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International Conference on Wireless Communications and Signal Processing (WCSP)

An SC-Fano Based Enumerator of Low-Weight Codewords for Polar/PAC Codes

Xingyu Zhou, Shaohua Wu, Junhua You, Yajing Deng, Qinyu Zhang

Published Oct 23, 2025• 0 citations
Publisher

Abstract

The maximum likelihood (ML) decoding performance of linear codes is strongly influenced by the weight distribution, especially the number of low-weight codewords. In this work, we propose a successive cancellation (SC) code weight analyzer for polar codes and polarization-adjusted convolutional (PAC) codes. By feeding raw BPSK codewords into the SC analyzer under a noiseless channel, the SC metrics are shown to effectively characterize the weight of codewords. This insight motivates a bit-flipping strategy for efficient enumeration of lowweight codewords. Furthermore, we present an SC-Fano based enumerator that accurately accounts the low-weight codewords of polar codes and PAC codes without restrictions on rateprofiling. The results demonstrate that the proposed enumerator significantly reduces computation time for short to moderate block lengths compared to state-of-the-art methods.

DOI
10.1109/wcsp68525.2025.1010133

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The maximum likelihood (ML) decoding performance of linear codes is strongly influenced by the weight distribution, especially the number of low-weight codewords. In this work, we propose a successive cancellation (SC) code weight analyzer for polar codes and polarization-adjusted convolutional (PAC) codes. By feeding raw BPSK codewords into the SC analyzer under a noiseless channel, the SC metrics are shown to effectively characterize the weight of codewords. This insight motivates a bit-flipping strategy for efficient enumeration of lowweight codewords. Furthermore, we present an SC-Fano based enumerator that accurately accounts the low-weight codewords of polar codes and PAC codes without restrictions on rateprofiling. The results demonstrate that the proposed enumerator significantly reduces computation time for short to moderate block lengths compared to state-of-the-art methods.
Polarization-adjusted convolutional (PAC) codes are modified polar codes in which a one-to-one convolutional transformation is employed before the classical polar transform. Fano decoding of PAC codes in the Shannon lecture at ISIT2019 showed an outstanding performance at the cost of a high time-complexity, particularly at low SNR regimes. In order to reduce this complexity, an adaptive heuristic metric is proposed that improves the comparability of the variable-length paths and adjusts itself in response to the channel noise level. This metric can significantly reduce the number of nodes visited on average in tree-traversal. Additionally, a partial rewinding of the successive cancellation process is proposed to efficiently compute the intermediate LLRs and partial sums when backtracking occurs in the Fano algorithm. This method avoids storing the intermediate results of the decoding process or restarting (full rewinding) the decoding process.
The number of minimum weight codewords, a.k.a error coefficient, is a good comparative measure for the block error rate (BLER) of a linear block code, in particular in a high SNR regime. The smaller the error coefficient, the lower the BLER. Unlike polar codes, the error coefficient of polarization-adjusted convolutional (PAC) codes cannot be determined easily due to the precoding stage. In this work, we propose an enumeration method that considers the impact of convolutional precoding on the minimum weight codewords of polar codes. This relative enumeration method simplifies the process significantly compared with conventional methods as the complexity analysis shows.
Polar codes have attracted the attention of numerous researchers in the past decade due to their excellent performance. However, their performance at short block lengths under standard successive cancellation decoding is far from desirable. An effective method to improve the performance at short lengths is CRC precoding followed by successive-cancellation list decoding. Later, Arikan presented polarization-adjusted convolutional (PAC) codes, which further improve the performance of polar codes. In fact, bit-flipping is another post-processing method that can improve decoding performance. In this paper, we propose a novel SCL Bit-Flipping of PAC Codes. We show that better performance can be achieved using list decoding when the list size is the same for PAC codes (N=128, K=64). The decoding performance of our newly proposed PAC-SCLF with a list size of 32 is 0.3 dB better than that of the traditional PAC-SCL with a list size of 32. We set the maximum number of bit flips to 5. The performance of the list size (L=32) for PAC-SCLF is almost the same as the performance of the list size (L=128) for PAC-SCL.
In this letter, we introduce an efficient method for estimating weight distributions of polar codes and polarization-adjusted convolutional (PAC) codes. Based on a recursive algorithm of computing the weight enumerating functions of polar cosets, this method focuses on two key objectives: accurately determining the number of low-weight codewords and quickly approximating the distribution of high-weight codewords. Simulation results demonstrate that this hybrid method maintains competitively low complexity while effectively achieving the objectives.
By adopting successive cancellation list decoding (SCL), polar codes demonstrate competitive error correction performance over LDPC and Turbo codes. However, SCL decoding suffers from high computational complexity and long decoding latency, especially when the list size is very large. Successive cancellation flip (SCF), as another decoding algorithm that can achieve high error correction performance, has a complexity that is close to that of successive cancellation (SC) decoding. With the observation that SCL and SCF decoding are similar at giving more chances to inspect possible codewords simultaneously or sequentially, a novel hybrid decoder is proposed in this paper, which essentially combines the ideas of SCF and SCL decoders. Moreover, in order to compensate for the degradation of performance caused by the reduction of path splitting and further reduce the decoding latency, the convolutional polar codes are adopted with a designed bit-flipping set. Simulation results demonstrate that the proposed decoder achieves the reduction of decoding latency while attaining better performance than conventional CRC-aided SCL decoder.
The decoding complexity of polarization-adjusted convolutional (PAC) codes and Polar codes is significantly affected by the efficiency of partial-sum updates. In this letter, we propose a fast partial-sum update method for the successive-cancellation decoding (SCD) of PAC and Polar codes by exploiting the Pascal-triangle structure of the Polar generator matrix. Simulation results show that the proposed method reduces the number of partial-sum updates compared with conventional SCD, particularly for short code lengths and in the low-rate regime. Compared with representative fast decoding methods, the proposed method also shows a complexity advantage in the low-rate regime.