The discovery of suitable automorphisms of polar codes gained a lot of attention by applying them in Automorphism Ensemble Decoding (AED) to improve the error-correction performance, especially for short block lengths. This paper introduces Successive Cancellation Automorphism List (SCAL) decoding of polar codes as a novel application of automorphisms in advanced Successive Cancellation List (SCL) decoding. Initialized with L permutations selected from the automorphism group, a superposition of different noise realizations and path splitting takes place inside the decoder. In this way, the SCAL decoder automatically adapts to the channel conditions and outperforms the error-correction performance of conventional SCL decoding and AED. For a polar code of length 128, SCAL performs near Maximum Likelihood (ML) decoding with L=8, in contrast to M=16 needed decoder cores in AED. Application-Specific Integrated Circuit (ASIC) implementations in a 12nm technology show that high-throughput, pipelined SCAL decoders outperform AED in terms of energy efficiency and power density, and SCL decoders additionally in area efficiency.
The discovery of suitable automorphisms of polar codes gained a lot of attention by applying them in Automorphism Ensemble Decoding (AED) to improve the error-correction performance, especially for short block lengths. This paper introduces Successive Cancellation Automorphism List (SCAL) decoding of polar codes as a novel application of automorphisms in advanced Successive Cancellation List (SCL) decoding. Initialized with L permutations selected from the automorphism group, a superposition of different noise realizations and path splitting takes place inside the decoder. In this way, the SCAL decoder automatically adapts to the channel conditions and outperforms the error-correction performance of conventional SCL decoding and AED. For a polar code of length 128, SCAL performs near Maximum Likelihood (ML) decoding with L=8, in contrast to M=16 needed decoder cores in AED. Application-Specific Integrated Circuit (ASIC) implementations in a 12nm technology show that high-throughput, pipelined SCAL decoders outperform AED in terms of energy efficiency and power density, and SCL decoders additionally in area efficiency.
In the short block length regime, pre-transformed polar codes together with successive cancellation list (SCL) decoding possess excellent error correction capabilities. However, in practice, the list size is limited due to the suboptimal scaling of the required area in hardware implementations. Automorphism ensemble decoding (AED) can improve performance for a fixed list size by running multiple parallel SCL decodings on permuted received words, yielding a list of estimates from which the final estimate is selected. Yet, AED is limited to appropriately designed polar codes. Subcode ensemble decoding (ScED) was recently proposed for low-density parity-check codes and does not impose such design constraints. It uses multiple decodings in different subcodes, ensuring that the selected subcodes jointly cover the original code. We extend ScED to polar codes by expressing polar subcodes through suitable pre-transformations (PTs). To this end, we describe a framework classifying pre-transformations for pretransformed polar codes based on their role in encoding and decoding. Within this framework, we propose a new type of PT enabling ScED for polar codes, analyze its properties, and discuss how to construct an efficient ensemble.
Automorphism ensemble decoding (AED) with the successive cancellation (SC) constituent decoder (AED-SC) can achieve similar decoding performance to the successive cancellation list (SCL) decoder when decoding short-to-medium-length polar codes. However, implementing automorphisms requires additional hardware for routing, leading to significant area overhead, especially with a large number of automorphisms. We propose methods for selecting automorphisms in SC decoders to reduce routing overhead. We establish the equivalence between automorphism selection and the NP-hard minimum K-union (MKU) problem. To maintain decoding performance under the selected automorphisms, we formulate the equivalence class property of AED-SC as a quadratic constraint. Compared to the state-of-the-art, our method achieves up to a 2.9× reduction in the number of routes needed to implement all automorphisms.
Automorphism ensemble decoding (AED) is a highly parallel approach that enables decoding of polar and Reed-Muller (RM) codes with automorphisms, offering a practical solution with near-maximum likelihood (ML) performance and manageable computational complexity. To meet the growing demands for high throughput and low latency in cloud and virtual random access networks, this paper presents a graphics processing unit (GPU)-based AED architecture for polar and RM codes, utilizing low-complexity successive cancellation (SC) and small list SC (SCL) decoders as the constituent of AED. The proposed architecture exploits the inherent parallelism of AED to optimize decoding tasks on the GPU, significantly enhancing throughput by efficiently harnessing the extensive parallel processing capabilities of the GPU. Additionally, improved thread mapping and data management techniques markedly reduce latency for automorphism ensemble SC (Aut-SC) decoding, while a low-latency sorting mechanism further accelerates automorphism ensemble SCL (Aut-SCL) decoding. Experimental results on an NVIDIA RTX 4090 demonstrate that the proposed Aut-SC decoder, with an ensemble size of 8, achieves a throughput exceeding 17 Gbps. Compared to the state-of-the-art softwarebased SCL decoders, the proposed GPU-based Aut-SC and Aut-SCL architectures outperform existing solutions by factors of up to 28× and 10×, respectively, in normalized throughput while maintaining the same or even superior error correction performance.
In this paper we deal with polar code automorphisms that are beneficial under low-latency automorphism ensemble (AE) decoding, and we propose polar code designs that have such automorphisms. Successive-cancellation (SC) decoding and thus SC-based AE decoding are invariant with respect to the only known polar code automorphisms, namely those of the lower-triangular affine (LTA) group. To overcome this problem, we provide methods to determine whether a given polar code has non-LTA automorphisms and to identify such automorphisms. Building on this, we design specific polar codes that admit automorphisms in the upper-diagonal linear (UTL) group, and thus render SC-based AE decoding effective. Demonstrated by examples, these new polar codes under AE decoding outperform conventional polar codes under SC list decoding in terms of error rate, while keeping the latency comparable to SC decoding. Moreover, state-of-the-art BP-based permutation decoding for polar codes is beaten by BP-based AE thanks to this design.
Automorphism ensemble (AE) decoding for polar codes was proposed by decoding permuted codewords with successive cancellation (SC) decoders in parallel and hence has lower latency compared to that of successive cancellation list (SCL) decoding. However, some automorphisms are SC-invariant, thus are redundant in AE decoding. In this paper, we find a necessary and sufficient condition related to the block lower-triangular structure of transformation matrices to identify SC-invariant automorphisms. Furthermore, we provide an algorithm to determine the complete SC-invariant affine automorphisms under a specific polar code construction.
Successive-cancellation list (SCL) decoding is a widely used and studied decoding algorithm for polar codes. For short blocklengths, empirical evidence shows that SCL decoding with moderate list sizes (say,L≤32) closely matches the performance of maximum-likelihood (ML) decoding. Hashemi et al. proved that on the binary erasure channel (BEC), SCL decoding actually coincides with ML decoding for list sizesL≥2γ, whereγis a new parameter we call the mixing factor. Loosely speaking, the mixing factor counts the number of information bits mixed-in among the frozen bits; more preciselyγ=∣{i∈Fc:i≤max{F}}∣, whereF⊂[n]denotes the set of frozen indices. Herein, we extend the aforementioned result of Hashemi et al. from the BEC to arbitrary binary-input memoryless symmetric channels. Our proof is based on capturing all2γdecoding paths that correspond to theγinformation bits appearing before the last frozen bit, and then finding the most-likely extension for each of these paths efficiently using a nearest coset decoding algorithm introduced herein. Furthermore, we present a hybrid successive-cancellation list (H -SCL) decoding algorithm, which is a hybrid between conventional SCL decoding and nearest coset decoding. We believe that the hybrid algorithm can outperform the conventional SCL decoder with lower decoding complexity.