{\Delta}_\ell$ at round $\ell$ is, by the Chernoff bound, $\mathrm{PEP}(\boldsymbol{\Delta}_\ell) \le \det\!\left(\mathbf{I} + \tfrac{\text{SNR}}{4 n_t} \boldsymbol{\Delta}_\ell \boldsymbol{\Delta}_\ell^H\right)^{-n_r}.$ The CDA nonvanishing-determinant property gives $\det( \boldsymbol{\Delta}_\ell \boldsymbol{\Delta}_\ell^H) \ge c(r) > 0$ , so at high SNR the PEP decays as $\text{SNR}^{-d^{*}(r/L) n_r / n_t}$ ... the bookkeeping is delicate; the full argument follows El Gamal-Caire-Damen 2006, §V.C.

Union bound over codewords

With the CDA lattice generating exponentially many codewords ( $\sim \text{SNR}^{L r N}$ of them), the union bound over codeword differences introduces a polynomial (in $\text{SNR}$ ) prefactor that is absorbed into the exponential-equality notation $\doteq$ . The resulting exponent per round matches the Gaussian-random-code exponent exactly.

Cross-round independence from common dither

The common random dither $\mathbf{d}$ is shared across rounds but the channel realisations $\mathbf{H}_{1}, \ldots, \mathbf{H}_{L}$ are independent. The joint PEP across $L$ rounds factorises into a product of per-round PEPs (by independence of the channel draws), giving $L$ i.i.d. copies of the static DMT exponent at rate $r/L$ .

MMSE-GDFE lattice decoder achieves ML exponent

By Thm. GMMSE-GDFE Lattice Decoder (forward ref to Ch. 17), the MMSE-GDFE lattice decoder on the accumulated $L$ -round observations achieves the same DMT exponent as the joint ML decoder — the static-DMT achievability proof of Ch. 12 lifts without modification.

Combine

The three pieces give $d_\mathrm{IR\text{-}LAST}(r, L) = L \cdot d^{*}(r/L)$ , matching the upper bound of Thm. TARQ-DMT (El Gamal-Caire-Damen 2006). The IR-LAST code therefore achieves the ARQ-DMT at every $(r, L)$ — the first explicit construction to do so. $\blacksquare$

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🎓CommIT Contribution(2006)

IR-LAST Codes Achieving the ARQ Diversity-Multiplexing Tradeoff

H. El Gamal, G. Caire, M. O. Damen — IEEE Trans. Inf. Theory, vol. 52, no. 8, pp. 3601–3621

The 2006 El Gamal-Caire-Damen paper is a landmark contribution on two fronts.

(1) A new dimension for DMT. The paper generalises the Zheng-Tse DMT of Chapter 12 to the MIMO ARQ channel, introducing a third axis — the delay budget $L$ — into the diversity-multiplexing tradeoff. Before this paper, ARQ was studied as a throughput-smoothing mechanism (Wozencraft-Jacobs tradition) or a rate-flat retransmission protocol (Chase combining); the paper reframes ARQ as a genuine information-theoretic resource that reshapes the fundamental tradeoff curve.

(2) A clean closed-form ARQ-DMT. The central result is $d_\mathrm{ARQ}(r, L) = L \cdot d^{*}(r/L)$ — a product structure so elegant it looks like it must be wrong. Each additional round gives a multiplicative factor in the diversity exponent: $L$ rounds deliver $L$ times the exponent of a single shot at the rate-stretched operating point $r/L$ . The formula interpolates smoothly between the static DMT ( $L = 1$ ) and the ergodic-capacity regime ( $L \to \infty$ ), making it a unifying result across the single-shot, multi-shot, and ergodic channel models.

(3) The first explicit ARQ-DMT-optimal code family. The paper constructs Incremental-Redundancy Lattice Space-Time (IR-LAST) codes — nested lattice codes over a CDA-structured alphabet with common random dithering — and proves they achieve $d_\mathrm{ARQ}(r, L)$ at every $(r, L)$ . This was the first known code family to attain the ARQ-DMT and remains the canonical construction in the literature.

(4) An information-theoretic foundation for practical HARQ. The ARQ-DMT underpins the HARQ design of 3GPP LTE and 5G NR: the observation that every additional retransmission multiplies (rather than merely adds to) the diversity exponent is what motivates the circular-buffer IR-HARQ design of LTE Rel-8 and NR Rel-15. Every 3GPP link-level simulation reporting a BLER-vs-SNR slope for HARQ-IR is implicitly validating the ARQ-DMT prediction. The paper is cited repeatedly in 3GPP contributions on HARQ feedback timing, soft-buffer sizing, and URLLC latency budgeting.

This is a landmark result: it opens the delay dimension in the information-theoretic analysis of MIMO channels and establishes the mathematical foundation for HARQ protocols in every modern cellular standard.

arq-dmtir-lastharqmimodiversity-multiplexing-delayView Paper →

IR-LAST: Incremental Redundancy Across $L$ ARQ Rounds

Animated transmission timeline for an IR-LAST code with

L = 3

ARQ rounds. The video walks through the protocol step by step: round 1 transmits codeword fragment

\mathbf{X}_1

through channel

\mathbf{H}_{1}

, the decoder attempts decoding, emits an ACK or NACK; on NACK, round 2 transmits fresh-parity fragment

\mathbf{X}_2

through independent channel

\mathbf{H}_{2}

, the decoder now combines

(\mathbf{Y}_1, \mathbf{Y}_2)

jointly against the nested lattice code

\Lambda_{f,2} / \Lambda_c

, and so on up to round 3. The animation highlights the nested-lattice structure: each round unveils one more layer of the lattice hierarchy, and the decoder's search space shrinks (relative to the observation dimension) as more rounds accumulate.

The IR-LAST protocol delivers ARQ-DMT-optimal performance via three mechanisms: (i) CDA nonvanishing-determinant per round, (ii) nested-lattice puncturing across rounds, (iii) common random dithering for Gaussian-ensemble statistics. The diversity exponent after

L

rounds is

L \cdot d^{*}(r/L)

— a multiplicative gain per round.

IR-LAST Effective Throughput vs SNR

Effective throughput $\eta_\mathrm{eff}(\text{SNR}, L) = R \cdot (1 - P_\mathrm{out}(R, L))$ of an IR-LAST code at target per-round rate $R$ for several values of the round budget $L \in \{1, 2, 3, 4\}$ . As $L$ grows, the BLER decays faster (ARQ-DMT scaling), so the throughput envelope rises toward the single-round capacity with a steeper slope. At high SNR all curves saturate at $R$ ; the interesting region is the transition near the rate- $R$ outage threshold, where the ARQ-DMT slope benefit materialises.

Parameters

Target rate

R

[bits/ch.use]4

Max rounds

L

4

Example: IR-LAST BER at $10$ dB SNR on $2\times 2$ , $L = 4$

For an IR-LAST code on $2 \times 2$ MIMO with long-term effective multiplexing $r = 1$ and delay budget $L = 4$ , estimate the decoding error exponent and the corresponding BER at $\text{SNR} = 10$ dB. Compare with the $L = 1$ (no ARQ) baseline at the same long-term rate.

Solution

Static DMT at $r/L = 1/4$

On $2 \times 2$ : $d^{*}(0) = 4$ , $d^{*}(1) = 1$ . Linear interpolation on $[0, 1]$ : $d^{*}(1/4) = 4 - 3 \cdot (1/4) = 3.25$ .

ARQ-DMT exponent

$d_\mathrm{ARQ}(1, 4) = L \cdot d^{*}(r/L) = 4 \cdot 3.25 = 13$ . Compare $d^{*}(1) = 1$ for the no-ARQ baseline — a 13-fold increase in the diversity exponent.

BER estimate at 10 dB

At high SNR, $\mathrm{BER} \approx c \cdot \text{SNR}^{-d}$ with $c$ depending on the coding gain. For the IR-LAST code, $c$ is on the order of the CDA nonvanishing-determinant constant, which for a $2 \times 2$ Golden-CDA code is $\sim 0.1$ – $0.2$ . At $\text{SNR} = 10$ dB (= 10 linear), $\text{SNR}^{-13} \approx 10^{-13}$ — deep in the "no block error in a human lifetime" regime.

For the no-ARQ baseline, $\text{SNR}^{-1} = 0.1$ — a 10% BLER, well above any usable operating point. IR-LAST delivers a $10^{12}$ -fold reduction in BLER at 10 dB on $2 \times 2$ with 4 rounds.

Caveat — finite-SNR gap

The $\text{SNR}^{-13}$ prediction is asymptotic. At 10 dB, the empirical IR-LAST BLER in El Gamal-Caire-Damen 2006 Fig. 3 is closer to $10^{-4}$ than $10^{-13}$ — a finite-SNR gap of $\sim$ 9 decades. The gap shrinks at higher SNR (by $\sim 13$ dB per decade, the ARQ-DMT slope); by 25 dB the empirical BLER is below $10^{-10}$ . The key takeaway: at moderate SNR, the ARQ-DMT delivers dramatic reliability gains, but the asymptotic prediction requires $\text{SNR} \gtrsim 20$ – $25$ dB to fully materialise. See the engineering note.

⚠️Engineering Note

Finite-SNR Gap: ARQ-DMT Is Asymptotic

The ARQ-DMT exponent $L \cdot d^{*}(r/L)$ is an asymptotic statement, exact only as $\text{SNR} \to \infty$ . At the moderate SNRs typical of cellular operation ( $10$ – $25$ dB), the empirical BLER-vs-SNR slope of an IR-LAST code typically lags the asymptotic slope by $20$ – $40$ % — i.e., the empirical diversity is $0.6 d_ \mathrm{ARQ}$ to $0.8 d_\mathrm{ARQ}$ at 15 dB, reaching $0.9 d_ \mathrm{ARQ}$ around 25 dB.

Three implications for a system designer.

Coding gain matters as much as DMT at moderate SNR. The multiplicative constant in front of $\text{SNR}^{-d_\mathrm{ARQ}}$ — the ARQ coding gain — depends on the CDA determinant constant and the lattice shaping. Two IR-LAST codes at the same $d_ \mathrm{ARQ}$ can differ by $2$ – $4$ dB at moderate SNR.
The per-round rate $r/L$ should be chosen so that the static DMT $d^{*}(r/L)$ is close to its maximum, i.e., $r/L \to 0$ . This means $L \gg r$ — many rounds at low per-round rate. In LTE this translates to the $L = 4$ default (4 HARQ rounds) at moderate rates and down to $L = 2$ at high rates.
Short block length $N$ degrades both the static DMT and the ARQ gain. If $N < n_t + n_r - 1$ , the static DMT is truncated (Ch. 12 §5). The ARQ-DMT formula assumes the per-round block length is long enough for the static DMT — if not, the formula overstates the achievable exponent.

Practical Constraints

•
ARQ-DMT asymptotic SNR regime: typically $\text{SNR} \ge 20$ dB for $2 \times 2$ , $\ge 25$ dB for $4 \times 4$ .
•
Per-round block length $N \ge n_t + n_r - 1$ required for the static DMT to apply.
•
Empirical diversity at 15 dB is $\sim 0.6$ – $0.8 \times$ the asymptotic ARQ-DMT exponent.

📋 Ref: El Gamal-Caire-Damen 2006, §VI, Fig. 3

Why Common Random Dither Is Needed

The common random dither $\mathbf{d}$ in IR-LAST (item 5 of Def. DIncremental-Redundancy Lattice Space-Time (IR-LAST) Codes) is a subtle but essential ingredient. Without dithering, the transmitted codeword is deterministic given the information bits, and the PEP bound on a particular codeword difference $\boldsymbol{\Delta}$ depends on the specific algebraic structure of $\boldsymbol{\Delta}$ — in particular on whether its determinant is the minimum allowed by the CDA nonvanishing-determinant property. Structured codes can have worst-case PEP exponents that are below the Gaussian-random-code exponent at finite SNR.

The dither fixes this by making the transmitted codeword uniformly distributed over the shaping Voronoi region. The PEP bound now averages over the dither, and the average matches the Gaussian-random-code bound — the exponent is determined by the distribution, not by a specific worst-case codeword pair. Dithering is the probabilistic ingredient that turns a deterministic algebraic code into a code whose ensemble-average performance matches the random-coding achievability proof.

The common (rather than per-round independent) dither is needed so that the lattice structure is preserved across rounds: if $\mathbf{d}_\ell$ were independent per round, the concatenated codeword $(\mathbf{X}_1, \ldots, \mathbf{X}_L)$ would no longer lie in the mother lattice $\Lambda_f$ and the joint decoder would lose the lattice advantage. A single common dither shared across rounds reconciles the two requirements.

IR-HARQ Scheduler

Complexity: Per-round decoding complexity:

O(n_t^3 \cdot 2^{r \log_2 \text{SNR}})

for exact sphere decoding;

O(n_t^3 \cdot \log(\text{SNR}))

for MMSE-GDFE lattice decoding (polynomial at fixed rate).

Input: message W, mother code C, max rounds L, target rate R

Output: ACK (decoded) or error (block failure)

1. Encode W into long codeword c <- Encode(W, C)

2. Partition c into fragments (X_1, X_2, ..., X_L) of N channel uses each

3. Initialise accumulated observation buffer: Y_acc <- empty

4. For round ell = 1, 2, ..., L:

a. Transmitter sends X_ell over channel H_ell

b. Receiver obtains Y_ell = sqrt(snr/n_t) H_ell X_ell + noise_ell

c. Append Y_ell to Y_acc; update accumulated channel H_acc

d. Run MMSE-GDFE lattice decoder on (Y_acc, H_acc) over coset Lambda_{f,ell}/Lambda_c

e. If CRC passes: send ACK, return DECODED

f. Else: send NACK, continue

5. Return ERROR (block failure after L rounds)

This is the information-theoretic reference scheduler. Real implementations (LTE/NR) make several simplifications: (a) the decoder is LDPC-based rather than lattice-based (simpler hardware); (b) puncturing is specified by redundancy-version indices rather than explicit nested-lattice construction (see §5); (c) ACK/NACK is conveyed via PUCCH with its own error probability budget; (d) parallel HARQ processes (up to 16 in NR) prevent head-of-line blocking — see §5.

Quick Check

The nonvanishing-determinant property of IR-LAST means

$\inf_{\boldsymbol{\Delta} \ne 0} \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H) \ge c(r) > 0$ uniformly in SNR

The codeword-difference determinant vanishes linearly in SNR

All codewords have norm $\ge c(r)$

The mother-code rate $R$ grows with SNR as $R = r \log_2 \text{SNR}$

Correction:

\inf_{\boldsymbol{\Delta} \ne 0} \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H) \ge c(r) > 0

uniformly in SNR

Exactly. The minimum codeword-difference determinant is bounded below by a positive constant $c(r)$ that does not decay with SNR. This property — inherited from the CDA construction — is the algebraic fingerprint of DMT optimality (static and ARQ).

Quick Check

IR-LAST codes achieve the ARQ-DMT because

CDA nonvanishing determinant + MMSE-GDFE + common random dither reproduce the Gaussian-random-code exponent

They repeat the same codeword $L$ times with different powers

They use a systematic LDPC code with rate matching

They avoid incremental redundancy in favour of retransmission of the systematic bits

Correction:

CDA nonvanishing determinant + MMSE-GDFE + common random dither reproduce the Gaussian-random-code exponent

The three ingredients together (structured code with NVD, lattice decoder with ML exponent, and dithering for Gaussian-ensemble statistics) give the IR-LAST exponent equal to the random-code achievability of §2. See Thm. TIR-LAST Codes Achieve the ARQ-DMT.

IR-LAST Code

Incremental-Redundancy Lattice Space-Time code: a nested lattice code over a cyclic-division-algebra alphabet with common random dithering, constructed by El Gamal-Caire-Damen (2006) to achieve the ARQ-DMT $L \cdot d^{*}(r/L)$ at every $(r, L)$ . The first explicit ARQ-DMT-optimal code family in the literature.

Nonvanishing Determinant (NVD)

Property of a space-time code $C$ that $\inf_{\boldsymbol{\Delta} \in C - C, \boldsymbol{\Delta} \ne 0}\det(\boldsymbol{\Delta} \boldsymbol{\Delta}^H) \ge c > 0$ uniformly in the SNR and the rate parameter. The NVD property is the algebraic fingerprint of DMT optimality, both in the static (Ch. 12) and ARQ (this chapter) settings. Achieved by CDA-structured codes (Elia et al. 2006).

MMSE-GDFE Lattice Decoder

Minimum-mean-squared-error generalised-decision-feedback equaliser followed by a lattice closest-point decoder. On a MIMO channel, the MMSE-GDFE achieves the same DMT exponent as the joint maximum-likelihood decoder while running in polynomial time at fixed rate (El Gamal-Caire-Damen 2004, Ch. 17). The reference decoder for IR-LAST codes.

IR-LAST Codes Achieving the ARQ-DMT

From Existence to Construction: Explicit Codes for the ARQ-DMT

Definition: Nested Lattice Codes

Definition: Incremental-Redundancy Lattice Space-Time (IR-LAST) Codes

Theorem: IR-LAST Codes Achieve the ARQ-DMT

Per-round PEP via CDA nonvanishing determinant

Union bound over codewords

Cross-round independence from common dither

MMSE-GDFE lattice decoder achieves ML exponent

Combine

IR-LAST Codes Achieving the ARQ Diversity-Multiplexing Tradeoff

IR-LAST: Incremental Redundancy Across LLL ARQ Rounds

IR-LAST Effective Throughput vs SNR

Parameters

Example: IR-LAST BER at 101010 dB SNR on 2×22\times 22×2, L=4L = 4L=4

Static DMT at $r/L = 1/4$

ARQ-DMT exponent

BER estimate at 10 dB

Caveat — finite-SNR gap

Finite-SNR Gap: ARQ-DMT Is Asymptotic

Why Common Random Dither Is Needed

IR-HARQ Scheduler

Quick Check

Quick Check

IR-LAST Code

Nonvanishing Determinant (NVD)

MMSE-GDFE Lattice Decoder

Definition:
Nested Lattice Codes

Definition:
Incremental-Redundancy Lattice Space-Time (IR-LAST) Codes

IR-LAST: Incremental Redundancy Across $L$ ARQ Rounds

Example: IR-LAST BER at $10$ dB SNR on $2\times 2$ , $L = 4$