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Threshold Successive Cancellation Flip Decoding Algorithm for Polar Codes: Design and Performance

Zhicheng Liu, Liuquan Yao, Shuai Yuan, Guiying Yan, Zhi-Ming Ma, Yuting Liu

Published Jun 12, 2025• 1 citation
Publisher

Abstract

In this paper, we propose the threshold successive cancellation flip (Th-SCF) decoding algorithm for polar codes, which enhances the performance of the SC decoder while maintaining low complexity. Theoretical analysis reveals that Th-SCF asymptotically delays the first error position (FEP, the first part where the SC decoder fails) with probability 1, ensuring high decoding performance. Simulation results show that the Th-SCF algorithm achieves performance comparable to the dynamic SC flip (D-SCF) algorithm, but with a reduction in complexity by eliminating the need for sorting operations. A key contribution of this work is the rigorous theoretical framework supporting the Th-SCF algorithm, distinguishing it from existing SC flip (SCF) decoding methods. This theoretical foundation not only explains the performance improvements but also provides insights into the underlying mechanisms of flipping. The proposed Th-SCF algorithm demonstrates strong performance across a wide range of code lengths and rates, and its performance remains stable within a certain threshold range, indicating its practical applicability in real-world communication systems. These results offer valuable perspectives for the design of efficient flip decoding strategies in 5G and future networks.

DOI
10.3390/e27060626

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In this paper, we propose the threshold successive cancellation flip (Th-SCF) decoding algorithm for polar codes, which enhances the performance of the SC decoder while maintaining low complexity. Theoretical analysis reveals that Th-SCF asymptotically delays the first error position (FEP, the first part where the SC decoder fails) with probability 1, ensuring high decoding performance. Simulation results show that the Th-SCF algorithm achieves performance comparable to the dynamic SC flip (D-SCF) algorithm, but with a reduction in complexity by eliminating the need for sorting operations. A key contribution of this work is the rigorous theoretical framework supporting the Th-SCF algorithm, distinguishing it from existing SC flip (SCF) decoding methods. This theoretical foundation not only explains the performance improvements but also provides insights into the underlying mechanisms of flipping. The proposed Th-SCF algorithm demonstrates strong performance across a wide range of code lengths and rates, and its performance remains stable within a certain threshold range, indicating its practical applicability in real-world communication systems. These results offer valuable perspectives for the design of efficient flip decoding strategies in 5G and future networks.
SC-Flip (SCF) decoding algorithm shares the attention with the common polar code decoding approaches due to its low-complexity and improved error-correction performance. However, the inefficient criterion for locating the correct bit-flipping position in SCF decoding limits its improvements. Due to its improved bit-flipping criterion, Thresholded SCF (TSCF) decoding algorithm exhibits a superior error-correction performance and lower computational complexity than SCF decoding. However, the parameters of TSCF decoding depend on multiple channel and code parameters, and are obtained via Monte-Carlo simulations. Our main goal is to realize TSCF decoding as a practical polar decoder implementation. To this end, we first realize an approximated threshold value that is independent of the code parameters and precomputations. The proposed approximation has negligible error-correction performance degradation on the TSCF decoding. Then, we validate an alternative approach for forming a critical set that does not require precomputations, which also paves the way to the implementation of the Fast-TSCF decoder. Compared to the existing fast SCF implementations, the proposed Fast-TSCF decoder has 0.240.240.24 to 0.410.410.41 dB performance gain at frame error rate of 10−310^{-3}10−3, without any extra cost. Compared to the TSCF decoding, Fast-TSCF does not depend on precomputations and requires 87%87\%87% fewer decoding steps. Finally, implementation results in TSMC 65nm CMOS technology show that the Fast-TSCF decoder is 20%20\%20% and 82%82\%82% more area-efficient than the state-of-the-art fast SCF and fast SC-List decoder architectures, respectively.
Polar codes are a class of error-correcting codes that provably achieve the capacity of practical channels. The successive-cancellation flip (SCF) decoder is a low-complexity decoder that was proposed to improve the performance of the successive-cancellation (SC) decoder as an alternative to the high-complexity successive-cancellation list (SCL) decoder. The SCF decoder improves the error-correction performance of the SC decoder, but the variable execution time and the high worst-case execution time pose a challenge for the realization of receivers with fixed-time algorithms. The dynamic SCF (DSCF) variation of the SCF decoder further improves the error-correction performance but the challenge of decoding delay remains. In this work, we propose a simplified restart mechanism (SRM) that reduces the execution time of SCF and DSCF decoders through conditional restart of the additional trials from the second half of the codeword. We show that the proposed mechanism is able to improve the execution time characteristics of SCF and DSCF decoders while providing identical error-correction performance. For a DSCF decoder that can flip up to 3 simultaneous bits per decoding trial, the average execution time, the average additional execution time and the execution-time variance are reduced by approximately 31%, 37% and 57%, respectively. For this setup, the mechanism requires approximately 3.9% additional memory.
SC-Flip (SCF) decoding algorithm shares the attention with the common polar code decoding approaches due to its low-complexity and improved error-correction performance. However, the inefficient criterion for locating the correct bit-flipping position in SCF decoding limits its improvements. Due to its improved bit-flipping criterion, Thresholded SCF (TSCF) decoding algorithm exhibits a superior error-correction performance and lower computational complexity than SCF decoding. However, the parameters of TSCF decoding depend on multiple channel and code parameters, and are obtained via Monte-Carlo simulations. Our main goal is to realize TSCF decoding as a practical polar decoder implementation. To this end, we first realize an approximated threshold value that is independent of the code parameters and precomputations. The proposed approximation has negligible error-correction performance degradation on the TSCF decoding. Then, we validate an alternative approach for forming a critical set that does not require precomputations, which also paves the way to the implementation of the Fast-TSCF decoder. Compared to the existing fast SCF implementations, the proposed Fast-TSCF decoder has 0.24 to 0.41 dB performance gain at frame error rate of 10-3, without any extra cost. Compared to the TSCF decoding, Fast-TSCF does not depend on precomputations and requires 87% fewer decoding steps. Finally, implementation results in TSMC 65nm CMOS technology show that the Fast-TSCF decoder is 20% and 82% more area-efficient than the state-of-the-art fast SCF and fast SC-List decoder architectures, respectively.
When the code length is finite, the successive cancellation list (SCL) decoding of polar codes shows as good error-correction performance as low-density parity-check codes and turbo codes and it will be used in high-speed communications. However, since the hardware complexity of the SCL increases linearly as the size of the list increases, it is difficult to use the high-complexity SCL decoding for devices with limited resource such as IoT devices. The successive cancellation flip (SCF) decoding algorithm has been proposed to solve the problem of high hardware complexity and to enhance the error-correction performance of successive cancellation decoding. However, the conventional SCF decoding has high computational complexity. In this paper, we propose the interleaver-aided successive cancellation flip (ISCF) decoding to reduce the computational complexity. The computational complexity of the ISCF decoding is less by at least 32% than the conventional SCF decoding for error frame at Eb/N0=3.5dB.
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.
Polar Codes achieve average error correction performance at finite code lengths, when decoded using Successive Cancellation (SC) decoding. Successive Cancellation Flip (SC-Flip) based decoders provide a viable low complexity substitute of the high performing high complexity Successive Cancellation List (SCL) decoders, providing a considerable improvement in the performance. A novel approach to identify and flip multiple erroneous bits in each decoding attempt has been aimed in this paper. Moreover, a faster version of the proposed decoder is also presented, which provides better performance throughout the entire decoding range. The performance of the proposed decoders have been evaluated with respect to frame error correction rate along with average normalized complexity and then compared with some of the existing works for validating the outperformance.