Soft-Output Successive Cancellation List Decoding

TL;DR

The work advances decoding for polar-like codes by introducing a codebook-probability-based soft-output framework (SO-SCL) that yields both blockwise and bitwise soft outputs compatible with SC-based decoding. A tractable SC-based approximation of codebook probability enables a closed-form blockwise soft-output and a dynamically weighted bitwise soft-output, improving generalized decoding, error-detection capability, and iterative decoding for product and GLDPC codes. Dynamically frozen Reed-Muller components demonstrate superior BLER and UER performance over CRC-concatenated polar codes under SCL, with misdetection rate controllable to desired levels. The results highlight the practical impact in reliable communications by combining decoding performance gains with robust error-detection and high-quality soft information for iterative receivers.

Abstract

We introduce an algorithm for approximating the codebook probability that is compatible with all successive cancellation (SC)-based decoding algorithms, including SC list (SCL) decoding. This approximation is based on an auxiliary distribution that mimics the dynamics of decoding algorithms with an SC decoding schedule. Based on this codebook probability and SCL decoding, we introduce soft-output SCL (SO-SCL) to generate both blockwise and bitwise soft-output (SO). Using that blockwise SO, we first establish that, in terms of both block error rate (BLER) and undetected error rate (UER), SO-SCL decoding of dynamic Reed-Muller (RM) codes significantly outperforms the CRC-concatenated polar codes from 5G New Radio under SCL decoding. Moreover, using SO-SCL, the decoding misdetection rate (MDR) can be constrained to not exceed any predefined value, making it suitable for practical systems. Proposed bitwise SO can be readily generated from blockwise SO via a weighted sum of beliefs that includes a term where SO is weighted by the codebook probability, resulting in a soft-input soft-output (SISO) decoder. Simulation results for SO-SCL iterative decoding of product codes and generalized LDPC (GLDPC) codes, along with information-theoretical analysis, demonstrate significant superiority over existing list-max and list-sum approximations.

Figures (10)

• Figure 1: Example of the SC decoding tree of a polar code with frozen bits $u_1=u_3=0$. The whole decoding tree consists of three parts: a) visited leaf: the SC output $\hat{u}^4=(0,1,0,0)$. b) invalid subtrees: the subtree rooted at $\hat{u}_1=1$ and the subtree rooted at $\hat{u}^3=(0,1,1)$. c) unvisited subtrees: the subtree rooted at $\hat{u}^2=(0,0)$ and the leaf $\hat{u}^4=(0,1,0,1)$.
• Figure 2: Approximated blockwise SO vs. BLER of polar-like codes with proposed scheme and Forney's approximation. The proposed method (solid) works with SO-SCL decoding of list size $L$, while the Forney's approximation (dashed) works with SCL decoding of list size $L^\prime$.
• Figure 3: $\text{E}\left[1-\Gamma^*\left(y^N, \mathcal{L}_U\right)\right]$ vs. LER of polar-like codes under SO-SCL decoding with list size $L$.
• Figure 4: BLER(solid), UER(dashed), MDR(dotted) vs. $E_b/N_0$ over the biAWGN channel for the $\left(64,42\right)$ dynamic RM code compared to a $(64,42+6)$ static 5G polar code with an outer CRC-$6$0x30. SCL with $L=4$, threshold $\epsilon=0.1$
• Figure 5: BLER(solid), UER(dashed), MDR(dotted) vs. $E_b/N_0$ over the biAWGN channel for the $\left(64,42\right)$ dynamic RM code compared to a $(64,42+11)$ static 5G polar code with an outer CRC-$11$0x710. SCL with $L=8$, threshold $\epsilon=0.005$