Analysis of Universal Decoding Techniques for 6G Ultra-Reliable and Low-Latency Communication Scenario

Abstract

Ultra-reliable and low-latency communication (URLLC) in 6G networks is characterized by very high reliability and very low latency to enable mission-critical applications. The ability of a coding scheme to support diverse use cases requires flexibility on the part of the decoder. High reliability and low latency require decoders with improved error rate performance and reduced complexity. This article investigates candidate universal decoding algorithms for 6G communication scenarios. Universal decoders work on a wide range of error-correcting codes, making them scalable for different communication protocols. This article undertakes the comparative analysis and performance evaluation of the code-agnostic decoding schemes, including automorphism ensemble (AED), guessing random additive noise (GRAND), ordered statistics (OSD), belief propagation (BPD), bit flipping (BFD), and their variants. Simulations are carried out in MATLAB (R2024a) for the error rate performance of decoders, and plots are provided for the comparative analysis from the results of inferred data. The key findings in this paper highlight the competitive advantage of universal decoding techniques in comparison to the standardized CA-SCL decoding of polar code. Consequently, this work will help in identifying more efficient decoding algorithms for potential 6G URLLC applications. We aim to provide an insight into the scalability of universal decoding techniques by exploring their key performance metrics and comparing their performances.

1. Introduction

The sixth-generation (6G) wireless network is posed to play a significant role in society’s digital transformation, and the ultra-reliable low-latency communication (URLLC) scenario will be one of its key enablers. Compared to fifth-generation new radio (5G NR), URLLC in 6G demands a much lower end-to-end (E2E) latency of (1 ms∼100 $\mathsf{\mu }$s) and provides a probability of successful message delivery on the order of 99.9999%, requiring a block error rate (BLER) of ${10}^{-5}$∼${10}^{-7}$ [1,2,3]. According to 3GPP 5G NR guidelines, while eMBB targets to achieve high throughput, URLLC and mMTC services are sensitive to latency, thereby necessitating short and highly reliable data transmissions. To this end, Release 16 included network slicing and service-based architecture (SBA), for enhancing URLLC for V2X (vehicle-to-everything) and industrial automation. Also, Release 17 incorporated IoT over non-terrestrial networks (NTNs) to provide broader coverage [4].

Channel coding is a key physical-layer technique used in digital communication systems for the reliable and efficient transmission of data. The main goal in channel coding theory is to approach channel capacity with minimal computational complexity [5]. Additionally, mission-critical applications have rekindled interest in the effectiveness of short-block-length error-correcting codes due to the need for high-reliability, low-latency communications. As 6G is targeted to support a wide range of applications, it has to cater to diverse data rates, latency, reliability, and massive connectivity requirements. The complex network topology of next-generation wireless communication requires a range of adaptable and state-of-the-art coding schemes. Further, applications utilizing short block lengths require an effective decoder of the coding technique employed. The capacity of the decoder to decode any linear code is known as universality. Different decoding techniques have been developed over the years, often unique to one family of error-correcting codes. Universal soft decoding techniques approximate maximum likelihood (ML) decoders for binary linear codes [6]. Also, for a 6G transmission environment where the channel state varies dynamically, universal decoders work effectively as they can operate without exact knowledge of the channel distribution. Some of the communication protocols where universal decoding techniques find their application are listed here:

• Wireless communication—5G NR, Wi-Fi [7], low-power wide-area networks (LPWANs). Universal decoders enhance the spectral efficiency of the communication system by enabling flexible decoding across different channel conditions.
• Satellite and space communication—DVB-S2/DVB-S2X (Digital Video Broadcasting—satellite/second-generation extension.) Universal decoders adapt to unknown channel conditions and burst error-prone environments like cosmic radiation. They also reduce power consumption by working with different coding schemes [8].
• Optical fiber communication—GPON (Gigabit Passive Optical Network), OTN (Optical Transport Network). Universal decoding techniques are used for the mitigation of transmission impairments like chromatic and polarization mode dispersion.
• Cellular IoT—MQTT (message queuing telemetry transport), Industrial IoT (IIoT) protocols (e.g., PROFIBUS, OPC-UA). Universal decoders find their application for data reliability and robustness.

Figure 1 classifies the universal decoding techniques for channel codes into five main categories, which are discussed in this article.

1.1. Impediments in Channel Coding for URLLC

One of the main characteristics of 5G NR is the use of two new error-correcting codes for control and data channels of the enhanced mobile broadband (eMBB) use case. Low-density parity-check (LDPC) codes are adopted for data channels replacing Turbo codes, whereas polar codes are adopted for control channels [9]. For URLLC applications, the channel coding requirement for low latency is short-block-length codes, while a low error rate is necessary for ultra-reliability. Although LDPC, Turbo, and polar codes perform near channel capacity at large block lengths, their performance degrades at small block lengths [10]. Further, in current over-the-air communication, data are transmitted in packets, and the encoding/decoding algorithms of contemporary codes are very time consuming. Also, most of the state-of-the-art decoders like successive cancellation list (SCL) can decode cyclic redundancy check assisted polar codes (CA-Polar) with near maximum likelihood (ML) performance, yet large list sizes result in strong data dependency and suffer from a huge decoding delay [11]. This contradicts the low-latency requirement in URLLC, thus necessitating the application of universal decoding techniques.

1.2. Related Work

Channel coding for 6G networks is one of the most explored research domains. Review articles on advanced channel coding schemes emphasize decoder performance analysis as a critical area for future research [12,13]. In general, the literature on error-correcting codes for short block lengths focuses on code comparison, but lacks an explicit comparison of decoders. Wonterghem et al., in [14], have carried out a performance analysis of LDPC, Reed Muller, polar, and other short-length codes with ordered statistics decoding (OSD) for 5G NR. They have proposed OSD as a potential candidate universal decoder for 5G use cases. One significant development in code-agnostic decoding is guessing random additive noise decoding (GRAND), which recovers the original message by estimating the noise that distorted the transmitted data [15]. This method can be applied to different linear block codes, making it feasible for 6G applications. We extend our contribution in the direction of obtaining a desirable network performance satisfying the strict requirements of latency and reliability for 6G URLLC use cases by providing a comparative evaluation of existing works.

Table 1. Contemporary work.

Year	Paper [Ref]	Contribution	Limitation
2023	Yue et al. [16]	Evaluation and comparison of many potential URLLC decoding methods	Variants of GRAND and other decoding techniques not considered
2024	Mohammad et al. [17]	Comprehensive review of primarily LDPC, Turbo, and polar coding schemes	Decoding algorithms scalable for 6G not covered
2023	Geiselhart et al. [18]	Highlights fundamental coding schemes that have been commercialized	No comparative analysis of code-agnostic decoding techniques
2023	H. Zhang and W. Tong [19]	Review on key performance indicators for 6G and trade-offs	Discussed KPIs, compared only for LDPC and polar codes
2024	Miao et al. [20]	Review of channel coding techniques from implementational perspective in 6G	Brief discussion on universal decoding techniques

lists recent review articles on code-agnostic decoding techniques in the 6G scenario. Taking into account the limitations of these articles, we discuss and present a comparative analysis of recent universal decoding techniques scalable toward the 6G scenario.

1.3. Contribution and Organization

In this work, we analyze, compare, and evaluate the performance of universal decoding techniques for different key performance indicators (KPIs). The results are inferred from the study of research articles on the decoding techniques. The potential contribution of our work lies in the analysis, summary, and interpretation of the results of a performance analysis of key decoder metrics. The rest of the contributions of this work are as follows:

• We discuss and present the algorithms of automorphism ensemble, guessing random additive noise, ordered statistics, bit flipping, and belief propagation decoding techniques, regarding their effective understanding and implementation.
• We summarize and analyze the key performance metrics of AED, GRAND, OSD, BFD, BPD, and their variants for their scalability toward the URLLC communication scenario of 6G.
• We provide a comparative analysis of the selected decoding techniques with the CA-SCL decoder of polar code as the benchmark, and we performed simulations in MATLAB for the error-correction performance of AED, GRAND, and OSD decoders on random linear code of dimension (128, 103).
• We discuss and list machine learning (ML) techniques that are implemented to enhance and optimize channel coding in 5G/6G networks.

The structure of the paper includes six sections. Section 2 discusses the communication system model for universal decoding and includes a table of symbols and abbreviations. Section 3 describes the universal decoding techniques along with their algorithms. Section 4 carries out the comparative and performance analysis, including a summary table and a comparison table. Section 5 discusses the application of AI/ML models for universal decoding in context to 6G URLLC. Section 6 concludes the paper.

Table 2. List of symbols and abbreviations.

Symbols and Abbreviations	Meaning	Application
$\pi$, ${\pi }^{-1}$	Permutation, inverse permutation	Automorphism ensemble decoding
p	Permutation	Ordered statistics decoding
u	Codeword	Encoder input
$\widehat{x}$	Estimated codeword	Decoder output
e, s	Error vector, syndrome	Checking validity of a codeword
T	Transpose of a vector	Parity-check equation
z	Noise sequence	GRAND
$\alpha$	Confidence value	Soft decoding
ms, $\mathsf{\mu }$s	Millisecond, microsecond	Measurement of latency
$E_b/N_0$	Ratio of energy per bit to noise power spectral density	Normalized SNR measure
C	Codebook/code	Set of valid codewords
$G,H$	Generator matrix, parity-check matrix	Encoding, decoding of codeword
eMBB	Enhanced mobile broadband	5G new radio
mMTC	Massive machine type communication	5G use case
SNR	Signal-to-noise ratio	Measure of channel quality
BER, BLER, FER, WER	Bit error rate, frame error rate, block error rate, word error rate	Reliability metrics
AWGN	Additive white Gaussian noise	Simulation channel
BPSK	Binary phase shift keying	Modulation technique for simulation
N, K, R	Code block length, information block length, code rate	Code parameters

lists the symbols and abbreviations used in text and algorithms.

2. System Model

This section presents an overview of the system setup at the physical layer for the reliable transmission of data across a wireless channel prone to noise effects. Figure 2 describes the system model for channel encoding at the transmitter and universal decoding at the receiver. The input to the encoder is a vector u of K number of message bits. After encoding, redundant bits get added to generate vector x of N bits. Under a BPSK modulation scheme, the modulator maps the encoded vector x onto the symbol vector s = {+1, −1}. In a point-to-point communication link for an AWGN channel, the received vector is obtained as y = s + n, where n is the Gaussian noise vector. Universal decoding is applied to the demodulated vector y. Universal decoders like GRAND are designed to decode any linear block code by systematically guessing the noise patterns instead of estimating every codeword bit, whereas OSD starts each decoding operation by permuting received symbols in a declining order of reliability. Ensemble decoding (AED) involves running several identical low-complexity decoders in parallel.

Short Block Length Codes

The design of channel coding schemes for URLLC presents a challenging trade-off between latency and reliability, due to the requirement of short-block-length codes for message transmission. Short packets, used to reduce latency, cause a significant loss in coding gain of the error-correction scheme employed. The normal approximation is used in practice to identify the limits of information for a code of block length N, and is considered the performance benchmark for comparison in a finite block-length regime [21]. For an AWGN channel, the maximal coding rate for finite-codeword-length transmission is given as

$$R=C-\sqrt{{\displaystyle \frac{V}{N}}}{Q}^{-1}\left(ϵ\right)+{\displaystyle \frac{1}{2N}}{log}_2\left(N\right)$$

(1)

where R is the channel coding rate, C is the channel capacity, V is the channel dispersion coefficient, and $ϵ$ is the average error probability [22]. The approximation (1) implies that to achieve a given error probability $ϵ$ for codes with finite block lengths, the rate loss is proportional to $1/\sqrt{N}$. For instance, achieving $ϵ<{10}^{-9}$ is extremely challenging for block lengths $N<50$ as the value of $Q^{-1}\left({10}^{-9}\right)$ is ≈5.998, which significantly increases the rate loss.

3. Candidate Universal Decoders for URLLC

A number of channel codes offering a certain reliability order have been developed for short block lengths, ranging in computational complexity. To clarify, while a code might be appropriate for a particular application, it might result in significant performance degradation in other cases. For code-specific decoders, every code has a unique decoder that is incompatible with other codes. For instance, a low-complexity BP (belief propagation) decoder is advantageous for LDPC or PBRL-LDPC codes, but it is not optimal for polar codes. Similarly, the decoding algorithm of successive cancellation (SC) is generic to polar codes only [23]. Consequently, it becomes reasonable to choose a universal (code-agnostic) channel decoder rather than a code-specific one to achieve maximum likelihood (ML) performance. This section discusses the universal decoding algorithms and also AED and bit-flip decoders in the context of a URLLC scenario.

3.1. Automorphism Ensemble Decoding (AED)

An automorphism decoder is a decoder that operates on a signal that has been received and permuted in accordance with a code automorphism. The original codeword is then recovered by rearranging the result estimation [24]. Codewords are permuted using automorphisms. A permutation of N elements that maps each codeword $x\in$ C onto another codeword $\pi \left(x\right)\in C$ is called an automorphism of a code C. Different permutations are applied to the channel output Log-likelihood ratio (LLR) vector in an AE decoder. To recover the initial codeword estimate, an automorphism decoder scrambles a signal that has been received and applies an automorphism. The outcome is then scrambled again. AED executes many low-complexity decodings, like successive cancellation (SC) in parallel, and can be applied to any code that has automorphisms. Geiselhart et al. [25] proposed AED as a universal decoding technique to enhance error rate performance through the execution of an ensemble of M-independent SC decoders with high implementation efficiency. Each decoder j operates on a permuted version ${\pi }_j\left(y\right)$ of the received sequence, and each decoding result is then unpermuted (applying ${\pi }_j^{-1}$ on decoder) as

$${\widehat{x}}_j={\pi }_j^{-1}(SC\left({\pi }_j\left(y\right)\right)$$

(2)

where ${\widehat{x}}_j$ is estimated codeword output. The permuted received sequences correlate to different noise patterns that are easier to decode. Therefore, choosing the most likely option from ${\widehat{x}}_j$ increases the probability of finding the correct codeword. Algorithm 1 is developed with reference to [25].

Algorithm 1: Automorphism ensemble decoding (AED)

1  Varaiable declaration   2  y: received vector of channel LLRs   3  A: set of M automorphisms   4  ${\pi }_j$: automophism ∈A, $j\in \{1,2,\dots ,M\}$   5  Step I       Permutation (Automorphism application)   6  Input: y   7  Output:  $y_j^{\prime }$   8  ${\pi }_j$: $y\to$$y_j^{\prime }$ for $j\in \{1,2,\dots ,M\}$   9  Step II       Decoding permuted words 10  Input:  $y_j^{\prime }$ 11  Output:  ${\widehat{x_j}}^{\prime }$ 12  1. Decode $y_j^{\prime }$ using a polar decoder 13  2. Obtain ${\widehat{x_j}}^{\prime }\forall j\in \{1,2,\dots ,M\}$ 14  Step III       Inverse permutation 15  Input:  ${\widehat{x_j}}^{\prime }$ 16  Output:  $\widehat{x_j}$ 17  1. Apply ${\pi }_j^{-1}$ to ${\widehat{x_j}}^{\prime }$ 18  2. Obtain $\widehat{x_j}$ for $j\in \{1,2,\dots ,M\}$

3.2. Guessing Random Additive Noise Decoding (GRAND)

GRAND is a noise guessing decoding algorithm that initially generates the most likely test error patterns (TEPs) (e) based on channel-induced noise. The error patterns are then applied to the hard demodulated received signal (y), and checked whether the resulting string is a member of the codebook. It makes use of ordered statistics of noise for decoding, and the codebook is only used for the look-up of the potential decoded codeword [26]. In contrast to other decoding algorithms, which find errors in the received vector by utilizing the code’s structure, GRAND uses hard-decision decoding to identify the most likely error pattern e or noise sequence z. The error patterns are rearranged from most probable to least probable, and the remaining sequence in the codebook is checked to see if it is still valid. An $(N,K)$ linear block code with K information bits characterizes a codebook denoted by C. The codewords in C have length N and rate R, with $2^K=2^{NR}$. The generator matrix for the code is given as $G_{K\times N}$ and the parity-check matrix as $H_{(N-K)\times N}$. A codeword u $\in C$ if and only if

$$H{u}^T=0$$

(3)

    The encoded signal x is transmitted through a channel, and the signal is obtained as $y=x\oplus z$ at the receiver, where z is the added noise and ⊕ denotes a modulo-2 operation. For the detected word $\widehat{y}=x\oplus e$, the syndrome, s is calculated as

$$s=H{\widehat{y}}^T=H{(x\oplus e)}^T=H{e}^T$$

(4)

which is the 0 vector only if e is 0 or e is a valid codeword. GRAND identifies the error pattern based on the syndrome. Since GRAND’s decoding complexity decreases with increasing code rate and is suitable for short codewords with a high code rate, it is appropriate for low-latency communications. The Algorithm 2 of the GRAND decoder is developed based on the study of [26].

Algorithm 2: Guessing random additive noise decoding (GRAND)

1  Variable declaration   2  $y^n$: channel output of length n   3  $z^{n,i}$: ith noise sequence   4  $G\left(z^n\right)$: list of noise sequences in descending order of likelihood, i.e., $G\left(z^{n,i}\right)\le G\left(z^{n,j}\right)$ for $z^{n,j}\ge z^{n,i}$   5  H: parity-check matrix   6  $C^n$: codebook   7  Step I        Test error pattern (TEP) generation   8  Input: y, $G\left(z^n\right)$   9  Output:  $x^n$ 10  for i = 1 to n 11  set $G\left(z^n\right)$ = i as the most likely noise sequence 12  end for 13  Compute $x^n=y^n\ominus z^n$ 14  Step II         Code membership 15  Input: $x^n$, H 16  Output:  ${\widehat{x}}^n$ 17  if $H{\left(x^n\right)}^T$ = ${\left(0^n\right)}^T$ then 18  ${\widehat{x}}^n\in C^n\leftarrow {\mathrm{x}}^n$ 19      else if $x_n=y_n\ominus z_n\notin C^n$ 20          do $i=i+1$ 21           go to line 10 22       end if 23  end if

3.3. Ordered Statistics Decoding (OSD)

For linear block codes, ordered statistics decoding (OSD) is a soft-decision decoder. In successive cancellation (SC) decoding, a single data bit cannot be decoded until all of its preceding bits have been received, which increases the data dependency. That is to say, when the block length increases, the decoding latency unavoidably rises. Ordered statistics decoding (OSD) was proposed by Fossorier and Lin [27] to prevent such data dependency. OSD works by parallelizing the bit flips and using the most dependable bits to get the most likely codeword. In the first step, the received vector y is sorted in order of descending confidence values ${\alpha }_i$. In OSD, two permutations are performed over the received signal vector y and the generator matrix G before decoding. Applying the first permutation $p_1$, the columns of G and y are sorted in the decreasing order of the confidence values ${\alpha }_i=\left|y_i\right|$. On applying the permutation $p_1$ to G, $G^{\prime }$ is obtained. Now, Gaussian elimination is applied to $G^{\prime }$ to get the augmented systematic matrix $G=\left[I_k\right|P]$, in which $I_k$ is a k-dimensional identity matrix and P is the parity sub-matrix. Using the permuted generator matrix, the channel output’s k most reliable values are converted into a codeword. The Algorithm 3 is developed based on reference [27].

Algorithm 3: Ordered statistics decoding (OSD)

1  Variable declaration   2  $\mathbf{y}$: channel output vector   3  G: $K\times N$ generator matrix   4  $p:$ permutation function   5  Step I        Reordering   6  Input:  $\mathbf{y}$   7  Output:  $\mathbf{r}$   8  Sort $\mathbf{y}$ in descending order of reliability (absolute LLRs)   9  Obtain the resultant sequence $\mathbf{r}$ 10  Generate permutation $p_1$ 11  Step II         Matrix operation 12  Input:  G 13  Output:  $\tilde{G}$ 14  Permute columns of G as $p_{1}G$ = $G^{\prime }$ having first K columns as most reliable first K columns of G 15  Construct $G^{″}$ by permuting $G^{\prime }$ as $G^{″}$ = $p_2\left(p_1\left(G\right)\right)$ 16  Obtain systematic form of $G^{″}$ as $\tilde{G}$ 17  Step III        MRI sequence 18  Input: r, $\tilde{G}$ 19  Output:  $\widehat{c}$ 20  Generate most reliable independent (MRI) sequence $z^K=p_2\left(p_1\left(\mathbf{r}\right)\right)$ 21  Perform hard decisions on the K MRI symbols of $z^K$ to get $a^K$ 22  Obtain $\widehat{c}=p_1^{-1}\left(p_2^{-1}\left(a^K\tilde{G}\right)\right)$

3.4. Bit-Flip Decoding (BFD)

Bit flipping is another approach to improve the reliability of a decoder, where the block error can be corrected by flipping unreliable bits. Authors in [28] introduced the SC flip (SCF) decoder, which corrects the first wrong estimate of an unfrozen bit and shows that flipping the first incorrect hard choice can significantly improve decoding performance. Using additional SC decoding attempts to execute flip operations in error-prone positions is the fundamental concept behind the flip decoder, which enables it to outperform the SC decoder in error-correction performance. In [29], the bit-flip technique has been introduced into the SCL decoding process to further improve the error-correction performance of the CA-SCL decoder. Authors in [30] have introduced a diversity flip decoder (DFD) as a universal decoding algorithm for preserving diversity order. The Algorithm 4 is given for SC bit flipping as proposed in [28].

Algorithm 4: Bit-Flip decoding (BFD)

1  Variable declaration   2  $y^N$: channel output of length N   3  A: information set   4  T: additional number of SC decoding attempts   5  $\mathit{U}$: set of least reliable values of T   6  Step I        SC-decoding   7  Input:  $y^N$   8  Output:  ${\widehat{u_1}}^N$   9  Perform SC decoding on $y^N$ to obtain ${\widehat{u_1}}^N$ 10  Step II        Bit flipping 11  Input:  ${\widehat{u_1}}^N$ 12  Output:  ${\widehat{u}}^N$ 13  Perform CRC check on ${\widehat{u_1}}^N$ 14  if CRC(${\widehat{u_1}}^N$) = success 15         Return ${\widehat{u_1}}^N$ 16     else 17       Perform SC decoding for T number of attempts 18       for k$\leftarrow 1$ to T ∈$\mathit{U}$ 19         flip $u_k$ 20          if CRC(${\widehat{u}}_k$) = success then 21           break; 22          end if 23        end for 24      end if 25  Return ${\widehat{u}}^N$

3.5. Belief Propagation Decoding (BPD)

The BP algorithm was first developed by R.G. Gallager, and, later, R.M. Tanner proposed a Tanner graph representation of it. As opposed to decoders based on successive cancellation, belief propagation (BP) decoding has low latency and high parallelism, making it a suitable choice for hardware implementation. BP decoding is based on message transmission between check nodes and variable nodes positioned on the right and left edges of the factor graph. A factor graph is defined as a bipartite graph with a vertex set $V\cup F$, where V represents the variable nodes and F is the set of factors. Polar code decoding using belief propagation (BP) offers the benefits of high parallelism and low latency, which makes it particularly scalable for URLLC applications. The given Algorithm 5 is developed based on the article [31].

Algorithm 5: Belief propagation decoding (BPD)

1  Variable declaration   2  y: received signal vector   3  $(i,j)$: edge between variable node i and check node j   4  $L_i^{\left(t\right)}$: LLR value of variable node i at t-th iteration   5  $N\left(i\right)$: set of check nodes connected to the variable node i   6  $N\left(j\right)$: set of variable nodes connected to the check node j   7  $M_{ij}$: message from each variable node i to check node j   8  $M_{ji}^{\left(t\right)}$: message from check node j to variable node i at t-th iteration   9  $M_{kj}$: message from each variable node k to check node i 10  Step I Initialization based on channel observations 11  Input:  y 12  Compute $L_i$ for each variable node 13  Each variable node sends $L_i$ to connected check nodes 14  Step II         Iterative message passing 15  Compute $M_{ij}$ = $L_i$ + $\mathrm{\Sigma }$ $M_{ki}$, $k\in N\left(i\right)$ excluding j 16  Compute $M_{ji}^t$ = 2 $tan{h}^{-1}$($\Pi tanh(M_{lj}^{t-1}$/2)), $l\in N\left(j\right)$ over all variable nodes excluding variable node i 17  Step III          Message update 18  Compute updated belief $L_i^{\prime }$ = $L_i$ + $\mathrm{\Sigma }{M}_{ji}$, $j\in N\left(i\right)$ 19  step IV           Hard decision 20  Input:  $L_i^{\prime }$ 21  Output:  ${\widehat{u}}_i$ 22  Compute ${\widehat{u}}_i$ = $\left\{\begin{array}{c}0,if{L}_{i^{\prime }}>0\hfill \\ 1,if{L}_{i^{\prime }}<0\hfill \end{array}\right.$

4. Performance Evaluation and Result Analysis

Sixth-generation networks are expected to meet high power efficiency, reliability, and throughput standards. Polar codes can outperform state-of-the-art low-density parity-check (LDPC) and Turbo codes when successive cancellation list (SCL) decoding is combined with cyclic redundancy check (CRC) code for error correction. However, the decoding complexity of the SC-List decoder is very high for large list sizes, making it less suitable for low-latency applications of URLLC. Also, URLLC requires precision up to the single-bit level in block lengths and coding rates to support scenarios with varying latency and capacity restrictions. Universal decoders can provide bit-level granularity for rate-matching schemes by decoding binary linear codes. The selection of the optimal decoding algorithm that achieves the latency–reliability trade-off efficiently would help in the design of adaptive error-correction schemes. In this section, we compare the performance of decoders listed in the summary in

Table 3. Summary of performance metrics for universal decoders.

Ref	Decoder	Parameters	Performance Indicators
			BLER/FER/BER	Latency	Throughput	Energy Efficiency	Computational Complexity
[29]	Successive cancellation list flip (SCL-flip) decoder	Polar code (PC) length N = 256, 1024, code rate R = 0.5, standard bit-flip metric T = 50 and L = 8	FER $\approx {10}^{-4}$ at 2.7 dB	Average extra attempts ≈ 7 at SNR 2 dB for T = 10, N = 256	NA	NA	Average normalized complexity $\approx 8$ at SNR 2 dB for L = 8, T = 10, N = 1024
[30]	Diversity flip decoder (DFD)	Rayleigh fading channel with unit variance, BPSK modulation, parity check, polar code (128, 113)	BER $\approx {10}^{-4}$ at 26 dB	NA	NA	NA	Worst-case complexity of the order of ${10}^1$
[32]	Automorphism ensemble decoding (AED)	Polar code (128, 60), AED ensemble M = 16 parallel SC decoders	FER $\approx {10}^{-4}$ at 3.6 dB	22.2 ns for AED-16	63.3 Gbps	AED-16 10.16 pJ/bit	NA
[33]	Automorphism ensemble decoder with SCL (AE-SCL), P = 8, L = 4	5G NR polar codes (512, 251), code rate = 0.5	BLER $\approx {10}^{-4}$ at 3.4 dB	AE-SCL, L = 4 comparable to SCL decoder, L = 16	NA	NA	NA
[34]	Successive cancellation automorphism list (SCAL)	AWGN, BPSK, PC (128, 60), SCAL = 16	≈${10}^{-6}$	64.0 ns for SCAL 8	64 Gbps	8.62 pJ/bit for SCAL 8	NA
[35]	Fast logarithmic successive cancellation stack (Fast Log-SCS)	Rayleigh fading, QPSK, polar code N = 1024, R = 1/2, stack size S = 128	≈${10}^{-3}$ at SNR 2dB	20 $\mathsf{\mu }$s for N = 1024	416.20 Mbps	Estimated value 0.24 pJ/bit, taking power consumption ≈ 0.1 W in Equation (8)	≈${10}^4$ function evaluations at BLER ${10}^{-3}$
[36]	Adaptive ordered statistics decoding (OSD)	BPSK, AWGN, PC (N = 64, R = 0.5)	BLER $\approx {10}^{-4}$ at 3 dB	≈2 clock cycles at SNR 2dB for N = 128, ≈20 ns	NA	NA	NA
[37]	Probability-based ordered statistics decoding (PB-OSD)	(128, 64, 22) eBCH code	FER $\approx {10}^{-4}$	Decoding time ≈ 1 ms at SNR 2 dB for PB-OSD (order 3)	NA	NA	Average 5808 TEPs at SNR 2dB
[38,39]	Guessing random additive noise with abandonment decoding (GRANDAB)	BPSK, AWGN, CA-Polar (128,105), AB bit-flips = 4	BLER $\approx {10}^{-4}$ at 9.8 dB	4098 clock cycles for Max Freq = 500 MHz, ≈8196 ns	9 Gbps for AB = 3	Estimated value 0.01 pJ/bit, taking consumed power $P_c$ ≈ 0.1 W in Equation (8)	≈10 queries per bit at SNR 9 dB
[40,41]	Ordered reliability bits guessing random additive noise decoding (ORBGRAND)	AWGN, BPSK, CA-Polar (128,106), R = 0.82, SCL -16	BLER ≈ ${10}^{-4}$	2.47 ns	42.5 Gbps for R = 0.82 CA-Polar code	Estimated value = 2.35 pJ/bit, taking consumed power $P_c$ ≈ 0.1 W in Equation (8)	≈3.5 codebook queries per bit at SNR 5.5 dB
[42]	Successive cancellation list (SCL)-GRAND	AWGN, BPSK, PC (64, 41), R = 0.65	BLER $\approx {10}^{-4}$ at SNR 5 dB	NA	NA	NA	Average number of queries ≈ 100 at SNR 4 dB
[43]	Fading-GRAND	Rayleigh fading channel, maximum Hamming weight = 4, RLC (128, 104), BCH code (127, 106), (127, 113), CRC (128, 104)	FER ${10}^{-7}$	NA	NA	NA	$1.5$ queries at $E_b/N_0$ = 26 dB for target FER of ${10}^{-7}$
[44]	Adaptive list flip (ALF) decoder	AWGN, BPSK, Flip metric = 5, PC (512, 384 + 24)	FER $\approx {10}^{-4}$	NA	NA	NA	Average complexity 1.292 at SNR 2.5 dB, T = 15 decoding attempts
[45]	Improved belief propagation list (IBPL) decoder	Polar code length = 128 bits, R = 1/2	BLER $\approx {10}^{-4}$ at 3.8 dB	NA	NA	NA

.

4.1. Performance Indicators

The International Telecommunication Union—Radiocommunication Sector (ITU-R) has established several key performance indicators (KPIs) that are linked to 5G standards [46]. For coding schemes, decoders determine a communication system’s compliance with ITU-R KPIs in mMTC, eMBB, and URLLC use cases. Processing latency, throughput, error correction, flexibility, area efficiency, and energy efficiency are the key decoding KPIs, which are discussed as follows:

• Error correction—The reliability of a communication system at the block level is determined by the BLER. The ratio of blocks received in error to the total blocks transmitted over a specified number of frames is known as the block error rate (BLER), which is a physical-layer error estimation technique. For mission-critical 6G URLLC applications, only one block out of every ${10}^9$ should experience significant transmission faults that the channel decoder cannot fix, i.e., reliability on the order of ${10}^{-9}$ [47].
• Throughput—A crucial performance indicator of the decoder is the throughput, which is the number of bits that are successfully decoded per second. Since the channel decoder processes all received data, a high level of parallel processing is necessary to achieve a high throughput. The throughput of a fully parallel decoder where each bit is decoded in one clock cycle is given as

  $$T=f\times N$$

  (5)

  where f is the clock frequency and N is the block length. For an iterative decoder, the effective throughput is given as

  $$T={\displaystyle \frac{f\times P}{I}}$$

  (6)

  where P is the number of bits per cycle and I is the number of iterations [48]. Throughput is expressed in bits per second (bps).
• Latency—The URLLC applications of 6G have the most stringent latency requirements. Moreover, other physical-layer processes, including channel estimation and demodulation, take place in this latency limit with channel decoding. The time needed to process a received codeword and produce decoded bits of information is known as the decoder latency. A decoder’s latency (L) in relation to throughput (T) and block length (N) can be expressed as

  $$L=N/T$$

  (7)

  where N is the block length in bits and T is the throughput in bits/second (bps); therefore, latency is expressed in seconds.
• Computational complexity—URLLC requires low-complexity decoders in addition to efficient encoders. A typical way to quantify the computational complexity of decoding methods is to look at how many operations, such as comparisons, additions, multiplications, or bit manipulations, are needed in relation to the block length and code parameters. Computational complexity is generally expressed as the number of operations per bit or the number of operations per codeword [49].
• Energy efficiency—It is estimated that the information and communications technology (ICT) sector is responsible for 5% of global carbon emissions. Energy consumption in the 5G domain, measured in bits/joule, has been seen as a crucial design parameter due to global energy-efficiency initiatives. The energy efficiency (EE) of a decoder in a communication system is expressed as the ratio of data transmission rate (throughput) to consumed power [50], and is expressed as

  $$EE=T/P_c$$

  (8)

  where T is the throughput and $P_c$ is the consumed circuit power. The unit of EE is bits/joule.
• Area efficiency—The area efficiency (AE) of a decoder is expressed as

  $$AE=T/A$$

  (9)

  where T is the information throughput and A is the chip area. The unit of $AE$ is (Mbps/mm²) or (Gbps/mm²) [48].

4.2. Summary Tables

Table 3 lists the key performance metrics, such as the error probability, latency, throughput, energy efficiency, and computational complexity, of the referenced decoding techniques. The attainable performance of the decoding schemes is with reference to the coding scheme given under the parameters column of Table 3.

Table 4. Summary of comparative analysis.

Decoder Parameter [Ref]	Benchmark (SCL Decoder)	Complexity/Area Efficiency (AE)	Latency	Energy Efficiency (EE)	Coding Gain (Over SCL Decoding)
SCL-flip, T = number of bit-flip attempts [29]	CA-SCL decoding of polar code of length N = 1024 and rate R = 0.5	At SNR 1.5 dB, T = 50 same as the SCL-16	NA	NA	≈0.2 dB gain at FER ${10}^{-4}$ over CA-SCL with L = 32, T = 50 for SCL-flip
AED-16 [32]	SCL decoding of 5G NR polar code (128, 60 + 11) for CRC = 11, L = 16	AE 8.9 times better than SCL-16	5.3 times less than SCL-16	4.5 times better than SCL-16	0.2 dB coding gain at FER ${10}^{-4}$, similar performance at higher SNR
AE-SCL P= 8, L = 4 [33]	SCL decoding of (512, 251) polar code, L = 32	NA	2.68 times less than SCL-32	NA	0.47 dB gain over SCL at WER ${10}^{-3}$
SCAL-8 [34]	Polar code (128, 60), $I_{min}$ = 27, under SCL-8 decoding	168.2 Gbps/mm2	2 ns more than SCAL 8	8.62 pJ/bit, for SCL-8	Coding gain of 0.024 dB over SCL-16 at FER ${10}^{-5}$
Fast Log-SCS [35]	NR PUCCH polar code N = 512, 1024, Log-SCL decoder, list = 11	Reduction in complexity in comparison to Log-SCL (L = 32)	Normal latency 21% lower than CA-SSCL decoder	NA	0.2 dB coding gain over fixed-point fast SSCL
Adaptive OSD [36]	CA-SCL, list = 8 for block lengths N = 64, R = 0.5	NA	Close to 1 clock cycle at SNR 4.6 dB	NA	≈0.4 dB coding gain over CA-SCL
ORB GRAND [40]	CA-SCL (list size L = 16) CA-Polar (128, 106), R = 0.82	${10}^2$ queries for BLER ${10}^{-3}$	2.47 ns	More energy efficient at higher SNR as 1 query/bit at 5.5 dB	0.5 dB gain over CA-SCL
SCL-GRAND [42]	Polar code (64, 43) CA-SCL (L = 12)	≈${10}^2$ queries /time	NA	NA	for R = 0.68, 0.7∼0.2 dB gain
IBPL [45]	SCL decoder for polar code length N = 128 bits, R = 1/2	NA	NA	NA	at BLER ${10}^{-3}$ SCL has a coding gain ≈$0.5$ dB

is the comparative analysis of the selected decoding schemes with the benchmark SCL decoding of 5G NR polar code with given parameters. Column six of Table 4 evaluates the coding gain of the decoders over the state-of-the-art SCL decoder. The abbreviation NA is used in tables for data that are not available in the referenced articles.

4.3. Graphical Analysis

This sub-section is the graphical analysis of the results from summary Table 3 and Table 4. It includes Figure 3, Figure 4, Figure 5 and Figure 6 for performance evaluation. With reference to Table 3, Figure 3 compares the required $E_b/N_0$ (SNR) values of different decoders to attain a BLER of ${10}^{-4}$ for the given values of the code lengths and code rates of the coding scheme. Analyzing the data from Table 4, Figure 4 gives the coding gain of different decoders in comparison with CA-SCL decoding of 5G NR polar code. Figure 5 gives the BLER performance of (128, 103) random linear code (RLC) under GRAND, AED, and OSD. It is evident that as the SNR increases, the error-correction performance of the decoder increases. For a target BLER of ${10}^{-4}$, GRAND has superior performance. Comparative analysis is done by providing MATLAB plots for Figure 3 and Figure 4. For Figure 5, Monte Carlo simulation is done for respective decoding techniques. For short codewords, the SNR requirement of URLLC applications is in the range of 10–20 dB.

• For URLLC applications, short-block-length codes, which are preferred for low latency, generally have a lower reliability, often requiring a higher SNR to achieve the same error-correction performance as compared to long-block-length codes. Thus, in a finite-block-length regime, there exists a trade-off between the required SNR and code block length to achieve a target BER/BLER. The scatter plot in Figure 3 compares the SNR required to achieve a target BLER of ${10}^{-4}$ for the decoding schemes. The data are collected from the performance analysis study of the decoders in the given reference articles, as listed in Table 3. The parameters of the coding scheme for which the decoding scheme attains the specific error rate are given in column three. The X-axis of the scatter plot has code block lengths ranging from 50 to 500. The Y-axis of the scatter plot gives the SNR range. Since the values of the SNR for achieving the target BLER have a wide range, the limits are set at 0–30 (dB). A lower SNR value for a short codewords is preferable for 6G URLLC and mMTC applications. By analyzing the plot, we can see that SCL-GRAND, adaptive-ordered statistics decoding (A-OSD), GRANDAB, and improved belief propagation list (IBPL) decoding are particularly scalable for URLLC use cases as the target BLER is achieved for shorter code lengths at lower SNR values for these decoding schemes.
• Coding gain is the reduction in signal-to-noise ratio (SNR) required to achieve a specific bit error rate (BER) or block error rate (BLER) when one coding scheme is compared with another. In other words, coding gain gives a measure of the energy efficiency of one coding scheme over another. 3GPP and IEEE standards consider coding gain a key performance metric to select candidate codes. The bar graph of Figure 4 gives the coding gain of the referenced decoding schemes listed in Table 4, compared to the standardized SCL decoder for CA-Polar codes with specified parameters mentioned in column two of the table. The data are collected and analyzed from the error rate performance plots of the compared schemes, as provided in the referenced articles. The coding gain values are provided at different error rate thresholds and, hence, are an approximate evaluation. The X-axis represents the decoders, and the Y-axis represents the coding gain (in dB) that each decoder on the X-axis achieves over the reference decoding scheme mentioned in the legend. The plotted range reflects the values in our analysis, and the Y-axis scale is adjusted accordingly. It is observed that GRAND variants achieve a higher coding gain in comparison with other decoders over the SCL decoder for polar code. A larger value of coding gain implies that the decoding scheme requires a lower SNR to achieve the target error rate as compared to the benchmarked scheme.
• The channel conditions and noise interference level are closely reflected by the BLER. A lower BLER indicates a cleaner radio channel or better SNR for a particular modulation scheme, which reduces the likelihood of the transport block being received in error. Random linear codes enable efficient error correction and offer flexibility for maintaining the low-latency requirements of URLLC applications in 6G [51]. The plot in Figure 5 highlights the error rate performance of AED, OSD, and GRAND decoders for random linear code with dimensions (128, 103) and (128, 64).

  Table 5. Simulation parameters.

  Parameter	Specification
  Channel, modulation	AWGN, BPSK
  Coding scheme	Random linear code
  Block length (N), information lengths (K)	128, 103, 64
  Code rates (R)	0.8047, 0.5
  Decoding algorithms	GRAND, AED, and OSD
  SNR range	0 dB to 12 dB
  Simulation method, number of iterations	Monte Carlo, ${10}^5$

  lists the parameters used for simulations. We evaluate the performance of the decoders for random linear code (128, 103) and (128,64), where N = 128 is the block length and K = 103, 64 are the information bits. For each decoding scheme, simulations are carried out in MATLAB 2024 for BPSK modulation and the AWGN channel. AWGN follows the normal distribution with zero mean and variance ${\sigma }^2$. Noise variance is expressed as

  $${\sigma }^2={\displaystyle \frac{1}{2R(E_b/N_o)}}={\displaystyle \frac{1}{SNR}}$$

  (10)

  where $R=K/N$ is the code rate. For simulation, the number of iterations per $E_b/N_o$ value is ${10}^5$. As shown in Figure 5, a BLER of the order of ${10}^{-5}$ is achieved at SNR ≈ 10 dB for the GRAND decoder for RLC with the code rate R ≈ 0.8. For the OSD decoder, the performance for both the code rates is similar, and a BLER of the order of ${10}^{-4}$ is attained at ≈11 dB. The AED decoder for code rate R = 0.5 achieves the BLER of ${10}^{-4}$ at an SNR ≈ 11.8 dB. The plot verifies better error correction by the GRAND decoder for a code with a high code rate, whereas the performance of the AED decoder degrades as the code rate increases.
• Figure 6 is the three-dimensional scatter plot characterizing the relationship between energy efficiency, latency, and throughput of the given decoders. On the X-axis, latency in nanoseconds (ns), on the Y-axis, throughput in giga bits per second (Gbps), and on the Z-axis, energy efficiency in pico joules per bit (pJ/bit) are marked, respectively. The color bar maps color to numeric values and indicates the variations in EE values. The performance metrics for AED, SCAL, GRANDAB, and Fast Log-SCS decoders are taken from Table 3. The value for energy efficiency is calculated using Equation (8) and given as an estimated value. For the given throughput value of the decoder, consumed power $P_c$ is taken to be ≈0.1 Watt and the unit for energy efficiency is taken as pJ/bit. From the plot, it is evident that the AED decoder shows better energy efficiency than other decoders, while Fast Log-SCS is the least energy efficient.

4.4. Key Observations in Trade-Offs

Energy efficiency vs. BLER—GRAND and its variants have better error correction for high-rate short-block-length codes but are less energy efficient.

Energy efficiency vs. coding gain—Automorphism ensemble decoders have higher energy efficieny but coding gain is less compared to GRAND decoders.

Throughput vs. latency—Successive cancellation decoders have higher throughput but lower latency compared to other decoders.

5. Machine Learning Application for Conventional and Universal Decoding Schemes

The next phase of fith-generation development, fifth-generation-advanced (5G-A), was unveiled by the Third Generation Partnership Project (3GPP) during Release 18, which aims to further enhance 5G performance by enabling AI-powered 5G [52]. Optimization of network energy efficiency, coverage area, mobility support, multiple-input multiple-output (MIMO), and broadcast services, can lead to further enhancement in 5G performance. Taking into account these parameters, Release 18 leverages machine learning (ML) and artificial intelligence (AI) technologies to effectively manage the ever-complex 5G networks. In order to attain optimal end-to-end performance in communication networks, machine learning methods are being developed nowadays. Although ML is categorized under AI, it serves as an extension of AI by incorporating probabilistic decision-making with deterministic decision-making performed by AI. The application of ML algorithms makes conventional decoding techniques more adaptable across different codes. Authors in [53] have categorized ML models into supervised, unsupervised, and reinforcement learning. They have discussed in detail how each ML technique can provide potential solutions to the 5G/B5G network requirements of each service class. The following machine learning models are being incorporated to support decoding for next-generation forward error-correction (FEC) techniques:

• Reinforcement learning (RL)—The RL-based decoder can dynamically adopt decoding strategies according to changing channel conditions.
• Deep learning (DL)—DL-based models excel in decoding as it is regarded as a classification problem. Deep neural network (DNN)-based adaptive decoders reduce computational complexity.
• Graph neural network (GNN)—Approximate decoding with fewer iterations, achieving ultra-low latency [54].
• Autoencoding neural network (ANN)—An unsupervised learning approach traditionally used for data reconstruction is used for signal processing and decoding of LDPC codes [55].

Reinforcement learning (RL) techniques have recently been studied to decode linear block codes in addition to the primary research technique that concentrates on employing supervised learning algorithms. A comprehensive survey of DL applications for channel coding problems is done in [56], where the authors have discussed state-of-the-art DL models for application in communication systems. They also have presented deep learning methods for model-free decoders, which employ neural networks for code-agnostic algorithms. It is demonstrated that model-free decoders are particularly suitable for low-latency applications. Further, a DL-based adaptive decoder is proposed by Wang et al. in [57], where the designed decoder can decode both polar and LDPC codes using the same decoding algorithm. Cammerer et al. [58] have proposed a graph neural network (GNN)-based universal decoder for binary linear codes based on bipartite graphs. Google introduced the federated learning (FL) concept, which avoids data leakage when developing machine learning models using data sets dispersed across numerous devices. AI and ML are displacing code-specific algorithms with decoders that are learning-based, flexible, and real-time. These universal decoders will be essential to 6G, allowing for reliable, low-latency, and energy-efficient communications in dynamic settings.

Table 6. Application of ML models in conventional coding techniques.

ML Algorithm Ref	Coding Technique/Decoder	Description	Advantage
Reinforcement learning [59]	BCH (63, 45) and Reed Muller (32, 16), (64, 42) Learned bit flipping (LBF) Bit-flipping decoding with Q-learning	Bit-flipping decoding mapped to Markov decision process for data-driven learning of optimal bit-flipping decision strategies	LBF improves error correction by employing RL to learn bit-flipping patterns, which results in a decreased error rate
Deep learning neural network (DNN) [60]	Polar (256, 128) Belief propagation (BP) decoding as trainable neural network (NN)	Polar code construction by application of iterative loop unrolling (deep unrolling) to replace frozen/unfrozen bit positions by trainable weights	Improved error correction and faster convergence to a stable BER through neural network training
Recurrent neural network (RNN) [61]	Short LDPC Rate 1/2 code length = 64, 128 BP-RNN decoding and OSD Post-Processing	Investigation of decoder diversity architecture	Improved decoding accuracy as different BP-RNN decoders are trained to decode errors in absorbing sets in bipartite graph
Convolutional neural network (CNN) [62]	Deep neural network decoder (NND)	Performance evaluation of NND built upon CNN and proposed saturation length for NND	Performance gain as CNNs are trained to extract features from large data set of transmitted bit sequences to decode noisy signals
Federated learning [63]	Maximum a posteriori probability (MAP) detector	Data-driven symbol detector exploring channel diversity through federated learning	More robust in scenarios with diverse channel conditions where fading is uncertain

summarizes different learning-based models for conventional coding techniques along with the corresponding applications in 5G/6G networks.

6. Conclusions

In this work, we discuss and perform a comparative analysis of code-agnostic decoding algorithms. The key performance metrics of reliability and energy efficiency for the decoding schemes are considered. The comparative analysis of the decoders is done with the CA-SCL decoder of 5G NR polar code, setting it as a benchmark. It is observed from the derived results that although GRAND and its variants provide a higher coding gain compared to the state-of-the-art SCL decoder, they require a high SNR value for performance optimization of short-length codes. Adaptive-OSD achieves the target BLER at relatively low SNR for short-block-length codes and also provides a significant coding gain over the SCL decoder for 5G NR polar codes. The energy efficiency of the successive cancellation automorphism list (SCAL) decoder is higher compared to other decoders. The performance enhancement of candidate URLLC decoders is a potential research direction; hence, a comparative analysis of the schemes provides a vision for their evaluation and effective implementation. The trade-offs between reliability, latency, and complexity for the decoders can be controlled by adjusting the decoding parameters, resulting in further improvements in their performance. A potential future direction of research work lies in balancing the performance parameter trade-offs of decoders to make them more energy efficient for a sustainable telecommunication ecosystem.