SC-Flip Decoding Research Articles

Efficient Partial Rewind of Successive Cancellation-Based Decoders for Polar Codes

The successive cancellation (SC) process in which symbols are decoded sequentially by processing some intermediate information is an essential component of various decoding algorithms used for polar codes and their variants. In some decoding schemes, we may need to redo this process from some specific symbol or from the first symbol. This operation is called rewinding. Rewinding the SC process seems trivial if we have access to all intermediate log-likelihood ratios (LLRs) and partial sums. However, as the block length increases, retaining all of the intermediate information becomes inefficient and impractical. Rewinding the SC process in a memory-efficient way is a problem that we address in this paper. As we store a fraction of all the intermediate information in the memory-efficient scheme, we may not be able to rewind the SC process to the target symbol index. The reason is that some of the stored intermediate information needed to decode the target symbol may have been overwritten. To recompute the lost information, we may need to rewind the process further. Before proposing the formal scheme for the rewinding process, we explore the known properties of the SC process based on the binary representation of the bit indices. Then, we introduce a new operator used for grouping the bit indices. This special grouping helps us in finding the closest bit index to the target index for rewinding. We also analytically prove that this approach gives access to the untouched intermediate information stored in the memory which is essential in resuming the SC process. Finally, we adapt the proposed approach to multiple rewinds and apply it to SC-flip decoding and shifted-pruning-based list decoding. The numerical evaluation of the proposed solution shows a significant reduction of ≥50% in the complexity of the additional decoding attempts at medium and high SNR regimes for SC-flip decoding and less for shifted-pruning based list decoding.

Parity Check Aided SC-Flip Decoding Algorithms for Polar Codes

When polar codes are decoded by the successive cancellation (SC) decoding algorithm, erroneously decoded non-frozen bits are caused by either channel noise or error propagation. SC-Flip algorithms aim to improve error performance by first identifying erroneous hard decisions due to channel noise and then flipping them during the decoding process to reduce error propagation. In existing SC-Flip algorithms, cyclic redundancy check (CRC) is used to check the decoded codeword to detect incorrect hard decisions. Differing from this detection approach based on CRC, we propose new SC-Flip decoders that take advantage of both the CRC and distributed parity checks (PCs) to detect, identify and flip erroneously decoded non-frozen bits. The proposed decoders terminate SC decoding early when a distributed PC is not satisfied. In addition, we propose a new metric to help locate the incorrect hard decisions. Finally, simulation results demonstrate that our SC-Flip decoders achieve better performance complexity tradeoffs than prior flipping algorithms, and approach the performance and complexity of the SC-Oracle algorithms.

Improved Belief Propagation Polar Decoders With Bit-Flipping Algorithms

Since the inherent serial nature of successive cancellation list (SCL) decoding results in a long latency, belief propagation (BP) decoding for polar codes has drawn attention for high-throughput applications. However, its error correction performance is inferior to that of SCL decoding. Therefore, the bit-flipping strategy has been recently applied to BP decoding, which can approach the SCL decoding performance through multiple additional decoding attempts. The original BP flip (BPF) decoding suffers from an inaccurate identification of erroneous bits by a fixed flip set (FS), which has been improved by the generalized BPF (GBPF) decoding. In this article, the GBPF decoding is extended to support multiple bits being flipped in one decoding attempt. In addition, for two types of decoding errors: detected errors and undetected errors, we propose two novel methods to more effectively identify erroneous bits. For detected errors, the concept of loop sets is defined and a loop-based identification method is introduced based on the study of error patterns of BP decoding. On the other hand, a method to generate a more accurate fixed FS is proposed for undetected errors, which considers the bit error distribution under BP decoding. Combining the two methods, the GBPF with merged sets (GBPF-MS) decoding can achieve the SCL-8 performance and outperforms the state-of-the-art BPF, BP list, and SC flip (SCF) decoding, for polar codes with length 1024 and information rate 1/2. Implemented by 40nm CMOS technology, the proposed GBPF-MS decoder with ten flips exhibits an average throughput of 4.19 Gbps at 2.5 dB, which is $1.6\times $ and $1.72\times $ faster than the state-of-the-art SCL-4 and SCF decoders, respectively.

Error-Aware SCFlip Decoding of Polar Codes

The successive-cancellation flip (SCFlip) decoder and its variants provide a significant coding gain with the average complexity practically identical to that of the successive cancellation (SC) decoder in a wide range of signal-to-noise ratios (SNRs). But, they suffer from high complexity and long latency when the SNR decreases, since the average number of extra decoding attempts becomes inevitably large. To mitigate this problem, we propose a novel SCFlip decoder, called an error-aware SCFlip (EA-SCFlip) decoder, for distributed cyclic-redundancy-check (CRC) polar codes. Based on the distributed CRC bits, it employs early termination at each extra decoding attempt so that it can reduce the decoding complexity and latency on the average. It also reduces the search space of candidate bit-flips in the dynamic building of the bit-flip list by exploiting the parity-check relationship (PCR) of the first error-detected CRC bit at each extra decoding attempt. Furthermore, we propose a greedy algorithm to design a distributed CRC code such that the obtained PCRs make the early-error-detection capability of the EA-SCFlip decoder as high as possible. Numerical results demonstrate that the EA-SCFlip decoder can indeed achieve an early termination gain as well as a complexity reduction, when a polar code is concatenated with the distributed CRC code designed by the proposed algorithm.

BER Evaluation Based SCFlip Algorithm for Polar Codes Decoding

Successive cancellation (SC) decoding of polar codes may bring about error propagation that needs to be mitigated. In this paper, we present a new SC Flipping (SCFlip) decoder, named bit error rate (BER) evaluation based SCFlip (BER-SCFlip), which can accurately target the first error bit and correct it with a high probability. Thus, a high error correction capability and a low decoding complexity can be achieved. First, we propose a new criterion to find out the most suspicious error bit. Those non-frozen bits that have higher decoding BERs derived from log-likelihood-ratios (LLRs) after SC decoding than the corresponding expected ones estimated via Gaussian Approximation (GA), are collected into the flip-bits set. These candidate bits will be flipped one by one according to their SC decoding orderings in extra decoding attempts until the decoded codeword passes cyclic redundancy check (CRC) or a predetermined maximum number of extra attempts is reached. We then propose an extended version of BER-SCFlip, named BER-SCFlip-w with the capability to correct up to w error bits in each extra decoding attempt. By combining our criterion for the flip-bits selection with that of Dynamic SCFlip (D-SCFlip), the proposed BER-SCFlip-w significantly reduces decoding complexity and latency while maintaining the superior error-correction performance close to that of D-SCFlip-w. The simulation results show that the proposed schemes are competitive among existing SCFlip algorithms and could achieve the error-correction performance approaching that of CRC-aided SCL decoding under list size L = 16 while maintaining low complexity.

Improved Multiple Bit-Flipping Fast-SSC Decoding of Polar Codes

Polar codes can provably achieve channel capacity and have been used in standards of the 5th generation wireless communication. As the first decoding algorithm of polar codes, successive cancellation (SC) suffers a significant error-correction performance loss at the moderate to short code length. The SC-Flip decoding aims to correct the first error in the SC decoding, and has the competitive error-correction performance compared with the SC decoding at the cost of high decoding latency. However, there may exist more than one errors in the SC decoding. Therefore, this paper proposes to enhance the fast-simplified SC (Fast-SSC) decoding to incorporate multiple bit-flipping, and further proposes two enhanced Fast-SSC-Flip decoding algorithms. One is a two-bit-flipping Fast-SSC (Fast-SSC-2Flip-E2) decoding algorithm based on the distribution of the second error ( $E_{2}$ ) in the Fast-SSC decoding, and is further expanded to flip multiple bits based on the distribution of multiple errors. The other is a partitioned Fast-SSC-Flip (PA-Fast-SSC-Flip) decoding algorithm. The Fast-SSC decoder tree is divided into several partitions, on which the Fast-SSC-Flip decoding is performed. Compared with the traditional Fast-SSC-Flip decoding, the proposed Fast-SSC-2Flip-E2 has an error-correction performance gain up to 0.2 dB while keeping the average complexity close to that of the traditional Fast-SSC-Flip. The decoding speed of the PA-Fast-SSC-Flip is up to 6 times faster than that of traditional Fast-SSC-Flip, and has an error-correction performance gain up to 0.16dB.

Energy-Efficient Hardware Architectures for Fast Polar Decoders

Interest in polar codes has increased significantly upon their selection as a coding scheme for the 5th generation wireless communication standard (5G). While the research on polar code decoders mostly targets improved throughput, few implementations address energy consumption, which is critical for platforms that prioritize energy efficiency, such as massive machine-type communications (mMTC). In this work, we first propose a novel Fast-SSC decoder architecture that has novel architectural optimizations to reduce area, power, and energy consumption. Then, we extend our work to an energy-efficient implementation of the fast SC-Flip (SCF) decoder. We show that sorting a limited number of indices for extra decoding attempts is sufficient to practically match the performance of SCF, which enables employing a low-complexity sorter architecture. To our knowledge, the proposed SCF architecture is the first hardware realization of fast SCF decoding. Synthesis results targeting TSMC 65nm CMOS technology show that the proposed Fast-SSC decoder architecture is 18% more energy-efficient, has 14% less area and 30% less power consumption compared to state-of-the-art decoders in the literature. Compared to the state-of-the-art available SC-List (SCL) decoders that have equivalent error-correction performance, proposed Fast-SCF decoder is 29% faster while being $2.7\times $ more energy-efficient and 51% more area-efficient.

Improved Fast-SSC-Flip Decoding of Polar Codes

In this letter, state-of-the-art fast simplified successive-cancellation (Fast-SSC) decoding is modified to cooperate with SC flip decoding. The resulting Fast-SSC-Flip decoding aims to flip the first error of invalid codewords to improve the performance of polar codes. Compared with existing Fast-SSC-Flip decoding, our decoder has about a 0.45-dB performance gain. For most undecodable codewords under Fast-SSC decoding, the proposed Fast-SSC-Flip decoding can successfully decode them much faster than prior Fast-SSC-Flip decoding.

Improved Bit-Flipping Algorithm for Successive Cancellation Decoding of Polar Codes

The interest in polar codes has been increasing significantly since their adoption for use in the 5th generation wireless systems standard. Successive cancellation (SC) decoding algorithm has low implementation complexity, but yields mediocre error-correction performance at the code lengths of interest. SC-Flip algorithm improves the error-correction performance of SC by identifying possibly erroneous decisions made by SC and re-iterates after flipping one bit. It was recently shown that only a portion of bit-channels are most likely to be in error. In this paper, we investigate the average log-likelihood ratio (LLR) values and their distribution related to the erroneous bit-channels, and develop the Thresholded SC-Flip (TSCF) decoding algorithm. We also replace the LLR selection and sorting of SC-Flip with a comparator to reduce the implementation complexity. Simulation results demonstrate that for practical code lengths and a wide range of rates, TSCF shows negligible loss compared with the error-correction performance obtained when all single-errors are corrected. At matching maximum iterations, TSCF has an error-correction performance gain of up to 0.45 dB compared with SC-Flip decoding. At matching error-correction performance, the computational complexity of TSCF is reduced by up to 40% on average and requires up to $5\times $ lower maximum number of iterations.

SC-Fano Decoding of Polar Codes

For finite-length polar codes, the standard successive cancellation (SC) decoding has been improved, such as SC-List (SCL), SC-Stack (SCS), and SC-Flip (SCF) decodings. In this paper, we present an alternative improvement of SC decoding by incorporating the Fano sequential decoding into SC decoding. This is referred to as SC-Fano decoding. Specifically, it can address the major drawback of SC decoding by enabling moving-backward when the reliability of an on-going path is not good enough. The SCS and SC-Fano decodings can be viewed as the sequential decoding for polar codes. In addition, for cyclic-redundancy-check (CRC) concatenated polar codes, we enhance SC-Fano decoding by leveraging the bit-flipping idea of SCF decoding. The simulation results demonstrate that the proposed SC-Fano decoding can provide better performance-complexity tradeoff than the existing decoding methods.