Improved Successive Cancellation Decoding of Polar Codes (1208.3598v2)

Abstract: As improved versions of successive cancellation (SC) decoding algorithm, successive cancellation list (SCL) decoding and successive cancellation stack (SCS) decoding are used to improve the finite-length performance of polar codes. Unified descriptions of SC, SCL and SCS decoding algorithms are given as path searching procedures on the code tree of polar codes. Combining the ideas of SCL and SCS, a new decoding algorithm named successive cancellation hybrid (SCH) is proposed, which can achieve a better trade-off between computational complexity and space complexity. Further, to reduce the complexity, a pruning technique is proposed to avoid unnecessary path searching operations. Performance and complexity analysis based on simulations show that, with proper configurations, all the three improved successive cancellation (ISC) decoding algorithms can have a performance very close to that of maximum-likelihood (ML) decoding with acceptable complexity. Moreover, with the help of the proposed pruning technique, the complexities of ISC decoders can be very close to that of SC decoder in the moderate and high signal-to-noise ratio (SNR) regime.

• The paper introduces improved successive cancellation decoding algorithms —SCL, SCS, and SCH—to enhance the finite-length performance of polar codes.
• The authors unify these decoding algorithms as path searching on the code tree and propose a pruning technique to reduce computational complexity, especially at high SNR.
• Simulations show that the improved decoders approach ML performance with manageable complexity, enabling practical implementation of polar codes in communication systems.

Improved Successive Cancellation Decoding of Polar Codes

Polar codes, introduced by Arıkan and proven to achieve the symmetric capacities of binary-input discrete memoryless channels (B-DMCs), have become a pivotal element in modern coding theory. This paper by Chen, Niu, and Lin expands upon this foundational concept by enhancing the successive cancellation (SC) decoding algorithm, which traditionally struggles with finite-length performance despite its asymptotic efficacy. The authors propose notable improvements through the development of successive cancellation list (SCL) decoding, successive cancellation stack (SCS) decoding, and a novel successive cancellation hybrid (SCH) decoding, collectively termed Improved Successive Cancellation (ISC) decoding algorithms.

Overview and Algorithmic Developments

The primary advancement lies in the formulation of these ISC algorithms as path searching procedures on the code tree of polar codes. This approach unifies SC, SCL, and SCS decoding algorithms under a single framework, allowing for a systematic comparison and development.

1. Successive Cancellation List (SCL) Decoding: By maintaining multiple candidate paths simultaneously, SCL decoding improves upon SC by allowing more paths to be explored per level of the code tree. The main challenge lies in balancing computational and space complexity, managed by keeping only the top paths with the largest metrics in a list. With a searching width of $L$, the SCL decoding adapts well to varying complexities across signal-to-noise ratio (SNR) regimes.
2. Successive Cancellation Stack (SCS) Decoding: Unlike SCL, SCS allows a single path to be expanded deeper into the code tree until its metric is not the largest anymore, minimizing computations needed when the correct path is evident early. However, space complexity is more demanding due to the stack structure needed to manage paths, which tends to be large, especially in moderate SNR environments.
3. Successive Cancellation Hybrid (SCH) Decoding: SCH combines the strengths of SCL and SCS by utilizing a stack for initial searching, then switching to list mode when necessary. This hybrid approach provides a balance between computational and space complexities, adapting to the conditions dynamically and eliminating unnecessary path expansions.

Complexity Reduction and Pruning Techniques

To further enhance ISC decoding performance, a pruning technique is proposed. This involves setting a probability ratio threshold, $\tau$, allowing the pruning of paths whose metrics fall significantly below the best path of equivalent length. The authors provide a conservative configuration for $\tau$ that offers substantial complexity reductions without considerable loss in error performance, particularly effective in moderate and high SNR regimes.

Performance and Complexity Analysis

Through extensive simulations, the paper demonstrates that ISC decoders, when properly configured, approach ML decoding performance closely, with manageable computational overhead. The computational complexities are particularly reduced in high SNR environments, making ISC decoders practical for realistic communication systems.

The implications of this research are broad; theoretically, it underscores the potential of polar codes in achieving reliable communication under finite-length constraints. Practically, it provides insights into implementing polar codes within next-generation wireless systems, striking a balance between performance and system constraints.

Future Directions

The paper suggests that further explorations might be geared towards refining the stack and list structures inherent to SCH decoding to enhance adaptability across varying code lengths and rates, and developing adaptive configurations for the pruning threshold based on real-time decoding feedback. Additionally, extensions of these decoding algorithms to non-binary or multipath communication channels present intriguing possibilities for future developments.

In conclusion, this research provides a robust framework for improving polar code decoding, addressing the perennial challenge of finite-length performance while maintaining a feasible complexity profile.