Unequal Error Protection of NACK in HARQ Systems

Updated 2 December 2025

The paper establishes that protecting NACK messages via UEP is vital to avoid misinterpretation errors that can lead to critical system outages.
It employs methods like asymmetric thresholding and joint source–channel coding to assign rigorous error limits to NACK while tolerating higher errors for ACK and data bits.
Advanced code designs, including multi-edge LDPC and red-alert signaling, demonstrate significant SNR gains and reduced power requirements in HARQ protocols.

Unequal error protection (UEP) of negative acknowledgments (NACK) is a specialized design imperative in modern feedback and retransmission-based communication schemes where NACK bits or messages must attain error rates orders of magnitude lower than those tolerated for positive acknowledgments (ACK) or typical data bits. This requirement stems from the system-level impact of NACK errors, which can trigger catastrophic outages or irremediable radio link failures, whereas ACK errors typically induce only redundant retransmissions or mild throughput penalties. This article surveys the theory, system architectures, coding methods, and performance implications of UEP for NACK, with focus on reliable feedback for HARQ protocols, advanced JSCC schemes, and code design under stringent reliability targets.

1. Motivation and System-Level Consequences

The classical HARQ (Hybrid Automatic Repeat reQuest) framework with incremental redundancy is particularly sensitive to feedback integrity. An erroneously received NACK—interpreted at the transmitter as an ACK—halts retransmissions prematurely and results in unrecoverable decoding failures, permanently impacting user-level throughput and link stability. In contrast, ACK errors induce unnecessary retransmissions but do not degrade reliability. Consequently, the protection of NACK outstrips that of ACK both in stringency and in the attainable SNR region, especially in systems such as 5G NR PUCCH where feedback operates under severe power or coverage constraints (Ding et al., 11 Jan 2024, Doshi et al., 25 Nov 2025).

A similar dichotomy surfaces in "red-alert" or high-priority signaling for one-out-of-many error patterns: when a single message (e.g., NACK, emergency alert) must be decoded with an exponentionally vanishing error probability, far below that required for the codebook average (Nazer et al., 2011). In all these settings, UEP of NACK is a structural component, not merely a coding artifact.

2. Asymmetric Feedback Detection: Threshold Design and Performance

In IR-HARQ systems with unreliable, typically one-bit feedback, UEP can be realized at the physical layer via asymmetric detection. The scalar matched-filter output $y$ from the AWGN uplink feedback passes through two distinct decision thresholds, $\tau_{\mathrm{ACK}}$ and $\tau_{\mathrm{NACK}}$ , instead of a conventional midpoint:

$\begin{cases} \text{decide ACK} &\text{if } y > \tau_{\mathrm{ACK}},\ \text{decide NACK} &\text{if } y < \tau_{\mathrm{NACK}},\ \text{declare erasure/repeat} &\text{otherwise.} \end{cases}$

By selecting an asymmetry index $\alpha>0$ so that $\tau_{\mathrm{ACK}}>\tau_{\mathrm{NACK}}$ , the error probability for NACK-to-ACK misdetection ( $P_N$ ) can be forced much lower than the reverse ( $P_A$ ):

$P_N = \tfrac12\,\mathrm{erfc}\bigl((1+\alpha)\sqrt{6\,\mathrm{SNR}_u}\bigr),\quad P_A = \tfrac12\,\mathrm{erfc}\bigl((1-\alpha)\sqrt{6\,\mathrm{SNR}_u}\bigr)$

Numerically, increasing $\alpha$ reduces $P_N$ sharply (to $10^{-3}$ at moderate SNR $_u$ ), while $P_A$ stays in the $10^{-1}$ to $10^{-2}$ range (Ding et al., 11 Jan 2024). This markedly dampens the system-level outage probability $P_{\mathrm{out}}$ , especially when system constraints require rigorous outage floors (e.g., $\epsilon\le1\%$ ). The resulting throughput gain versus symmetric detection or brute-force repetition is 8–12% under Rayleigh fading ( $\mathrm{SNR}_d=3\,\mathrm{dB}$ , $\mathrm{SNR}_u=-10\,\mathrm{dB}$ ), with optimal operational regimes guided by feedback SNR, bandwidth, and dynamic programming-based rate adaptation (Ding et al., 11 Jan 2024).

3. UEP of NACK via Joint Source–Channel Coding and Decoder Rule Engineering

Feedback payloads (HARQ-ACK/NACK bit-vectors) are inherently non-uniform (e.g., $p\approx 0.9$ for ACK), opening the path for UEP using joint source–channel coding (JSCC) and "bitwise" decoding rules. Transformer-based encoders induce codebooks and per-codeword power shaping that naturally exploit this non-uniformity (Doshi et al., 25 Nov 2025).

A principled approach employs an extension of the Neyman–Pearson (NP) hypothesis test for joint bitwise decoding under codebook-induced symbol likelihoods. For each bit $b_i$ , the decoder evaluates

$\Lambda_i(\mathbf y) = \frac{p(\mathbf y|b_i=0)}{p(\mathbf y|b_i=1)} \gtrless \frac{\beta}{1-\beta}$

where $\beta$ is a tunable "fake prior" chosen to enforce the NACK error constraint $\delta_0$ and accordingly relax ACK error to $\delta_1$ ( $\beta/(1-\beta)\approx \delta_0/\delta_1$ ). This coding-theoretic inversion of the Bayesian prior sharply lowers NACK error rates without sacrificing coverage: SNR requirements fall by $3$–$6$ dB, and the max power needed to hold per-NACK error $\le0.1\%$ is reduced by $2$–$3$ dB over 5G NR baselines (Doshi et al., 25 Nov 2025). Deep JSCC codebooks, joint with power shaping—such as entropy-weighted, arithmetic, or step schemes—reinforce this UEP principle, while preserving robust operation to changes in source priors.

4. Code Design for Persistent UEP: Multi-Edge LDPC and Structural Separation

Conventional block codes with irregular variable-node degrees can provide some level of UEP, but without careful design this effect disappears as belief propagation (BP) iterations grow. Multi-edge-type LDPC codes prevent this collapse via an explicit partition of variable nodes into high-priority (NACK), data, and parity classes, classified as $C_1$ , $C_2$ , $C_3$ .

Each check node's degree profile is carefully engineered via edge fractions $\rho_d^{(j)}$ , such that the average number of type- $j$ edges per check node is minimized for higher priority classes. The design ensures monotonic convergence of mutual information in density evolution for each class across BP iterations, locking in the UEP gap even at high numbers of BP passes (Neto et al., 2011). Table 1 summarizes the results for varying the cap parameter (which limits the interaction between classes) on the achieved UEP SNR gap:

cap $_2$	$\Delta$ UEP SNR (NACK-vs-Data, dB)
0.35	1.5
0.55	1.0
0.75	0.7

By minimizing the connections of NACK bits to lower-priority checks, this approach provides a recipe for UEP in block-code based feedback and control message encoding.

5. The Red Alert Problem: Information-Theoretic Exponents for NACK UEP

The AWGN red alert problem formalizes the limit of UEP for a single "red-alert" codeword (i.e., NACK), requiring its missed-detection probability to vanish exponentially faster than for other symbols, subject to average error and false alarm constraints (Nazer et al., 2011). Under block-length $n$ , the exponent $E_{\mathrm{red}}(R)$ for missed-detection at rate $R$ is

$E_{\mathrm{red}}(R) = \frac{P_0 + P +2 \sqrt{P_0(P + \sigma^2(1-e^{2R}))}}{2 \sigma^2} - R$

where $P$ is the average power per standard message and $P_0\geq P$ is the permitted (possibly higher) power for NACK. The codebook places standard messages on an offset Gaussian shell and the NACK codeword antipodal to this region, widening the geometric separation and enabling a sharp, cone-shaped test at the decoder. This configuration achieves the optimal tradeoff between NACK missed-detection (outage) and ordinary message error, and the construction maps directly to UEP in practical feedback and control signaling.

6. Implementation Guidelines and System Trade-Offs

System designers must regulate UEP by balancing SNR, threshold asymmetry, code construction, and decoder complexity. In low-SNR feedback settings (e.g., $\mathrm{SNR}_u\leq-10$ dB), large asymmetry indices $\alpha\in[0.7,1]$ are recommended to virtually eliminate NACK errors (Ding et al., 11 Jan 2024), while in moderate SNR a smaller degree suffices. JSCC codes with per-codeword power shaping and NP-based UEP decoders can extend coverage by 4–6 dB while guaranteeing NACK error rates at $0.1\%$ and allowing relaxed ACK error rates (Doshi et al., 25 Nov 2025).

From a code design perspective, multi-edge-type LDPC codes should minimize NACK-class edge involvement and, where possible, restrict the encoded NACK population to a small subset of variable nodes. For scenario-driven, one-of-many UEP, the red-alert geometry provides the ultimate theoretical bound for attainable error exponents (Nazer et al., 2011).

7. Outlook and Ongoing Research Directions

Current research addresses practicalities such as robust threshold learning under time-varying priors, adaptive thresholding for multi-bit feedback, and integration with autoencoder-based codebooks or quantum-inspired detectors. Transformer-based JSCC, free-lunch training across source priors, and low-complexity coherent receiver approximations are under active investigation for integrating UEP of NACK into standards-compliant feedback mechanisms (Doshi et al., 25 Nov 2025). A plausible implication is that deep learning-based decoder parameterization may soon allow fine-grained, context-aware UEP of NACK in dense, multi-user networks, enforcing ultra-reliable radio link control with minimal power or bandwidth penalties.

References:

(Ding et al., 11 Jan 2024) Optimized Asymmetric Feedback Detection for Rate-adaptive HARQ with Unreliable Feedback.
(Doshi et al., 25 Nov 2025) AI/ML based Joint Source and Channel Coding for HARQ-ACK Payload.
(Neto et al., 2011) Multi-Edge type Unequal Error Protection LDPC codes.
(Nazer et al., 2011) The AWGN Red Alert Problem.