Algebraic Decoding Research Articles

An Improvement for Error-Correcting Pairs of Some Special MDS Codes

The error-correcting pair is a general algebraic decoding method for linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extensions, we focus on MDS linear codes. Recently, He and Liao showed that for an MDS linear code [Formula: see text] with minimum distance [Formula: see text], if it has an [Formula: see text]-error-correcting pair, then the parameters of the pair have three possibilities. Moreover, for the first case, they gave a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and for the other two cases, they only gave some counterexamples. For the second case, in this paper, we give a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and then basing on the Product Singleton Bound, we prove that there are two cases for such pairs, and then give some counterexamples basing on twisted generalized Reed–Solomon codes for these cases.

Proximity Gaps for Reed–Solomon Codes

A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are δ-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are δ-close to the property. In particular, no set in the collection has roughly half of its members δ-close to the property and the others δ-far from it. We show that the collection of affine spaces displays a proximity gap with respect to Reed–Solomon (RS) codes, even over small fields, of size polynomial in the dimension of the code, and the gap applies to any δ smaller than the Johnson/Guruswami–Sudan list-decoding bound of the RS code. We also show near-optimal gap results, over fields of (at least) linear size in the RS code dimension, for δ smaller than the unique decoding radius. Concretely, if δ is smaller than half the minimal distance of an RS code V ⊂ 𝔽 q n , then every affine space is either entirely δ-close to the code or, alternatively, at most an ( n/q )-fraction of it is δ-close to the code. Finally, we discuss several applications of our proximity gap results to distributed storage, multi-party cryptographic protocols, and concretely efficient proof systems. We prove the proximity gap results by analyzing the execution of classical algebraic decoding algorithms for Reed–Solomon codes (due to Berlekamp–Welch and Guruswami–Sudan) on a formal element of an affine space. This involves working with Reed–Solomon codes whose base field is an (infinite) rational function field. Our proofs are obtained by developing an extension (to function fields) of a strategy of Arora and Sudan for analyzing low-degree tests.

Soft Decision Decoding with Cyclic Information Set and the Decoder Architecture for Cyclic Codes

The soft decision decoding algorithm for cyclic codes, especially the maximum likelihood (ML) decoding algorithm, can obtain significant performance superior to that of algebraic decoding, but the complexity is much higher. To deal with this problem, an improved soft decision decoding algorithm based on a cyclic information set and its efficient implementation architecture are proposed. This algorithm employs the property of the cyclic codes to generate a series of cyclic information sequences by circularly shifting, constructing the cyclic information set. Then, a limited number of candidate information sequences are efficiently generated using an iterative computation method, and the candidate codewords are generated using the very concise encoding method of the cyclic codes. Furthermore, the efficient hardware architecture based on systolic arrays is also proposed to generate candidate information sequences and to select the optimal candidate codewords. An emulation platform is constructed to verify the error correction performance and to determine the optimal decoder parameters. Emulation results indicate that, with appropriate parameter selection, the proposed decoding algorithm can achieve a bit error rate approaching the ML performance while maintaining low complexity.

The error-correcting pair for TGRS codes

Twisted generalized Reed-Solomon codes are the generalization of generalized Reed-Solomon codes, due to their good algebraic structure and applications in cryptography, which have been widely studied in recent years. As a general algebraic decoding method for linear codes, the error-correcting pair exists in many classical linear codes. In this paper, we extend the error-correcting pair to twisted generalized Reed-Solomon codes. For twisted generalized Reed-Solomon codes with twist 1 or 2, we constructively show that the existence of the error-correcting pair. Moreover, we introduce the corresponding decoding procedure of twisted generalized Reed-Solomon codes by ℓ-error-correcting pairs and give several examples.

On the error-correcting pair for MDS linear codes with even minimum distance

The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extension, we focus our study on MDS linear codes. It is well-known that an MDS linear code with minimum distance 2ℓ+1 has an ℓ-error-correcting pair if and only if it is a generalized Reed-Solomon code. In this paper, we show that for an MDS linear code C with minimal distance 2ℓ+2, if it has an ℓ-error-correcting pair, then the parameters of the pair are three cases. For one case, we give a necessary condition that C is a generalized Reed-Solomon code, and then give some counterexamples that C is a non-generalized Reed-Solomon code for the other two cases.

Efficient Decoding Algorithm for Binary Quadratic Residue Codes Using Reduced Permutation Sets

The Quadratic Residue (QR) codes have a rich mathematic structure. Unfortunately, their Algebraic Decoding (AD) is not generalizable for all QR codes. In this study, an efficient hard decoding algorithm is proposed to generalize the decoding of the binary systematic Quadratic Residue (QR) codes. The proposed decoder corrects t erroneous bits or less, in the received word, based on a reduced set of permutations derived from the large automorphism group of QR codes. This set of permutations is applied to the received word to move the error positions and trap all of them in redundancy. Then, to evaluate the proposed method, we applied it to many binary QR codes of moderate code length starting with 17 until 113 with reducible and irreducible generator polynomials. The proposed decoder was validated by inserting all possible error patterns, that have t or less erroneous positions, as input of the proposed decoder and the output is always a correct codeword. The complexity study, in terms of the number of operations used, reveals that the light permutation decoding LPD algorithm significantly decreases decoding complexity without performance loss. So, it is qualified to be a good competitor to decode QR codes with lower lengths but is the best for QR codes with higher lengths.

Reduced complexity hard‐ and soft‐input BCH decoding with applications in concatenated codes

Error correction coding for optical communication and storage requires high rate codes that enable high data throughput and low residual errors. Recently, different concatenated coding schemes were proposed that are based on binary BCH codes with low error correcting capabilities. In this work, low-complexity hard- and soft-input decoding methods for such codes are investigated. We propose three concepts to reduce the complexity of the decoder. For the algebraic decoding we demonstrate that Peterson's algorithm can be more efficient than the Berlekamp–Massey algorithm for single, double, and triple error correcting BCH codes. We propose an inversion-less version of Peterson's algorithm and a corresponding decoding architecture. Furthermore, we propose a decoding approach that combines algebraic hard-input decoding with soft-input bit-flipping decoding. An acceptance criterion is utilised to determine the reliability of the estimated codewords. For many received codewords the stopping criterion indicates that the hard-decoding result is sufficiently reliable, and the costly soft-input decoding can be omitted. To reduce the memory size for the soft-values, we propose a bit-flipping decoder that stores only the positions and soft values of a small number of code symbols. This method significantly reduces the memory requirements and has little adverse effect on the decoding performance.

Algebraic Decoding of Quasi-Reversible BCH Codes Using Band Matrices

A Bose-Chaudhuri-Hocquenghem (BCH) is called quasi-reversible if there are consecutive elements a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,⋯,a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> in the defining set, where a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> is a negative integer and a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> is a positive integer. A matrix in band form presented in this article involves the use of the Newton identities and symmetric polynomial identities. This matrix is the coefficient matrix of a linear system. The solution to such a linear system is the coefficients of an error-locator polynomial for quasi-reversible BCH codes. Its computational complexity is significantly lower than BCH decoding using the linear system with coefficient matrix in column echelon form. This article also proposes a new partial syndrome matrix with radical locators and a lot of zeros. Such a matrix is still in band form. The expansion of its matrix determinant is exactly a radical-locator polynomial. Compared to the former radical-locator polynomials in the literature, the newly discovered radical-locator polynomial not only requires less storage memory but also results in computational benefits. Surprisingly, the resulting polynomials are much sparser than those for the narrow-sense BCH codes. Finally, a complete algebraic decoding algorithm for quasi-reversible BCH codes is provided. The error-locator and radical-locator polynomials of degree less than or equal to 5 are listed.

Estimation of hardware and time redundancy when using an additive error vector to correct burst errors in digital data transmission systems

In digital data transmission systems, to improve noise immunity, cyclic codes are widely used, detecting and correcting byte (packet) errors. An error packet is understood to mean errors whose multiplicity does not exceed the number of bits b of the information block. Cyclic codes are used to correct byte errors. The most effective method for correcting byte errors are Reed-Solomon codes, which allow correcting errors in a given number of bytes of information. The main problem of using cyclic (sequential) codes is a long delay time associated with the need to perform a division operation to obtain the remainder, which is not acceptable for digital data transmission systems operating in real time. For example, when using the Reed-Solomon code with a code set length of 69 information bits, the implementation of decoding according to the Euclidean algorithm requires 96 clock cycles, which cannot ensure the channel operation in real time. To eliminate this drawback, one should use codes that correct burst errors that implement an algebraic coding procedure with syndromic decoding of information. However, replacing the cyclic procedure for encoding (decoding) information with a syndromic one leads to a sharp increase in hardware costs associated with the use of a memory unit in the decoder for storing error vector values and a decoder for generating error addresses in accordance with the resulting syndrome. Thus, there is a need to develop a methodological apparatus associated with the construction of a code that corrects errors in a given number of bytes of information and an estimate of the hardware and time costs associated with this purpose. In this work, the need to develop a methodological apparatus associated with the construction of a code that corrects errors in a given number of bytes of information with algebraic syndromic decoding and an assessment of the hardware and time costs associated with this purpose is substantiated. The paper presents the rules for constructing a correcting code that corrects errors in a given number of bytes of information, which implements a linear procedure for constructing a correcting code with syndromic decoding and using an additive error vector, which made it possible to reduce hardware costs for constructing a decoding device (to reduce the amount of memory for storing error vector values). For the developed method for correcting byte errors, expressions for evaluating the number are obtained: checking discharges; additive error vectors that do not require their storage in a memory block; error vectors, for burst errors that occur in adjacent bytes at the same time and require their values to be stored in a memory block. A comparative assessment of hardware and time redundancy in the implementation of the proposed method for correcting packet errors with existing methods is carried out. The proposed method of error correction in a given number of bytes of information with additive formation of the error vector differs from the existing ones in that it allows: carry out the correction of burst errors with algebraic-syndromic decoding (exclude the cyclical procedure for encoding and decoding information); to reduce hardware costs for building a decoding device, since in most cases does not require hardware costs for storing error vectors; to reduce the time spent on encoding and decoding information and to ensure the operation of the data transmission channel in real time; to increase the reliability of the transmitted information by detecting uncorrectable byte errors. Thus, the proposed method for correcting errors in a given number of bytes of information with additive formation of an error vector has a regular and relatively simple procedure for constructing a code, which allows one to reduce the hardware and time costs for encoding and decoding information.

BCH CODES DECODER BASED ON EUCLID ALGORITHM

In the process of algebraic decoding of BCH codes over the field GF(q) with the word length n = qm-1, correcting t errors, both in the time and frequency domains, it is necessary to find the error locator polynomial ?(x) as the least polynomial for which the key equation. Berlekamp proposed a simple iterative scheme, which was called the Berlekamp-Messi algorithm, and is currently used in most practical applications. Comparative statistical tests of the proposed decoder and decoder using the Berlikamp-Messi algorithm showed that they differ slightly in decoding speed. The proposed algorithm is implemented in the environment in Turbo Pascal and can be used for the entire family of BCH codes by replacing the primitive Galois polynomial.

On Decoding Algebraic Codes Using Radical Locators

It is well-known that the decoding of algebraic codes with error locators have been extensively conceived in the literature over a half century. The radical locator, which is an $\omega $ th root of error locator, has recently been discovered. This paper focuses on the two classes of radical locators. The first is complete radical locators , where all the locators are assigned to radical locators. The second is partial radical locators in that there are only a few locators being radical locators. In the former case based on complete radical locators, a new square matrix whose determinant is a univariate radical-locator polynomial is proposed. In particular, this matrix is modified to allow a large square matrix. It can be transformed by Gaussian elimination to the matrix in a row-echelon form in which zeros appearing in the diagonal entries of the current matrix are able to determine errors implicitly. Furthermore, this work further extends the previous results from the univariate case to multivariate cases. The matrix methods found herein enable one to decode a large class of cyclic codes efficiently. In the latter case, the cyclotomic cosets are a practical approach to select a small subset of positive integers. They are employed to develop partial radical locators. Finally, in contrast with the complete radical locators, the algebraic decoding methods make a natural use of partial radical locators more widely and flexibly to arbitrary cyclic codes.

Peterson–Gorenstein–Zierler algorithm for differential convolutional codes

Differential Convolutional Codes with designed Hamming distance are defined, and an algebraic decoding algorithm, inspired by Peterson–Gorenstein–Zierler’s algorithm, is designed for them.

Coding Scheme for High-Order QAM Modulations in the Manhattan Metric

In this letter, a code construction in the Manhattan metric with a polynomial complexity algebraic decoding algorithm is proposed. It is shown that the code construction combined with a high-order QAM modulation enjoys a better code rate compared to a binary BCH code along with a competitive decoding procedure complexity. A performance gain from the use of the proposed code construction increases with the growth of an order of a QAM modulation. The code construction is mostly perspective in scenarios with high signal-to-noise ratios and decoding latency constraints, which are required for current and further generations of communications systems.

Power error locating pairs

We present a new decoding algorithm based on error locating pairs and correcting an amount of errors exceeding half the minimum distance. When applied to Reed–Solomon or algebraic geometry codes, the algorithm is a reformulation of the so-called power decoding algorithm. Asymptotically, it corrects errors up to Sudan’s radius. In addition, this new framework applies to any code benefiting from an error locating pair. Similarly to Pellikaan’s and Kotter’s approach for unique algebraic decoding, our algorithm provides a unified point of view for decoding codes with an algebraic structure beyond the half minimum distance. It permits to get an abstract description of decoding using only codes and linear algebra and without involving the arithmetic of polynomial and rational function algebras used for the definition of the codes themselves. Such algorithms can be valuable for instance for cryptanalysis to construct a decoding algorithm of a code without having access to the hidden algebraic structure of the code.

Algebraic decoding of the (41, 21, 9) quadratic residue code without determining the unknown syndromes

Quadratic residue (QR) codes have a high prospect on error correction for reliable data conveyance over channel with noise. This paper presents a fast algebraic scheme for decoding the binary (41, 21, 9) QR code for correcting up to 4 errors on an one-case-by-one-case basis, in which the calculation of unknown syndromes is eliminated and the conditions for checking for the various numbers of errors that exist in the received word are also simplified compared to those from Lin T. C. et al.’s algorithm, which is a traditional algebraic decoding algorithm (ADA). The computational complexity of the decoder performing the binary (41, 21, 9) QR code is analyzed thoroughly, which demonstrates that the proposed decoding scheme is faster, simpler, and more suitable for implementation than Lin T. C. et al.’s algorithm. Numerical emulation results demonstrate that the proposed decoding scheme achieves the same error-rate performance as Lin T. C. et al.’s algorithm, but it has a significantly decreased the complexity of this decoder in terms of the CPU time.

Module minimisation based low‐complexity soft decoding of Reed–Solomon codes

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, the algebraic Chase decoding and the Koetter-Vardy decoding. Their computationally expensive interpolation is solved by the module minimisation (MM) technique which consists of basis construction and basis reduction. Compared with Koetter's interpolation, the MM interpolation yields a smaller computational cost for the two ASD algorithms. Re-encoding transform is further applied to reduce the decoding complexity by reducing the degree of module generators. Based on assessing the degree of module seeds, a complexity reducing approach is introduced to further facilitate the two ASD algorithms. Computational cost of the two algorithms as well as their re-encoding transformed variants will be analysed. Performance of the two ASD algorithms will be compared under decoding expenditure benchmark, providing more practical insights of their applications.

Дослідження ефективності адаптивного методу декодування алгебраїчних згорткових кодів перемежування

It is shown that algebraic interleaved convolutional codes are widely used in the construction of telecommunication systems. The mathematical model of telecommunication system has been developed to evaluate the performance of adaptive and algebraic decoding of these codes in the communication channel with faction of errors. The scheme and purpose of the basic elements of this mathematical model are presented. The developed model takes into account the features of the transmission of information using algebraic interleaved convolutional codes depending on the parameters of the selected code, characteristics of interference in the communication channel and other factors. The process of information processing in each element of the presented mathematical model of telecommunication system is considered. The settings for the software implementation of the developed model are given: code type, code parameters, signal/noise ratio, number of information messages, parameters of the adaptive decoding method. As a result of the conducted researches it is determined that for given values of error coefficient and selected algebraic interleaved convolutional codes the energy gain from encoding when applying adaptive decoding method in comparison with algebraic decoding method is from 4.1 dB to 6.5 dB. However, reducing the encoding speed and increasing the depth of interleaving leads to a decrease in the energy efficiency of the encoding using the adaptive decoding method. It also follows from the obtained results that in the specified signal/noise ratio for the communication channel with Rayleigh scattering the adaptive decoding method compared to the algebraic method provides an average energy gain from encoding of about 5.5 dB.

Generalized Subspace Subcodes With Application in Cryptology

Most codes with an algebraic decoding algorithm are derived from Reed-Solomon codes. They are obtained by taking equivalent codes, for example, generalized Reed-Solomon codes, or by using the so-called subfield subcode method, which leads to alternant codes over the underlying prime field, or over some intermediate subfield. The main advantage of these constructions is to preserve both the minimum distance and the decoding algorithm of the underlying Reed-Solomon code. In this paper, we explore in detail the subspace subcodes construction. This kind of codes was already studied in the particular case of cyclic Reed-Solomon codes. We extend this approach to any linear code over the extension of a finite field. We are interested in additive codes who are deeply connected to subfield subcodes. We characterize the duals of subspace subcodes. We introduce the notion of generalized subspace subcodes. We apply our results to generalized Reed-Solomon codes which leads to codes with interesting parameters, especially over a large alphabet. To conclude this paper, we discuss the security of the use of generalized subspace subcodes of Reed-Solomon codes in a cryptographic context.

Analysis of efficiency of the bioinspired method for decoding algebraic convolutional codes

It has been shown that convolutional codes are widely used, along with various decoding methods, to improve the reliability of information transmission in wireless telecommunication systems. The general principles of synthesis and the parameters and algebraic non-systematic convolutional codes with arbitrary coding rate and maximum achievable code distance have been shown. The basic stages of the bioinspired method for decoding algebraic convolutional codes using a random shift mechanism have been presented. It has been shown that the essence of the presented decoding method implies applying the procedure of differential evolution with the heuristically determined parameters. In addition, this method uses information about the reliability of the adopted symbols to find the most reliable basis for the generalized generator matrix. The mechanism of random shift for the modification of the accepted sequence is additionally applied for the bioinspired search based on various most reliable bases of a generalized generator matrix. The research results established that the bioinspired method for decoding algebraic convolutional codes ensures greater efficiency compared with the algebraic decoding method in the communication channel with additive white Gaussian noise. Depending on the parameters of the algebraic convolutional code and the necessary error coefficient, the energy gain from encoding ranges from 1.6 dB to 3 dB. It was shown that the presented bioinspired decoding method can be used for convolutional codes with a large code constraint length. In doing so, the presented method for decoding algebraic convolutional codes is less efficient than the Viterbi decoding method and turbo codes at a sufficient number of decoding iterations

Soft‐input bit‐flipping decoding of generalised concatenated codes for application in non‐volatile flash memories

Error correction coding based on soft-input decoding can significantly improve the reliability of non-volatile flash memories. This work proposes a soft-input decoder for generalised concatenated (GC) codes. GC codes are well suited for error correction in flash memories for high reliability data storage. The authors propose GC codes constructed from inner extended binary Bose–Chaudhuri–Hocquenghem (BCH) codes and outer Reed–Solomon (RS) codes. The extended BCH codes enable an efficient hard-input decoding. Furthermore, a low-complexity soft-input decoding method is proposed. This bit-flipping decoder uses a fixed number of test patterns and an algebraic decoder for soft-decoding. An acceptance criterion for the final candidate codeword is proposed. Combined with error and erasure decoding of the outer RS codes, this acceptance criterion can improve the decoding performance and reduce the decoding complexity. The presented simulation results show that the proposed bit-flipping decoder in combination with outer error and erasure decoding can outperform maximum-likelihood decoding of the inner codes.