An algebraic soft-decision decoder for Hermitian codes is presented. We apply Koetter and Vardy's soft-decision decoding framework, now well established for Reed-Solomon codes, to Hermitian codes. First we provide an algebraic foundation for soft-decision decoding. Then we present an interpolation algorithm finding the Q-polynomial that plays a key role in the decoding. With some simulation results, we compare performances of the algebraic soft-decision decoders for Hermitian codes and Reed-Solomon codes, favorable to the former.
Maximum Hermitian rank metric codes were introduced by Schmidt in 2018 and in this paper we propose both interpolation-based encoding and decoding algorithms for this family of codes when the length and the minimum distance of the code are both odd.
We propose a new partial decoding algorithm for $h$-interleaved one-point Hermitian codes that can decode-under certain assumptions-an error of relative weight up to $1-(\tfrac{k+g}{n})^{\frac{h}{h+1}}$, where $k$ is the dimension, $n$ the length, and $g$ the genus of the code. Simulation results for various parameters indicate that the new decoder achieves this maximal decoding radius with high probability. The algorithm is based on a recent generalization of Rosenkilde's improved power decoder to interleaved Reed-Solomon codes, does not require an expensive root-finding step, and improves up…
We propose an improved soft-aided decoding scheme for product codes that approaches the decoding performance of conventional soft-decision TPD with only a 0.2 dB gap while keeping the complexity and internal decoder data flow similarly low as in hard decision decoders.
Guessing Random Additive Noise Decoding (GRAND) is a universal decoding algorithm that can be used to perform maximum likelihood decoding. It attempts to find the errors introduced by the channel by generating a sequence of possible error vectors in order of likelihood of occurrence and applying them to the received vector. Ordered reliability bits GRAND (ORBGRAND) integrates soft information received from the channel to refine the error vector sequence. In this work, ORBGRAND is modified to produce a soft output, to enable its use as an iterative soft-input soft-output (SISO) decoder. Three t…
We propose a novel soft-aided low-complexity decoder for product codes based on dynamic reliability scores and error-and-erasure decoding. We observe coding gains of up to 1.2 dB compared to conventional hard-decision decoders.
The successive cancellation list decoding algorithm for polar codes yields near-optimal decoding performance at the cost of high implementation complexity. The successive cancellation stack algorithm has been shown to provide similar decoding performance at a much lower computational complexity, but software implementations report a sub-par T/P performance. In this technical report, the benefits of the fast simplified successive cancellation list decoder are extended to the stack algorithm, resulting in a throughput increase by two orders of magnitude over the traditional stack decoder.
This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right choice for these codes and then we provide easily reversible encoding methods for each family. Later unique decoding algorithms for the codes are described. The decoding algorithms are interpolation-based and can uniquely correct errors for each code with rank up to $\lfloor(d-1)/2\rfloor$ in polynomial-time, where $d$ is the minimum distance of the code.
The interpolation based algebraic decoding for Reed-Solomon (RS) codes can correct errors beyond half of the code's minimum Hamming distance. Using soft information, the algebraic soft decoding (ASD) further improves the decoding performance. This paper presents a unified study of two classical ASD algorithms in which the computationally expensive interpolation is solved by the module minimization (MM) technique. An explicit module basis construction for the two ASD algorithms will be introduced. Compared with Koetter's interpolation, the MM interpolation enables the algebraic Chase decoding a…
Node-Based Soft-Output Fast Successive Cancellation List Decoding of Polar Codes
Shanghai Jiao Tong University（上海交通大学）；Beijing Institute of Technology（北京理工大学）；State Key Laboratory of CNS/ATM（通信与网络系统国家重点实验室）；Advanced Technology Research Institute, BIT（北京理工大学先进技术研究院）；Purple Mountain Laboratories（紫金山实验室）
This paper presents a generalized construction of RS-SPC product codes. A low-complexity joint-decoding scheme is proposed for these codes, in which a BP-based iterative decoding is performed based on the binary expansion of the whole parity-check matrix. Various powerful RS codes can be used as the component codes for RS-SPC product codes, which gives a good performance for local decoding (decode a single component codeword). The proposed BP-based iterative decoding is a global decoding, and it achieves an error-correcting capability comparable to codes of large blocklengths. This two-phase d…
The binary primitive BCH codes are cyclic and are constructed by choosing a subset of the cyclotomic cosets. Which subset is chosen determines the dimension, the minimum distance and the weight distribution of the BCH code. We construct possible BCH codes and determine their coderate, true minimum distance and the non-equivalent codes. A particular choice of cyclotomic cosets gives BCH codes which are, extended by one bit, equivalent to Reed-Muller codes, which is a known result from the sixties. We show that BCH codes have possibly better parameters than Reed-Muller codes, which are related i…
Maximum Likelihood (ML) decoding of forward error correction codes is known to be optimally accurate, but is not used in practice as it proves too challenging to efficiently implement. Here we introduce a ML decoder called SGRAND, which is a development of a previously described hard detection ML decoder called GRAND, that fully avails of soft detection information and is suitable for use with any arbitrary high-rate, short-length block code. We assess SGRAND's performance on CRC-aided Polar (CA-Polar) codes, which will be used for all control channel communication in 5G NR, comparing its accu…
In this paper, we design the joint decoding (JD) of non-orthogonal multiple access (NOMA) systems employing short block length codes. We first proposed a low-complexity soft-output ordered-statistics decoding (LC-SOSD) based on a decoding stopping condition, derived from approximations of the a-posterior probabilities of codeword estimates. Simulation results show that LC-SOSD has the similar mutual information transform property to the original SOSD with a significantly reduced complexity. Then, based on the analysis, an efficient JD receiver which combines the parallel interference cancellat…
Polar codes are able to achieve the capacity of memoryless channels under successive cancellation (SC) decoding. Soft Cancellation (SCAN) is a soft-output decoder based on the SC schedule, useful in iterative decoding and concatenation of polar codes. However, the sequential nature of this decoder leads to high decoding latency compared to state-of-the-art codes. To reduce the latency of SCAN, in this paper we identify special nodes in the decoding tree, corresponding to specific frozen-bit sequences, and propose dedicated low-latency decoding approaches for each of them. The resulting fast-SC…
Power decoding is a partial decoding paradigm for arbitrary algebraic geometry codes for decoding beyond half the minimum distance, which usually returns the unique closest codeword, but in rare cases fails to return anything. The original version decodes roughly up to the Sudan radius, while an improved version decodes up to the Johnson radius, but has so far been described only for Reed--Solomon and one-point Hermitian codes. In this paper we show how the improved version can be applied to any algebraic geometry code.
Products codes (PCs) are conventionally decoded with efficient iterative bounded-distance decoding (iBDD) based on hard-decision channel outputs which entails a performance loss compared to a soft-decision decoder. Recently, several hybrid algorithms have been proposed aimed to improve the performance of iBDD decoders via the aid of a certain amount of soft information while keeping the decoding complexity similarly low as in iBDD. We propose a novel hybrid low-complexity decoder for PCs based on error-and-erasure (EaE) decoding and dynamic reliability scores (DRSs). This decoder is based on a…
In this work, we investigate guessing random additive noise decoding (GRAND) with quantized soft input. First, we analyze the achievable rate of ordered reliability bits GRAND (ORBGRAND), which uses the rank order of the reliability as quantized soft information. We show that multi-line ORBGRAND can approach capacity for any signal-to-noise ratio (SNR). We then introduce discretized soft GRAND (DSGRAND), which uses information from a conventional quantizer. Simulation results show that DSGRAND well approximates maximum-likelihood (ML) decoding with a number of quantization bits that is in line…
Reed-Muller (RM) codes are conjectured to achieve the capacity of any binary-input memoryless symmetric (BMS) channel, and are observed to have a comparable performance to that of random codes in terms of scaling laws. On the negative side, RM codes lack efficient decoders with performance close to that of a maximum likelihood decoder for general parameters. Also, they only admit certain discrete sets of rates. In this paper, we focus on subcodes of RM codes with flexible rates that can take any code dimension from 1 to n, where n is the blocklength. We first extend the recursive projection-ag…
We review the recently introduced soft-aided bit-marking (SABM) algorithm and its suitability for product codes. Some aspects of the implementation of the SABM algorithm are discussed. The influence of suboptimal channel soft information is also analyzed.
