Search Results (28)

The newly developed automorphism ensemble decoder (AED) leverages the rich automorphisms of Reed–Muller (RM) codes to achieve near maximum likelihood (ML) performance at short code lengths. However, the performance gain of AED comes at the cost of high complexity, as the ensemble size required for near ML decoding grows exponentially with the code length. In this work, we address this complexity issue by focusing on the factor graph permutation group (FGPG), a subgroup of the full automorphism group of RM codes, to generate permutations for AED. We propose a uniform partitioning of FGPG based on the affine bijection permutation matrices of automorphisms, where each subgroup of FGPG exhibits permutation invariance (PI) in a Plotkin construction-based information set partitioning for RM codes. Furthermore, from the perspective of polar codes, we exploit the PI property to prove a subcode estimate convergence (SEC) phenomenon in the AED that utilizes successive cancellation (SC) or SC list (SCL) constituent decoders. Observing that strong SEC correlates with low noise levels, where the full decoding capacity of AED is often unnecessary, we perform path pruning to reduce the decoding complexity without compromising the performance. Our proposed SEC-aided path pruning allows only a subset of constituent decoders to continue decoding when the intensity of SEC exceeds a preset threshold during decoding. Numerical results demonstrate that, for the FGPG-based AED of various short RM codes, the proposed SEC-aided path pruning technique incurs negligible performance degradation, while achieving a complexity reduction of up to 67.6%. Full article

The enhanced Loran (eLoran) system, a critical terrestrial backup for the Global Satellite Navigation System (GNSS), traditionally utilizes a Reed-Solomon (RS) code for its data communication, which presents limitations in error performance, particularly due to its decoding method. This paper introduces a significant advancement by proposing the replacement of the conventional RS code with a 128-ary polar code, which is designed to maintain compatibility with the established 128-ary Pulse Position Modulation (PPM) scheme integral to eLoran’s positioning function. A Soft–Soft (SS) demodulation method, based on a correlation receiver, is developed to provide the requisite soft information for the effective Successive Cancellation List (SCL) decoding of the 128-ary polar code. Comprehensive simulations demonstrate that the proposed 128-ary polar code with SS demodulation achieves a substantial error performance improvement, yielding an approximate 9.3 dB gain at the 0.01 FER level over the RS code in eLoran data communication with EPD-MD demodulation. Additionally, the proposed scheme improves data transmission efficiency—either reducing transmission duration by 2/3 or increasing message bit number by 250% for comparable error performance—without impacting the system’s primary positioning capabilities. Full article

This paper proposes a modified polar coding-based selected mapping (PC-SLM) scheme to reduce the peak-to-average power ratio (PAPR) in orthogonal frequency-division multiplexing (OFDM) systems. In the proposed transmitter, modulated signal vector for a subset of frozen bits, termed PAPR bits, are precomputed, enabling a single polar encoder and modulator to generate multiple modulation symbols, thereby significantly reducing the hardware complexity compared to existing PC-SLM schemes. To achieve side information (SI)-free transmission, a novel belief propagation (BP)-based receiver is introduced, incorporating a G-matrix-based early termination criterion and a frozen bit check (BP-GF) for joint detection and decoding. Simulation results show that the proposed scheme significantly reduces PAPR across various code lengths, with greater gains as the number of PAPR bits increases. Furthermore, for PC-SLM schemes employing the partially frozen bit method, the BP-GF-based receiver achieves a PAPR reduction and error correction performance comparable to that of the successive cancellation (SC)-based receiver. Additionally, the BP-GF-based receiver exhibits lower decoding latency than the successive cancellation list (SCL)-based receiver. Full article

By exploiting the rich automorphisms of Reed–Muller (RM) codes, the recently developed automorphism ensemble (AE) successive cancellation (SC) decoder achieves a near-maximum-likelihood (ML) performance for short block lengths. However, the appealing performance of AE-SC decoding arises from the diversity gain that requires a list of SC decoding attempts, which results in a high decoding complexity. To address this issue, this paper proposes a novel quasi-optimal path convergence (QOPC)-aided early termination (ET) technique for AE-SC decoding. This technique detects strong convergence between the partial path metrics (PPMs) of SC constituent decoders to reliably identify the optimal decoding path at runtime. When the QOPC-based ET criterion is satisfied during the AE-SC decoding, only the identified path is allowed to proceed for a complete codeword estimate, while the remaining paths are terminated early. The numerical results demonstrated that for medium-to-high-rate RM codes in the short-length regime, the proposed QOPC-aided ET method incurred negligible performance loss when applied to fully parallel AE-SC decoding. Meanwhile, it achieved a complexity reduction that ranged from 35.9% to 47.4% at a target block error rate (BLER) of ${10}^{-3}$, where it consistently outperformed a state-of-the-art path metric threshold (PMT)-aided ET method. Additionally, under a partially parallel framework of AE-SC decoding, the proposed QOPC-aided ET method achieved a greater complexity reduction that ranged from 81.3% to 86.7% at a low BLER that approached ${10}^{-5}$ while maintaining a near-ML decoding performance. Full article

Polar codes concatenated with a cyclic redundancy check (CRC) code have been selected in the 5G standard with the successive-cancellation list (SCL) of list size L = 8 as the baseline algorithm. Despite providing great error-correction performance, a large list size increases the hardware complexity of the SCL decoder. Alternatively, flip decoding algorithms were proposed to improve the error-correction performance with a low-complexity hardware implementation. The combination of list and flip algorithms, the successive-cancellation list flip (SCLF) and dynamic SCLF (DSCLF) algorithms, provides error-correction performance close to SCL-32 with a list size L = 2 and $T_{\max }$ = 300 maximum additional trials. However, these decoders have a variable execution time, a characteristic that poses a challenge to some practical applications. In this work, we propose a restart mechanism for list–flip algorithms that allows us to skip parts of the decoding computations without affecting the error-correction performance. We show that the restart location cannot realistically be allowed to occur at any location in a codeword as it would lead to an unreasonable memory overhead under DSCLF. Hence, we propose a mechanism where the possible restart locations are limited to a set and propose various construction methods for that set. The construction methods are compared, and the tradeoffs are discussed. For a polar code of length N = 1024 and rate ¼, under DSCLF decoding with a list size L = 2 and a maximum number of trials $T_{\max }$ = 300, our proposed approach is shown to reduce the average execution time by 41.7% with four restart locations at the cost of approximately 1.5% in memory overhead. Full article

In the context of the high-speed development of 5G communications, high-performance decoding schemes for polar codes are a hot spot in channel coding research. Shift pruning successive cancellation list (SP-SCL) decoding aims to recover the correct path by shift pruning in the extra SCL decoding. However, the current SP-SCL decoding is inflexible in determining the shift positions. In this paper, a flexible shift pruning SCL (FSP-SCL) decoding is proposed. Firstly, the reasons for movement and the eliminated states of the correct path are analyzed in detail using the path metric range $\left(PMR\right)$, and on this basis, the validity of the method adopted in this paper for determining the shift priority of the information bits is verified. Secondly, the FSP-SCL decoding proposes two methods for determining the shift positions. One is the log-likelihood ratio (LLR) threshold method, which compares the LLR values of the eliminated paths on the shift bit with the corresponding LLR threshold to determine the shift positions. The other is the path distance method. It combines the minimum distance between the eliminated paths and the received vector with the path metrics to determine the shift positions. Both methods are more flexible and practical, as they can calculate the corresponding shift positions online based on a specific shift bit, avoiding the high complexity caused by the simulation method. Finally, this paper designs various experimental schemes to verify the decoding performance of the FSP-SCL. The experimental results show that in terms of error-correction performance, the LLR threshold-based FSP-SCL (FSPL (LLR threshold)) decoding, the path distance-based FSP-SCL (FSPL (path distance)) decoding and the existing SP-SCL decoding are roughly equal overall. In terms of decoding complexity, FSPL (LLR threshold) decoding is slightly better than FSPL (path distance) decoding, and the decoding complexity of both is lower than that of SP-SCL decoding, with the difference being more pronounced in the medium to high SNR regions. Full article

Relaying with network coding forms a basis for a variety of collaborative communication systems. A linear block coding framework for multi-way relaying using network codes introduced in the literature shows great promise for understanding, analyzing, and designing such systems. So far, this technique has been used with low-density parity check (LDPC) codes and belief propagation (BP) decoding. Polar codes have drawn significant interest in recent years because of their low decoding complexity and good performance. Our paper considers the use of polar codes also as network codes with differential binary phase shift keying (DBPSK), bypassing the need for channel state estimation in multi-way selective detect-and-forward (DetF) cooperative relaying. We demonstrate that polar codes are suitable for such applications. The encoding and decoding complexity of such systems for linear block codes is analyzed using maximum likelihood (ML) decoding for LDPC codes with log-BP decoding and polar codes with successive cancellation (SC) as well as successive cancellation list (SCL) decoding. We present Monte-Carlo simulation results for the performance of such a multi-way relaying system, employing polar codes with different lengths and code rates. The results demonstrate a significant performance gain compared to an uncoded scheme. The simulation results show that the error performance of such a system employing polar codes is comparable to LDPC codes with log-BP decoding, while the decoding complexity is much lower. Furthermore, we consider a hard threshold technique at user terminals for determining whether a relay transmits or not. This technique makes the system practical without increasing the complexity and can significantly reduce the degradation from intermittent relay transmissions that is associated with such a multi-way relaying protocol. Full article

Polar codes have garnered significant attention since they were proposed due to their capacity-achieving properties and their low-complexity decoding process, and they were selected as the coding scheme for the control channel in 5G mobile communication technology. A successive cancellation list (SCL) decoder, aided by a cyclic redundancy check (CRC), performs the competitive error correcting performance for polar codes compared to other sophisticated codes. The SCL-Flip decoding algorithm is proposed to enhance the error correction performance of the SCL. However, the SCL-Flip is hampered by a high decoding complexity due to the necessary selection and sorting process of path metrics (PMs). Additionally, it requires a large number of iterations to achieve a good error correction performance. In this paper, we propose the error segment bit-flip (ES-SCL-Flip) decoding algorithm, a simplified bit-flip approach for the SCL decoder based on analyzing the distributions of the first error bit. Firstly, we construct an index set with a high probability of the first erroneous bit. Second, a bit-flip criterion is introduced to prevent the costly operation of the PM selection and sorting without compromising the error correction performance. Finally, a search strategy is employed that utilizes segmented error indication to target and rectify the initial error segment sequentially. The simulation results demonstrate that the ES-SCL-Flip decoder provides an improved error correction performance with only a minimal increase in decoding complexity compared to the conventional CRC-aided SCL decoder. This algorithm attains a commendable equilibrium between the error correction performance and decoding complexity. Full article

Addressing the challenges of high decoding latency, reduced spectral efficiency, and substantial storage requirements in a Cyclic Redundancy Check (CRC)Aided Successive Cancellation List (CA-SCL) polar decoder, this paper proposes a chaotic phase modulation direct-sequence spread spectrum (CPMDSSS)-assisted adaptive serial cancellation list decoding method for underwater acoustic communication. The method involves segmenting the information bits to be transmitted, computing CRC for each segment, and mapping all CRCs to a CPMDSSS, which is then modulated onto the pilot subcarriers of underwater acoustic orthogonal frequency-division multiplexing (OFDM) to increase spectral efficiency. At the receiving end, CRCs are obtained by demodulating the CPMDSSS to verify the segmented information and adaptively select the number of decoding paths. The theoretical analysis and simulation results demonstrate that compared to CA-SCL, the proposed method effectively reduces the required storage units and improves spectral efficiency, with an average reduction of approximately 80% in the decoding paths. Sea trials further indicate that the proposed method reduces the average decoding paths by approximately 71% and decreases the average decoding delay by approximately 64% compared to CA-SCL. Full article

Over binary-input memoryless symmetric (BMS) channels, the performance of polar codes under successive cancellation list (SCL) decoding can approach maximum likelihood (ML) algorithm when the list size L is greater than or equal to $2^{MF}$, where MF, known as mixing factor of code, represents the number of information bits before the last frozen bit. Recently, Yao et al. showed the upper bound of the mixing factor of decreasing monomial codes with length $n=2^m$ and rate $R\le \frac{1}{2}$ when m is an odd number; moreover, this bound is reachable. Herein, we obtain an achievable upper bound in the case of an even number. Further, we propose a new decoding hard-decision rule beyond the last frozen bit of polar codes under BMS channels. Full article

Polar codes, as the coding scheme for the control channel in fifth-generation mobile communication technology (5G), have attracted widespread attention since their proposal. As a mainstream decoding algorithm for polar codes, the successive cancellation list (SCL) decoder usually improves the error correction performance by increasing the list size, but this method suffers from the problems of high decoding complexity. To address this problem, this paper proposes a layered-search bit-flipping (LS-SCLF) decoding algorithm based on SCL decoding. Firstly, a new flip-bit metric is proposed, which derives a formula to approximate the probability of an error occurring in an information bit. This formula introduces a perturbation parameter to improve the calculation accuracy. Secondly, a compromise scheme for determining the perturbation parameter is proposed. The scheme uses Monte Carlo simulation to determine an optimized parameter for the precise positioning of the first erroneous decoded bit under different decoding conditions. Finally, a layered search strategy is adopted to sequentially search the erroneous decoded bits from the low order to high order, which can correct up to multiple bits at the same time. Simulation results show that the proposed algorithm achieves improved error correction performance with a slight increase in decoding complexity compared to the generalized SCL-Flip (GSCLF) decoding algorithm. This algorithm also achieves a good balance between the error correction performance and decoding complexity. Full article

Wireless covert communication is an emerging communication technique that prevents eavesdroppers. This paper presents a novel approach to wireless covert communication based on polar codes with dirty constellation polarization. The feasibility of dirty constellation polarization is analyzed, and the impact of covert transmission power on undetectability is investigated. We polarize the covert channel and decode the covert information using a successive-cancellation decoder. Simulation results demonstrate that polar codes can significantly improve the undetectability and reliability of covert communication when compared to BCH codes. Additionally, we find that polarized covert channels achieve better undetectability and reliability when using a successive-cancellation list decoder. The proposed algorithm shows significant improvement in the undetectability and reliability of wireless covert communication with dirty constellations. This research provides a promising direction for future development in the field of covert communication, with potential applications in secure communication systems. Full article

CRC-aided successive cancellation list (CA-SCL) decoding is a powerful algorithm that dramatically improves the error performance of polar codes. Path selection is a major issue that affects the decoding latency of SCL decoders. Generally, path selection is implemented using a metric sorter, which causes its latency to increase as the list grows. In this paper, intelligent path selection (IPS) is proposed as an alternative to the traditional metric sorter. First, we found that in the path selection, only the most reliable paths need to be selected, and it is not necessary to completely sort all paths. Second, based on a neural network model, an intelligent path selection scheme is proposed, including a fully connected network construction, a threshold and a post-processing unit. Simulation results show that the proposed path-selection method can achieve comparable performance gain to the existing methods under SCL/CA-SCL decoding. Compared with the conventional methods, IPS has lower latency for medium and large list sizes. For the proposed hardware structure, IPS’s time complexity is $O\left(k{log}_2\left(L\right)\right)$ where k is the number of hidden layers of the network and L is the list size. Full article

Designing an efficient decoder is an effective way to improve the performance of polar codes with limited code length. List flip decoders have received attention due to their good performance trade-off between list decoders and flip decoders. In particular, the newly proposed dynamic successive cancellation list flip (D-SCLF) decoder employs a new flip metric to effectively correct high-order errors and thus enhances the performance potential of present list flip decoders. However, this flip metric introduces extra exponential and logarithmic operations, and the number of these operations rises exponentially with the increase in the order of error correction and the number of information bits, which then limits its application value. Therefore, we designed an adaptive list flip (ALF) decoder with a new heuristic simplified flip metric, which replaces these extra nonlinear operations in the original flip metric with linear operations. Simulation results show that the simplified flip metric does not reduce the performance of the D-SCLF decoder. Moreover, based on the in-depth theoretical analyses of the combination of the adaptive list and the list flip decoders, the ALF decoder adopts the adaptive list to further reduce the average complexity. Full article

Successive cancellation flip decoding requires a large number of extra successive cancellation decoding attempts at low signal-to-noise ratios (SNRs), resulting in high decoding complexity. In addition, it has a long decoding latency. Although modifications have been proposed in successive cancellation flip decoding, these still have high computational complexity at low SNRs due to a huge number of additional successive cancellation decoding attempts. It is desirable to detect the unsuccessful successive cancellation decoding process at an early stage in the additional successive cancellation flip attempts and stop it that can reduce the decoding complexity. This paper combines the parity-check-CRC concatenated polar codes with the low-latency simplified successive cancellation decoding and proposes a parity-check-CRC concatenated polar codes simplified successive cancellation flip (PC-CRC-SSCFlip) decoder. It further employs the parity-check vector to identify the unsuccessful simplified successive cancellation flip decoding at an early stage and terminates so that it can minimize the decoding complexity on average. Additionally, this work proposes an error-prone flipping list by incorporating the empirically observed indices based on channel-induced error distribution along with the first bit of each Rate-1 node. The proposed technique can identify more than one error-prone bit through a flipping list and correct them. In addition, the parity-check vector further narrows down the search space for the identification of erroneous decisions. Simulation results show that 60% of unsuccessful additional successive cancellation decoding attempts terminate early rather than decode the whole codeword. The proposed PC-CRC-SSCFlip decoder has approximately 0.7 dB and 0.3 dB gains over successive cancellation and successive cancellation flip decoders, respectively, at a fixed block error rate (BLER) = ${10}^{-3}$. Additionally, it reduces the average computational complexity and decoding latency of the successive cancellation flip decoder at low-to-medium SNRs while approaching successive cancellation decoding complexity at medium-to-high SNRs. Full article