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Permutation-Based Trellis Optimization for a Large-Kernel Polar Code Decoding Algorithm

Chunjuan Diao, Zhenling Wang, Ying Xiao, Feifei Zhang, Zhiliang Huang

Published Jan 29, 2026• 0 citations

Abstract

Compared to Arikan’s G2 kernel, large-kernel polar codes exhibit higher polarization rates and superior error correction performance. The critical steps of exact successive cancellation (SC) decoding for such codes can be implemented via trellis-based computations to reduce complexity. However, the complexity remains high for large kernels. This paper proposes a permutation-based trellis optimization scheme. The approach builds on the Massey minimal trellis and reorders its time axis to find a permutation that minimizes the number of trellis edges, thereby further reducing the exact SC decoding complexity. For smaller kernels (G3–G12), an exhaustive search is conducted to identify the optimal trellis. For larger kernels (G13–G16), where an exhaustive search becomes infeasible due to the factorial growth of the permutation space, an ant colony optimization (ACO)-based method is employed to find a near-optimal permutation. Simulation results show that the permutation-optimized trellis lowers the direct SC decoding complexity drastically. Furthermore, compared to the l-expression, the W-formula and original Massey trellis methods, it achieves multiplication operation reductions of up to 99.2%, 58.1%, and 56.5%, respectively. The improvement is particularly beneficial for large kernels, where traditional decoding methods become computationally prohibitive.

DOI: 10.3390/info17020127

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Compared to Arikan’s G2 kernel, large-kernel polar codes exhibit higher polarization rates and superior error correction performance. The critical steps of exact successive cancellation (SC) decoding for such codes can be implemented via trellis-based computations to reduce complexity. However, the complexity remains high for large kernels. This paper proposes a permutation-based trellis optimization scheme. The approach builds on the Massey minimal trellis and reorders its time axis to find a permutation that minimizes the number of trellis edges, thereby further reducing the exact SC decoding complexity. For smaller kernels (G3–G12), an exhaustive search is conducted to identify the optimal trellis. For larger kernels (G13–G16), where an exhaustive search becomes infeasible due to the factorial growth of the permutation space, an ant colony optimization (ACO)-based method is employed to find a near-optimal permutation. Simulation results show that the permutation-optimized trellis lowers the direct SC decoding complexity drastically. Furthermore, compared to the l-expression, the W-formula and original Massey trellis methods, it achieves multiplication operation reductions of up to 99.2%, 58.1%, and 56.5%, respectively. The improvement is particularly beneficial for large kernels, where traditional decoding methods become computationally prohibitive.

A trellis decoding scheme based on Massey trellis for polar codes with a ternary kernel is proposed. First, we construct trellises for kernel computations of the successive cancellation (SC) decoder based on the Massey trellis. Then, the scheme calculates the bit-channel transition probability through these trellises, which provides an appropriate framework to take advantage of the distributive law and can effectively reduce the number of operations. Simulation results show that for a G5⊗3 polar code of length 243 and rate 0.5, our scheme can save 14.2% of the computational cost with no sacrifice in error performance under the SC decoder.

A reduced complexity algorithm is presented for computing the log-likelihood ratios (LLRs) arising in the successive cancellation (SC) decoder for polar codes with large kernels of arbitrary dimension. The proposed algorithm exploits a recursive trellis representation of the codes generated by submatrices of the polarization kernel and enables polar codes based on large kernels to be decoded with lower complexity compared to the codes based on the Arıkan kernel with the same performance. Extensions to the case of non-binary kernels are presented.

As two common successive cancellation (SC)-based decoding algorithms of polar codes, the SC list (SCL) and SC stack (SCS) decoder can achieve satisfactory error correction performance, especially with increased list size or stack depth. Nevertheless, a large list size or stack depth will lead to high computational complexities and hardware resources. To this end, successive cancellation priority (SCP) decoding with priority- first searching strategy and trellis-like storage is proposed to offer one solution. In this paper, an efficient SCP decoder is first proposed to verify its advantages over SCL and SCS decoders. Furthermore, an adaptive node-inserting scheme is proposed to reduce the number of bits insert into the priority queue. Numerical results have shown that for the polar code with transmission length 1024 and rate 1/2, the proposed adaptive SCP (ASCP) decoder can achieve significant time complexity reduction on average compared with the standard SCL decoder. The hardware architecture of SCP decoding is implemented using 65-nm CMOS technology and the results show better throughput compared with the SCS decoder.

Polar codes have been recently shown to have an automorphism group larger than the lower-triangular affine group. Permutations from this group can be used in an automorphism successive cancellation (SC) decoding algorithm providing a lower latency compared to that of successive cancellation list decoding. In this paper, we demonstrate that any permutation from the automorphism group of Reed-Muller codes is suitable for permutation-based SC decoding of polar codes. Specifically, a code obtained by a permutation that is not in the automorphism group of the code, but is in the automorphism group of Reed-Muller codes, is shown to be a polar subcode. Simulations results show that these additional permutations allow improving performance of the automorphism SC decoding algorithm limited to the permutations from the automorphism group of the code.