Near-Optimal Generalized Decoding of Polar-like Codes

TL;DR

The paper addresses achieving low block error and undetected error rates for polar-like codes by introducing a codebook-probability based generalized decoding framework that remains compatible with SC-based decoders. It develops a practical SC-tree based approximation to the codebook probability, enabling accurate BLER prediction and a threshold-based decision rule that constrains the MDR. Results show that dynamic frozen-bit schemes, particularly dynamic Reed-Muller codes, outperform CRC-concatenated polar codes with SCL in both BLER and UER, and the framework supports applications in coded pilot-free channel estimation, bitwise soft-output decoding, and turbo product decoding. Together, these contributions provide a principled method to quantify decoding reliability and to design more reliable polar-like coding schemes for practical communications.

Abstract

We present a framework that can exploit the tradeoff between the undetected error rate (UER) and block error rate (BLER) of polar-like codes. It is compatible with all successive cancellation (SC)-based decoding methods and relies on a novel approximation that we call codebook probability. This approximation is based on an auxiliary distribution that mimics the dynamics of decoding algorithms following an SC decoding schedule. Simulation results demonstrates that, in the case of SC list (SCL) decoding, the proposed framework outperforms the state-of-art approximations from Forney's generalized decoding rule for polar-like codes with dynamic frozen bits. In addition, dynamic Reed-Muller (RM) codes using the proposed generalized decoding significantly outperform CRC-concatenated polar codes decoded using SCL in both BLER and UER. Finally, we briefly discuss three potential applications of the approximated codebook probability: coded pilot-free channel estimation; bitwise soft-output decoding; and improved turbo product decoding.

Figures (8)

• 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: Predicted BLER vs. simulated BLER of polar-like codes with proposed scheme and Forney's approximation. The proposed method (solid) works with SCL decoding of list size $L$, while the Forney's approximation (dashed) works with list size $L^\prime$.
• Figure 3: 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 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 $(64,42+11)$ static 5G polar code with an outer CRC-$11$0x710. SCL with $L=8$, threshold $\epsilon=0.005$
• Figure 5: Predicted LER vs. simulated LER of polar-like codes under SCL decoding with proposed scheme.