BCH Codes Research Articles

Construction of reversible self-dual codes

We study a construction method of binary reversible self-dual codes in this paper. Reversible codes have good properties in applications, and it is interesting to note that a class of reversible codes is closely connected to BCH codes and LCD codes. We first characterize binary reversible self-dual codes. Using these characteristics of reversible self-dual codes, we find an explicit method for constructing all the binary reversible self-dual codes up to equivalence. Furthermore, using this construction, we obtain nine optimal reversible self-dual codes of length 70 which are all inequivalent, and these codes are all new with respect to binary self-dual codes; they all have the same parameter [70,35,12] and their automorphism groups have the same order two.

The properties and parameters of generic Bose – Chaudhuri – Hocquenghem codes

The Bose – Chaudhuri – Hocquenghem type of linear cyclic codes (BCH codes) is one of the most popular and widespread error-correcting codes. Their close connection with the theory of Galois fields gave an opportunity to create a theory of the norms of syndromes for BCH codes, namely, syndrome invariants of the G-orbits of errors, and to develop a theory of polynomial invariants of the G-orbits of errors. This theory as a whole served as the basis for the development of effective permutation polynomial-norm methods and error correction algorithms that significantly reduce the influence of the selector problem. To date, these methods represent the only approach to error correction with non-primitive BCH codes, the multiplicity of which goes beyond design boundaries. This work is dedicated to a special error-correcting code class – generic Bose – Chaudhuri – Hocquenghem codes or simply GBCH-codes. Sufficiently accurate evaluation of the quantity of such codes in each length was produced during our work. We have investigated some properties and connections between different GBCH-codes. Special attention was devoted to codes with constructive distances of 3 and 5, as those codes are usual for practical use. Their almost complete description is given in the range of lengths from 7 to 107. The paper contains a fairly clear theoretical classification of GBCH-codes. Special attention is paid to the corrective capabilities of the codes of this class, namely, to the calculation of the minimal distances of these codes with various parameters. The codes are found whose corrective capabilities significantly exceed those of the well-known GBCH-codes with the same design parameters.

Two Families of Optimal Linear Codes and Their Subfield Codes

In this paper, a family of $[{q}^{2}-1, 4, {q}^{2}-{q}-2]$ cyclic codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2}-1,{q}^{2}-5,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of cyclic codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$ . Some of the subfield codes are optimal and some have the best known parameters. It is shown that the subfield codes are equivalent to a family of primitive BCH codes and thus the parameters of the BCH codes are solved. The duals of the subfield codes are also optimal with respect to the sphere-packing bound. A family of $[{q}^{2}, 4, {q}^{2}-{q}-1]$ linear codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2},{q}^{2}-4,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of linear codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$ . Five infinite families of 2-designs are also constructed with three families of linear codes of this paper.

Comparison of communication performance of block coding methods in different channels using DWT

In this study, the audio transmission performance of some block coding methods in wavelet and non-wavelet cases is compared. For this purpose, a communication system has been established in the MATLAB environment for the transmission of the audio signal for 5 s. The comparison of the transmission performance of the block codes was performed separately in different parameters of additive white Gaussian noise, Rician and Rayleigh channels. At the same time, it is determined which discrete wavelet transform (DWT) family is suitable in terms of transmission performance for block coding methods and how DWT level affects transmission performance is investigated. Four DWT families at different levels were applied to the audio signal for linear block coding, Hamming, Cyclic, Bose–Chaudhuri–Hocquenghem, Reed–Solomon, encoded channels, and bit error rate and mean squared error performances were compared.

ON THE DESIGN OF BCH CODES POLYNOMIALS FOR DIGITAL TELEVISION DVB-T2 BROADCASTING SYSTEMS

Digital Video Broadcasting Terrestrial{Second Generation (DVB-T2) requires a high coding rate to transmit data of high-quality video. This paper evaluates Bose-Chaudhuri-Hocquenghem (BCH) codes for the DVB-T2 to measure the gain of BCH codes. This paper evaluates the BCH codes as outer coding to be combined with other different inner encoding schemes such as Low Density Parity Check (LDPC) codes and convolutional codes to measure the best suitable inner encoding scheme. This paper also studies the performances of BCH codes for DVB-T2 specified by two different standards, i.e., (a) The European Telecommunications Standards Institute (ETSI) Technical Specification (TS) 102 831 and (b) ETSI European Standard (EN) 302 755. To obtain better error correcting capability, we propose new BCH polynomials based on the general guideline from ETSI TS 102 831 for Galois Field GF(214). We perform computer simulations to evaluate bit-error-rate (BER) performances under additive white Gaussian noise (AWGN) channel and Indonesia DVB-T2 channel model. We revealed the superiority of BCH codes in high data rate transmission, which is required for DVB-T2, and found that BCH codes are better suited to the LDPC codes as inner encoding rather than to the convolutional codes. We also confirmed that BCH codes of DVB-T2 from ETSI TS 102 831 using the proposed BCH polynomials have better performances compared to the standard polynomial of ETSI EN 302 755. We are expecting that the obtained polynomials can be adopted by the BCH codes of Indonesia DVB-T2 system. These results are expected to support the Indonesian government in determining the parameters of the BCH codes of DVB-T2 for Indonesia.

Quantum BCH codes with maximum designed distance

In this paper, we investigate all coset leaders of primitive BCH codes for $$\delta$$ in the range $$1\le \delta \le q^\frac{m+7}{2}$$ , which extends Liu and Shi’s results. Besides, we also generalize Shi’s results by proposing the maximum designed distance of non-narrow-sense( $$b=k_2q^2+k_1q+k_0$$ ) primitive BCH codes which can contain their Euclidean dual. At the end, we calculate the dimension of the Euclidean dual containing non-narrow-sense primitive BCH codes and construct some quantum BCH codes.

The automorphisms and error orbits of Reed – Solomon codes

The purpose of this work with its results presented in the article was to develop and transfer to the class of Reed – Solomon codes (RS-codes) the basic provisions of the theory of syndrome norms (TNS), previously developed for the noise-resistant coding of the class of Bose – Chaudhuri – Hocquenghem codes (BCH-codes), which is actively used in theory and practice. To achieve this goal, a transition has been made in the interpretation of the theory of RS-codes from polynomial to matrix language. This approach allows you to fully use the capabilities of Galois field theory. The main difficulty of RS-codes is that they rely on a non-binary alphabet. The same factor is attractive for practical applications of RS-codes. The matrix language allows you to break the syndromes of errors into components that are elements of the Galois field – the field of definition of RS-codes. The TNS for BCH codes is based on the use of automorphisms of these codes – cyclic and cyclotomic substitutions. Automorphisms of RS-codes are studied in detail. The cyclic substitution belongs to the categories of automorphisms of RS-codes and generates a subgroup Г of order N (code length). The cyclotomic substitution does not belong to the class of automorphisms of RS-codes – the power of the alphabet greater than 2 prevents this. When expanding the concept of automorphism of a code beyond substitutions of coordinates of vectors to automorphisms of RS-codes, homotheties or affine substitutions can be attributed, since they also form a cyclic group A of order N. It is shown that cyclic and affine substitutions commute with each other, which, generally speaking, is not typical for linear operators and substitutions. The group Г of cyclic substitutions, the group A of affine substitutions, and the combined AГ group of order N2 generate 3 types of error orbits in RS-codes. The structure of the orbits of errors with respect to the action of groups A, Г and the combined group AГ is studied {231 words}.

A Deep Learning Assisted Node-Classified Redundant Decoding Algorithm for BCH Codes

This paper proposes a node-classified redundant decoding (NC-RD) algorithm based on the received sequence’s channel reliability for high-density parity-check (HDPC) codes. Two preprocessing steps are proposed prior decoding. The variable nodes of the parity-check matrix are firstly classified by the $k$ -median algorithm based on the number of shortest cycles associated with each variable node before decoding. Then, by searching among the automorphism group of the HDPC codes, we generate a list of permutations for bit positions by computing and sorting the permutation reliability metrics. The redundant decoder conducts the message-passing decoding according to the sorted permutations, which limit the unreliable information propagation for each permutation. Besides proposing a list decoding algorithm on top of the NC-RD algorithm to augment the decoder’s performance, we show that the NC-RD algorithm can be transformed into a neural network system. More specifically, multiplicative tuneable weights are attached to the decoding messages to optimize the decoding performance. Simulation results of BCH codes over the AWGN channels show that the NC-RD algorithm provides a performance gain compared to the random redundant decoding algorithm. Additional decoding performance gain can be obtained by both the list decoding method and the neural network “learned” NC-RD algorithm.

Reduced-Complexity Key Equation Solvers for Generalized Integrated Interleaved BCH Decoders

Generalized integrated interleaved (GII) codes nest linear block codewords to generate codewords belonging to stronger linear block codes. They can achieve very high throughput with excellent error-correcting capability. GII codes can be based on either Reed-Solomon (RS) or BCH codes. GII-BCH codes are the most promising candidate for error correction in next-generation terabit/s Flash and storage class memories, and optical communications. On the other hand, GII decoder implementation faces many challenges. In particular, the key equation solver (KES) in the nested decoding process causes not only clock frequency bottleneck but also large area. Although techniques have been developed recently to address these issues for GII-RS codes, they cannot be directly extended for binary GII-BCH codes if every other iteration of the nested KES is skipped to reduce the latency. Two instead of one higher-order syndromes need to be incorporated into each nested KES iteration and this makes the implementation much more challenging. In this paper, algorithmic reformulations and architectural modifications are developed to eliminate the clock frequency bottleneck and reduce the area of GII-BCH nested KES. Additionally, the number of processing elements is substantially reduced by analyzing the minimum number of coefficients to keep for the involved polynomials without undesirable degradation on the error-correcting performance. The critical path of the proposed scaled nested BCH KES architecture with reduced processing elements is only one multiplier and three adders. For an example GII-BCH code over ${GF}(2^{12})$ that has a ${t}_{v}=58$ -error-correcting code as the most powerful code generated by the nesting, our design also further reduces the area by 33%.

Deterministic construction of compressed sensing matrices from constant dimension codes

Compressed sensing theory provides a new approach to acquire data as a sampling technique and makes sure that sparse signals can be reconstructed from few measurements. The construction of compressed sensing matrices is a central problem in compressed sensing theory. In this paper, the deterministic sparse compressed sensing matrices from constant dimension codes are constructed and the coherence of the matrices are computed. Furthermore, the maximum sparsity of recovering the sparse signals by using our matrices is obtained. Meanwhile, the numerical simulations are made among our matrices from constant dimension codes, R. DeVore’s matrices, matrices from p-ary BCH codes and random matrices. Moreover, the deterministic sparse compressed sensing matrices can be constructed from certain Steiner structures. Finally, a general analysis indicates that the deterministic sparse compressed sensing matrices from constant dimension codes can recover signals of arbitrary sparsity order with any number of budged rows in some Euclid spaces with proper dimensions.

ON DECODING ALGORITHMS FOR GENERALIZED REED — SOLOMON CODES WITH ERRORS AND ERASURES

The article is devoted to the decoding algorithms for generalized Reed Solomon codes with errorsand erasures. These algorithms are based on Gao algorithm, Sugiyama algorithm, Berlekamp Massey algorithm (Peterson Gorenstein Zierler algorithm). The first of these algorithms belongs to syndrome-free decoding algorithms, the others to syndrome decoding algorithms. The relevance of these algorithms is that they are applicable for decoding Goppa codes, which are the basis of some promising post-quantum cryptosystems. These algorithms are applicable for Goppa codes over an arbitrary field, as opposed to the well-known Patterson decoding algorithm for binary Goppa codes.

Medical data transmission using the product of TLDPC and BCH error control coding systems with two interleavers

SummaryMedical data are very precious since it has the most sensitive data of patient which can save them. If it is handled wrongly or it is manipulated, then it could result in permanent problems or even it can result in fatal. So medical data widely used in a digital format with error detection and correction codes (EDC) in its transmission and storage process. EDC is a method of adding redundant bits to the static message bits to enhance the reliability of binary digital data broadcast while the medical data communication medium adds the errors in the course of transmission. In the proposed method, we use Bose–Chaudhuri–Hocquenghem codes (BCH) as an outline algorithm and trellis Low‐Density Parity Check code (TLDPC) as an inner algorithm along with two interleavers. Product of BCH code and TLDPC code with interleavers can control more errors as compared with its plain code and product of RS‐Hamming code. MATLAB used as a simulation tool. Plain BCH code is simulated first, and then plain LDPC code is simulated for the same information. Finally, the product of BCH and TLDPC with two interleavers is imposed on the same information. The result of our research work shows enhancement in bit error rate (BER) compared to other plain and product codes from 10−2 to 10−3. Also, frame error rate (FER) is improved in 0.5 scaling factors when compared with other trellis decoders. Hence, this proposed system can be utilized in an application where less error rate and high accuracy of information transmission between medical data processing systems are required.

A novel BCS code in a downlink LTE system over an LTE-MIMO channel

In general, a concatenated RS/BCH code consists of an outer RS code and an inner BCH code with separation by an interleaver. We create a novel BCS code using a new methodology by combining an outer BCH code with an inner RS code appended by an interleaver to randomize burst errors and help the RS code in correcting the errors. Generally, BCH codes handle binary data, whereas RS codes handle non-binary data. Hence, RS (7, 1) inner code is proposed to achieve compatibility with BCH (15, 5) outer codes and ultimately create a novel BCS code to handle binary data. The proposed BCS code is compared with familiar concatenated RS/BCH codes and single codes (RS and BCH) for BPSK and QPSK modulation schemes. Results show the downlink LTE system performance using the proposed BCS code is significantly better than the uncoded LTE system and single codes (RS and BCH). Moreover, the proposed BCS code also outperforms familiar concatenated RS/BCH codes for both schemes. In contrast, the system performance is improved further when the number of antennas in the MIMO channel is increased from $$2 \times 2$$ to $$4 \times 4$$ . Therefore, the proposed BCS code can be considered a stronger code with high reliability in wireless communication systems.

Syndrome-Coupled Rate-Compatible Error-Correcting Codes: Theory and Application

Rate-compatible error-correcting codes (ECCs), which consist of a set of extended codes, are of practical interest in both wireless communications and data storage. In this work, we first study the lower bounds for rate-compatible ECCs, thus proving the existence of good rate-compatible codes. Then, we propose a general framework for constructing rate-compatible ECCs based on cosets and syndromes of a set of nested linear codes. We evaluate our construction from two points of view. From a combinatorial perspective, we show that we can construct rate-compatible codes with increasing minimum distances, and we discuss decoding algorithms and correctable patterns of errors and erasures. From a probabilistic point of view, we prove that we are able to construct capacity-achieving rate-compatible codes, generalizing a recent construction of capacity-achieving rate-compatible polar codes. Applications of rate-compatible codes to data storage are considered. We design two-level rate-compatible codes based on Bose-Chaudhuri-Hocquenghem (BCH) and low-density parity-check (LDPC) codes which are two popular codes widely used in the data storage industry, and then we evaluate the performance of these codes in multi-level cell (MLC) flash memories. We also examine code performance on binary and $q$ -ary symmetric channels. Finally, we briefly discuss two variations of our main construction and their relative performance.

Some new results on dimension and Bose distance for various classes of BCH codes

In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over Fq with designed distance δ=aqm−1−1(resp. δ=aqm−1q−1) for all 1≤a≤q−1, where q is a prime power and m>1 is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results of Yue and Hu [16], and we give the dimension and, in some cases, the Bose distance for a large designed distance in the range [aqm−1q−1,aqm−1q−1+T] for 0≤a≤q−2, where T=qm+12−1 if m is odd, and T=2qm2−1 if m is even.

Порівняння стійкості стеганографічних методів Кохо-Жао та DWT[10] до різних типів спотворення програмними засобами

Наведено результати практичних досліджень щодо використання алгоритмів вбудовування для приховування інформації в цифрових відео файлах, отримані за допомогою спеціально розробленої програми на мові Java. Оцінена різниця між початковим відео з вбудованими даними та спотвореним відео за показником індексу структурної подібності індексу структурної подібності SSIM, який вважається більш точним ніж параметр пікового відношення сигналу до шуму (PSNR), та завжди лежить у діапазоні від –1 до 1. Проаналізовані результати, отримані на трьох етапах досліджень, оцінки відео з різними методами вбудовування; з різними методами завадостійкого кодування; з різними моделями каналів зв’язку. Оцінка стійкості до завад в залежності від вибору стеганографічного метода проводилася методами Кохо-Жао і DWT, використовувалися два пороги вбудовування, без завадостійких кодів та без імітації каналів зв’язку. Вплив використання різних методів завадостійкого кодування на стійкість відео проводився за допомогою кодів Хеммінга, БЧХ, РС. Оцінка стійкості до завад в залежності від каналів зв’язку проводилася для 4-х моделей, на які можна розкласти реальний канал зв’язку: каналу з простим шумом, каналу з чорно-білим шумом, каналу з адитивним білим гауссівським шумом та каналу з дискретним каналом зі стиранням. Дослідним шляхом було з’ясовано, що різниця в стійкості до завад для різних стегонографічних методів несуттєва; збільшення порогу вбудовування інформації приводить до зростання стійкості; ускладнення алгоритму кодування підвищує стійкість як до атак, так і для завад; виявлено, що найгіршим для передачі відео є дискретний канал зі стиранням.

An Infinite Family of Linear Codes Supporting 4-Designs

The question as to whether there exists an infinite family of near MDS codes holding an infinite family of t-designs for t ≥ 2 was answered in the recent paper [Infinite families of near MDS codes holding t-designs, IEEE Trans. Inf. Theory 66(9) (2020)], where an infinite family of near MDS codes holding an infinite family of 3-designs and an infinite family of near MDS codes holding an infinite family of 2-designs were presented, but no infinite family of linear codes holding an infinite family of 4-designs was presented. Hence, the question as to whether there is an infinite family of linear codes holding an infinite family of 4-designs remains open for 71 years. This paper settles this longstanding problem by presenting an infinite family of BCH codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+1</sup> +1 over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m+1</sup> ) holding an infinite family of 4-(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m+1</sup> + 1, 6, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sup> - 4) designs. This paper also provides another solution to the first question, as some of the BCH codes presented in this paper are also near MDS. Moreover, an infinite family of linear codes holding the spherical geometry design S(3, 5, 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> + 1) is presented. The new direction of searching for t-designs with elementary symmetric polynomials will be further advanced.

The Hamming distances of repeated-root cyclic codes of length [formula omitted

Due to the wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. In this paper, let p be a prime with p≥7. We determine the Hamming distances of all repeated-root cyclic codes of length 5ps over Fq, where q=pm and both s and m are positive integers. Furthermore, we find all MDS cyclic codes of length 5ps and take quantum synchronizable codes from repeated-root cyclic codes of length 5ps. By comparing the minimum distances of 5ps-length repeated-root cyclic codes to BCH codes of close lengths, we illustrated that quantum synchronizable codes constructed from repeated-root cyclic codes have in general better performance in correcting Pauli errors than those from BCH codes.

Experimental demonstration and analysis of underwater wireless optical communication link: Design, BCH coded receiver diversity over the turbid and turbulent seawater channels

AbstractIn this article, we demonstrate an experimental underwater wireless optical communication (UWOC) system in the presence of air bubbles and weak turbulence for varying turbidity levels of the aquatic optical medium. The major limiting factors of the UWOC system are: absorption, scattering, and beam fluctuations; these effects can be mitigated by employing transmitter/receiver diversity schemes and channel codes. In this proposed system, we have employed receiver diversity (selection combining (SC), majority logic combining (MLC), and equal gain combining (EGC)) techniques augmented with Bose‐Chaudhuri‐Hocquenghem (BCH) codes to improve the performance of on‐off keying modulated UWOC system. The bit error rate (BER) expressions are derived for the proposed system and results are validated using analytic and experimental means. The results show that the proposed system, that is, the receiver employing SC, MLC, and EGC receiver combining techniques augmented with the BCH code provides a transmit power gain of 4, 6, and 8 dB respectively, when compared with the uncoded single‐input single‐output system, at a BER of 10−5.

Performance analysis of concatenated BCH and convolutional coded OFDM system

ABSTRACT Orthogonal Frequency Division Multiplexing (OFDM) is a widely used technique for wireless communications. But uncoded OFDM is not sufficient by itself, that is why channel coding is included to increase the system performance. In this study, concatenated Bose Chaudhuri Hocquenghem (BCH) and Convolutional Coded (CC) OFDM system is investigated for multipath fading channel with Additive White Gaussian Noise (AWGN). The simulation results show that the proposed concatenated code needs lower Signal-to-Noise Ratio (SNR) when compared with single BCH code, single convolutional code and even with other concatenated systems. Throughout the simulations BCH coding is performed with 128, 256, 512 Fast Fourier Transform (FFT) lengths; whereas convolutional coding is performed with 1/2, 1/3 coding rates. Furthermore, interleavers are added to the system to prevent the burst errors that occur over the channel. With the proposed system, the best result is obtained by using BCH(511,340) and CC(3,1,7) concatenation which is 8.2 dB SNR value for 10−3 Bit Error Rate (BER). This result is very close to ideal AWGN channel value, which is 8 dB for 10−3 BER.