[Paper Review] A variant of list plus CRC concatenated polar code

Gregory Bonik, Sergei A. Goreinov+1 more|arXiv (Cornell University)|Jul 19, 2012

Error Correcting Code Techniques1 references3 citations

TL;DR

This paper proposes a novel polar code construction combining soft concatenation, list decoding, and CRC checks to achieve performance competitive with industry standards and the Tal-Vardy scheme, while reducing decoder complexity. By applying successive cancellation to rows and list+CRC decoding to columns with adaptive list sizes, the method achieves lower average complexity than traditional list-plus-CRC decoding, particularly due to reduced average list size and inherent parallelism.

ABSTRACT

A new family of codes based on polar codes, soft concatenation and list+CRC decoding is proposed. Numerical experiments show the performance competitive with industry standards and Tal, Vardy approach.

Motivation & Objective

• To develop a polar code decoding scheme with lower complexity than the Tal-Vardy list-plus-CRC approach while maintaining competitive error-rate performance.
• To reduce the decoder’s computational overhead by leveraging channel polarization to lower the average list size $L_{\text{av}}$ instead of using a fixed large $L$.
• To explore the use of short internal codes (e.g., CRC-4) in a soft concatenation framework to improve decoding efficiency and performance.
• To demonstrate that the proposed scheme achieves performance comparable to state-of-the-art codes like 802.11ad and LDPC codes in AWGN channels.
• To lay the groundwork for future theoretical analysis of optimal parameter selection and performance bounds.

Proposed method

• The scheme uses a matrix $X = VG$, where $V$ is a binary matrix with columns constrained to codewords of short linear codes $C_k$, and $G$ is a polar code generator matrix.
• Each column of $X$ is decoded using a list decoding algorithm with CRC-based candidate selection, reducing error propagation.
• The decoder alternates between successive cancellation on rows and list+CRC decoding on columns, exploiting the fact that $L_{\text{av}} < L_{\text{max}}$ due to channel polarization.
• The method introduces parallelism: column decoding can be performed simultaneously across rows, reducing overall latency and complexity.
• The decoding complexity is estimated as $O(NM(L_{\text{av}}\log M + \log N))$, which is lower than the $O(LNM\log(NM))$ complexity of Tal-Vardy when $L_{\text{av}} \ll L$.
• The scheme uses CRC-4 for each column to validate list candidates, improving reliability without significant overhead.

Experimental results

Research questions

• RQ1 Can a soft concatenation framework with list+CRC decoding on columns outperform the Tal-Vardy scheme in terms of error rate while reducing decoder complexity?
• RQ2 Does the use of adaptive list sizes based on channel polarization significantly reduce average decoding complexity compared to fixed large list sizes?
• RQ3 To what extent does the proposed scheme match or exceed the performance of industry-standard codes like 802.11ad and LDPC codes in AWGN channels?
• RQ4 Is the observed performance degradation at high SNR a fundamental limitation of the scheme or a result of suboptimal parameter choices?
• RQ5 Can the proposed method be generalized to use different internal codes or LDPC codes in the column decoding stage?

Key findings

• The proposed scheme achieves block error rate performance comparable to the Tal-Vardy scheme with $L=8$ on a $N=512$, rate $13/16$ code over an AWGN channel.
• For the same code length and rate, the new scheme performs comparably to 802.11ad and LDPC codes, as shown in numerical experiments.
• The average list size $L_{\text{av}}$ used in simulations was 8, significantly lower than the maximum list size, indicating effective utilization of channel polarization.
• The decoder complexity is reduced due to $L_{\text{av}} < L_{\text{max}}$, and the method supports natural parallelism, further lowering effective complexity.
• At high SNR, the proposed scheme shows a slight performance degradation compared to other tested codes, suggesting potential for further optimization.
• The scheme demonstrates that combining soft concatenation with list+CRC decoding enables high performance with lower complexity than traditional list decoding approaches.