Non-binary linear block codes (NB-LBCs) are an important class of error-correcting codes that are especially competent in correcting burst errors. They have broad applications in modern communications and storage systems. However, efficient soft-decision decoding of these codes remains to be further developed. This paper proposes successive cancellation list (SCL) decoding for NB-LBCs that are defined over a finite field of characteristic two, i.e., F2r, whereris the extension degree. By establishing a one-to-r mapping between the binary composition of each non-binary codeword andrbinary polar codewords, SCL decoding of therpolar codes can be performed with a complexity that is sub-quadratic in the codeword length. A simplified path sorting is further proposed to facilitate the decoding. Simulation results on short-length extended Reed-Solomon (eRS) and non-binary extended BCH (NB-eBCH) codes show that SCL decoding can outperform their state-of-the-art soft-decision decoding with fewer finite field arithmetic operations. For length-16 eRS codes, their maximum-likelihood (ML) decoding performances can be approached with a moderate list size.
Non-binary linear block codes (NB-LBCs) are an important class of error-correcting codes that are especially competent in correcting burst errors. They have broad applications in modern communications and storage systems. However, efficient soft-decision decoding of these codes remains to be further developed. This paper proposes successive cancellation list (SCL) decoding for NB-LBCs that are defined over a finite field of characteristic two, i.e., F2r, whereris the extension degree. By establishing a one-to-r mapping between the binary composition of each non-binary codeword andrbinary polar codewords, SCL decoding of therpolar codes can be performed with a complexity that is sub-quadratic in the codeword length. A simplified path sorting is further proposed to facilitate the decoding. Simulation results on short-length extended Reed-Solomon (eRS) and non-binary extended BCH (NB-eBCH) codes show that SCL decoding can outperform their state-of-the-art soft-decision decoding with fewer finite field arithmetic operations. For length-16 eRS codes, their maximum-likelihood (ML) decoding performances can be approached with a moderate list size.
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.
This paper aims to take advantage of the performances of polar decoding techniques for the benefit of binary linear block codes (BLBCs) with the main objective is to study the performances of the SSCL decoding for short-length BLBCs. Polar codes are one of the most recent error-correcting codes to be invented, and they have been mathematically demonstrated to be able to correct all errors under a specific situation, using the successive-cancellation decoder. However, their performances for real-time wireless communications at short block lengths remain less attractive. To take advantage of the performance of these codes in favor of error correction codes of short block length, an adaptation of the simplified successive-cancellation list as a decoder for polar codes for the benefit of short block length binary linear block codes is presented in this paper. This adaptation makes it possible to take advantage of the performances of less complex decoding methods for polar codes for BLBCs with latency and complexity optimization of the standard successive-cancellation list decoder. The experiment shows that the method can achieve the performances of the most famous order statistic decoder for binary linear block codes, which can achieve the performances of maximum-likelihood decoding with computational complexity and memory constraints.
In this paper, a new soft-decision decoding for extended Bose-Chaudhuri-Hocquenghem (eBCH) codes, referred to as CPC-SCL, is proposed, and it can achieve error-correction performance close to that of polarization-adjusted convolutional (PAC) codes in [1] when the code length n and dimension k are 128 and 64, respectively. Specifically, this paper first decomposes the eBCH code into a concatenated structure comprising an outer code and a Reed-Muller inner code. The outer code has a parity-check matrix characterized by a special block structure, which reveals that the positions of all frozen bits (including frozen zero bits and dynamic frozen bits) in the eBCH code are closely related to the index weights of elements from the perspective of the polar code. Subsequently, the CPC-SCL decoding of the eBCH codes is proposed by utilizing the cyclic property of codewords and parity-check-aided successive cancellation list (PC-SCL) decoding. Simulations also demonstrate that, over an additive white Gaussian noise (AWGN) channel with binary phase-shift keying (BPSK) modulation, the proposed decoding can achieve near maximum-likelihood (ML) performance at n = 64, k = 24 or 45.
Error correctiong codes (ECC) are widly used in applications to correct errors in data transmission over unreliable or noisy communication channels. Recently, two kinds of promising codes attracted lots of research interest because they provide excellent error correction performance. One is non-binary LDPC codes, and the other is polar codes. This dissertation focuses on efficient decoding algorithms and decoder design for thesetwo types of codes.Non-binary low-density parity-check (LDPC) codes have some advantages over their binary counterparts, but unfortunately their decoding complexity is a significant challenge. The iterative hard- and soft-reliability based majority-logic decoding algorithms are attractive for non-binary LDPC codes, since they involve only finite field additions and multiplications as well as integer operations and hence have significantly lower complexity than other algorithms. We propose two improvements to the majority-logic decoding algorithms. Instead of the accumulation of reliability information in the ex-isting majority-logic decoding algorithms, our first improvement is a new reliability information update. The new update not only results in better error performance and fewer iterations on average, but also further reduces computational complexity. Since existing majority-logic decoding algorithms tend to have a high error floor for codes whose parity check matrices have low column weights, our second improvement is a re-selection scheme, which leads to much lower error floors, at the expense of more finite field operations and integer operations, by identifying periodic points, re-selectingintermediate hard decisions, and changing reliability information.Polar codes are of great interests because they provably achieve the symmetric capacity of discrete memoryless channels with arbitrary input alphabet sizes an explicit construction. Most existing decoding algorithms of polar codes are based on bit-wise hard or soft decisions. We propose symbol-decision successive cancellation (SC) and successive cancellation list (SCL) decoders for polar codes, which use symbol-wise hard or soft decisions for higher throughput or better error performance. Then wepropose to use a recursive channel combination to calculate symbol-wise channel transition probabilities, which lead to symbol decisions. Our proposed recursive channel combination has lower complexity than simply combining bit-wise channel transition probabilities. The similarity between our proposed method and Arıkan's channel transformations also helps to share hardware resources between calculating bit- and symbol-wise channel transition probabilities. To reduce the complexity of the list pruning, atwo-stage list pruning network is proposed to provide a trade-off between the error performance and the complexity of the symbol-decision SCL decoder. Since memory is a significant part of SCL decoders, we also propose a pre-computation memory-saving technique to reduce memory requirement of an SCL decoder.To reduce the complexity of the recursive channel combination further, we propose an approximate ML (AML) decoding unit for SCL decoders. In particular, we investigate the distribution of frozen bits of polar codes designed for both the binary erasure and additive white Gaussian noise channels, and take advantage of the distribution to reduce the complexity of the AML decoding unit, improving the throughput-area efficiency of SCL decoders.Furthermore, to adapt to variable throughput or latency requirements which exist widely in current communication applications, a multi-mode SCL decoder with variable list sizes and parallelism is proposed. If high throughput or small latency is required, the decoder decodes multiple received words in parallel with a small list size. However, if error performance is of higher priority, the multi-mode decoder switches to a serialmode with a bigger list size. Therefore, the multi-mode SCL decoder provides a flexible tradeoff between latency, throughput and error performance at the expense of small overhead.
For polar codes, successive cancellation list (SCL) decoding algorithm significantly improves finite-length performance compared to SC decoding. SCL-flip decoding can further enhance the performance but the gain diminishes as code length increases, due to the difficulty in locating the first error bit position. In this work, we introduce an SCL-perturbation decoding algorithm to address this issue. A basic version of the algorithm introduces small random perturbations to the received symbols before each SCL decoding attempt, and exhibits non-diminishing gain at large block lengths. Its enhanced version adaptively performs random perturbations or directional perturbation on each received symbol according to previous decoding results, and managed to correct more errors with fewer decoding attempts. Extensive simulation results demonstrate stable gains across various code rates, lengths and list sizes. To the best of our knowledge, this is the first SCL enhancement with non-diminishing gains as code length increases, and achieves unprecedented efficiency. With only one additional SCL-L decoding attempt (in total two), the proposed algorithm achieves SCL-2L-equivalent performance. Since the gain is obtained without increasing list size, the algorithm is best suited for hardware implementation.
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.