Fast Decoding Algorithms Research Articles

Lossless text compression by means of binary-coded ternary number representation

A lossless data compression code based on a binary-coded ternary number representation is investigated. The code is complete, universal, synchronizable, and supports a direct search in a compressed file. A simple monotonous encoding and very fast decoding algorithms are constructed owing to code properties. Experiments show that in natural language text compression, the new code outperforms byte-aligned codes SCDC and RPBC either in compression ratio or in decoding speed.

Fast list decoders for polarization‐adjusted convolutional (PAC) codes

AbstractA latest coding scheme named polarization‐adjusted convolutional (PAC) codes is shown to approach the dispersion bound for the code (128,64) under list decoding. However, to achieve the near‐bound performance, the list size of list decoding needs to be excessively large, which leads to insufferable time complexity. In this paper, to improve the speed of list decoding, fast list decoders for PAC codes are proposed. Four types of constituent nodes are defined and fast list decoding algorithms are provided for each of them. Simulation results present that fast list decoding with three types of constituent nodes can yield exactly the same error‐correction performance as list decoding, and reduce more than 50% time steps for the code (128,64). Moreover, fast list decoding with four types of constituent nodes can further reduce decoding complexity with negligible performance degradation.

A Low-Complexity Sorting Network for a Fast List Polar Decoder

Fast list decoding algorithms for polar codes have been proposed to achieve low latency and high error correction performance. In the Rate-1 and SPC nodes of fast list decoding, a high-complexity sorter is required if multiple bits are to be decoded simultaneously in a single clock cycle. This paper proposes a partitioned sorting network (PSN) that reduces the complexity of metric sorter in Rate-1 and single parity check (SPC) decoders for fast list decoding. The proposed PSN consists of two sorting networks. For the first sorting network, we analyze the order of the path metrics (PMs) and reduce the number of input candidate paths. Furthermore, for the second sorting network, we analyze the cumulative results of path selection and identify relatively frequently selected paths. The proposed PSN has up to 90% fewer compare-and-swap units (CASUs) and up to 137% higher operating frequencies than existing sorting networks for Rate-1 and SPC decoders with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L=8$ </tex-math></inline-formula> .

Reinforcement learning decoders for fault-tolerant quantum computation

Topological error correcting codes, and particularly the surface code, currently provide the most feasible road-map towards large-scale fault-tolerant quantum computation. As such, obtaining fast and flexible decoding algorithms for these codes, within the experimentally realistic and challenging context of faulty syndrome measurements, without requiring any final read-out of the physical qubits, is of critical importance. In this work, we show that the problem of decoding such codes can be naturally reformulated as a process of repeated interactions between a decoding agent and a code environment, to which the machinery of reinforcement learning can be applied to obtain decoding agents. While in principle this framework can be instantiated with environments modelling circuit level noise, we take a first step towards this goal by using deepQ learning to obtain decoding agents for a variety of simplified phenomenological noise models, which yield faulty syndrome measurements without including the propagation of errors which arise in full circuit level noise models.

Fast Block Sequential Decoding of Polar Codes

A reduced complexity sequential decoding algorithm for polar (sub) codes is described. The proposed approach relies on a decomposition of the polar (sub) code being decoded into a number of outer codes, and on-demand construction of codewords of these codes in the descending order of their probability. Construction of such codewords is implemented by fast decoding algorithms, which are available for many codes arising in the decomposition of polar codes. Further complexity reduction is achieved by taking hard decisions of the intermediate LLRs, and avoiding decoding of some outer codes. Data structures for sequential decoding of polar codes are described. The proposed algorithm can be also used for decoding of polar codes with CRC and short extended BCH codes. It has lower average decoding complexity compared with the existing decoding algorithms for the corresponding codes.

Randomized symmetric McEliece cryptosystem based on generalized Reed-Solomon codes

One of the actual problems of modern cryptography is the design of practical post-quantum cryptosystems whose security is based on the complexity of solving one computationally challenging problem, in the same way as the security of the RSA cryptosystem is based on the complexity of integer factorization. A promising class of such cryptosystems is formed by code-based cryptosystems the first asymmetric of which is the McEliece cryptosystem. The purpose of this work is to design and research a symmetric version of the McEliece cryptosystem based on generalized Reed-Solomon codes. These codes were chosen because they exist for all natural values of the parameters (the length and dimension of the code) and they are maximal distance separable allowing a wide range to change the characteristics of the relevant cryptosystems. In addition, very fast decoding algorithms are known for these codes. Asymmetric cryptosystems based on the generalized Reed-Solomon codes are not secure because for them there are efficient algorithms for recovering private keys from public keys.A symmetric code cryptosystem is proposed that is more efficient (in terms of the length of the secret key for given security requirements) compared to the LPN-C cryptosystem. An estimate of the security of the proposed cryptosystem relative to an attack with the chosen plaintext is obtained and an algorithm for selecting parameters for constructing this cryptosystem is proposed. The proposed cryptosystem is compared with the LPN-C cryptosystem along the key length for a given lower limit of security to the attack in question.

Multi-bit sliding stack decoding algorithm for OVXDM

Overlapped x domain multiplexing (OVXDM) is a promising encoding technique to obtain high spectral efficiency by utilizing inter-symbol interference (ISI) intelligently. However, the computational complexity of maximum likelihood sequence detection (MLSD) increases exponentially with the growth of spectral efficiency in OVXDM, which is unbearable for practical implementations. In this paper, based on a novel path metric associating adjacent symbols, we propose a multi-bit sliding stack decoding (Multi-Bit SSD) algorithm to achieve multiple-bit decoding simultaneously in OVXDM. Theoretical analysis is provided for the algorithm, which indicates the relationship between the performance and parameters including multiplexing waveform, overlapping fold and sliding window size. Simulation results show that the proposed algorithm can achieve better decoding performance and higher spectral efficiency than conventional fast decoding algorithms.

Fast decoding algorithms for geometric coded apertures

Fast decoding algorithms are described for the class of coded aperture designs known as geometric coded apertures which were introduced by Gourlay and Stephen. When compared to the direct decoding method, the algorithms significantly reduce the number of calculations required when performing the decoding for these apertures and hence speed up the decoding process. Experimental tests confirm the efficacy of these fast algorithms, demonstrating a speed up of approximately two to three orders of magnitude over direct decoding.

Fast decoding algorithms for coded aperture systems

Fast decoding algorithms are described for a number of established coded aperture systems. The fast decoding algorithms for all these systems offer significant reductions in the number of calculations required when reconstructing images formed by a coded aperture system and hence require less computation time to produce the images. The algorithms may therefore be of use in applications that require fast image reconstruction, such as near real-time nuclear medicine and location of hazardous radioactive spillage. Experimental tests confirm the efficacy of the fast decoding techniques.

A Novel High Spectral Efficiency Waveform Coding-OVTDM

Different from other Coding, based on the Overlapped Multiplexing Principle discovered by author， a novel OVTDM (Overlapped Time Division Multiplexing) Waveform Coding is proposed. Instead of the encoding matrix and mapped signal constellation, any engineering sense band-limited Multiplexing Waveform can be employed. By its data weighted shift overlapped versions, the coding gain and spectral efficiency are both achieved. The heavier the overlap of the data weighted Multiplexing Waveform, the higher the coding gain and spectral efficiency as well as the closer the output to the optimum complex Gaussian distribution. The encoder structures, parameters, optimum and fast decoding algorithms, pre-coding, some implementation problems as well as the bit error performance are estimated and discussed. Simulations show that OVTDM is suitable for high spectral efficiency applications and its spectral efficiency is roughly proportional to SNR.

Fast decoding algorithms for variable-lengths codes

Data compression has been widely applied in many data processing areas. Compression methods use variable-length codes with the shorter codes assigned to symbols or groups of symbols that appear in the data frequently. There exist many coding algorithms, e.g. Elias-delta codes, Fibonacci codes and other variable-length codes which are often applied to encoding of numbers. Although we often do not consider time consumption of decompression as well as compression algorithms, there are cases where the decompression time is a critical issue. For example, a real-time compression of data structures, applied in the case of the physical implementation of database management systems, follows this issue. In this case, pages of a data structure are decompressed during every reading from a secondary storage into the main memory or items of a page are decompressed during every access to the page. Obviously, efficiency of a decompression algorithm is extremely important. Since fast decoding algorithms were not known until recently, variable-length codes have not been used in the data processing area. In this article, we introduce fast decoding algorithms for Elias-delta, Fibonacci of order 2 as well as Fibonacci of order 3 codes. We provide a theoretical background making these fast algorithms possible. Moreover, we introduce a new code, called the Elias–Fibonacci code, with a lower compression ratio than the Fibonacci of order 3 code for lower numbers; however, this new code provides a faster decoding time than other tested codes. Codes of Elias–Fibonacci are shorter than other compared codes for numbers longer than 26 bits. All these algorithms are suitable in the case of data processing tasks with special emphasis on the decompression time.

A class of generalized LDPC codes with fast parallel decoding algorithms

Given the Tanner graph of a generalized low-density parity-check (GLDPC) code, the decoding complexity is mainly dominated by the decoding algorithm of subcodes. In this paper, we propose a class of GLDPC codes with fast parallel decoding algorithm. The parity-check matrices of the newly constructed subcodes are composed of several square matrices, which can be viewed as elements of a finite field. Therefore the FFT-based a posteriori probability (APP) algorithm for nonbinary codes can be applied to decode the subcodes. When compared with the trellis-based APP algorithm, the FFT-based APP algorithm can be implemented in parallel and has lower complexity. Simulation results show that the proposed GLDPC codes perform well on AWGN channels.

Enhanced Multiple-Bit Differential Detection of DOQPSK

In this paper, we consider multiple-bit differential detection (MBDD) of differential offset quadrature phase-shift keying (DOQPSK) with conventional bit-wise differential encoding (DE). We propose two novel MBDD schemes that outperform Simon's MBDD and approach the performance of coherent detection (CD) as the observation window size N increases. The first scheme uses the full available detection window spanning N-1 bit intervals, and is referred to as F-MBDD. A careful bit-error rate (BER) analysis shows that a significant performance improvement can be realized by reducing the detection window size to N-3 bit intervals. The resulting improved MBDD (I-MBDD) of DOQPSK with bit-wise DE achieves a similar performance as Phoel's MBDD of DOQPSK with more complex symbol-wise DE. For both F-MBDD and I-MBDD, tight upper bounds and accurate BER approximations are derived. In addition, since the complexity per bit decision of brute-force decoding of MBDD of DOQPSK grows exponentially with N, we develop optimum and suboptimum fast decoding algorithms with complexities per bit decision of orders log(N+1) and N, respectively.

Codes in the Vandermonde F -Metric and Their Application

F-metrics are metrics based on projective sets. In this paper, construction of optimal codes for a special F-metric associated with a generalized Vandermonde matrix is given. Encoding and fast decoding algorithms are described. A public-key cryptosystem is considered as an example of a possible application of codes constructed.

Fast decoding algorithms for first order Reed-Muller and related codes

Fast decoding algorithms for first order Reed-Muller and related codes

Fast Decoding Algorithms for First Order Reed-Muller and Related Codes

Fast decoding algorithms for short codes based on modifications of maximum likelihood decoding algorithms of first order Reed-Muller codes are described. Only additions-subtractions, comparisons and absolute value calculations are used in the algorithms. Soft and hard decisions maximum likelihood decoding algorithms for first order Reed-Muller and the Nordstrom-Robinson codes with low complexity are proposed.

Classical codes as ideals in group algebras

The theory of algebraic codes is a perfect illustration of the fact that the more mathematical structure one is able to add to a system, the better are the descriptions one obtains: Changing from codes to linear codes, it is no longer necessary to compute all distances between any two codewords, only the codeword weights, in order to find the minimum distance. Going from linear codes to cyclic codes then, the linear structure is replaced by a much richer algebraic structure, and general results on properties of codes such as the BCH-bound, fast decoding algorithms, etc., may be developed.

Real-valued error control coding by using DCT

The discrete cosine theorem (DCT)-based real-valued linear code is derived for the first time in the literature. A BCH-like subclass of DCT linear codes is also developed, for which fast decoding algorithms exist. It is shown that the DCT-based error control codes proposed by the authors can be viewed as a bridge to link the fields of source coding and channel coding.