BCH Codes Research Articles

ON THE BERLEKAMP — MASSEY ALGORITHM AND ITS APPLICATION FOR DECODING ALGORITHMS

The paper is devoted to the Berlekamp Masssey algorithm and its equivalent version based on the extended Euclidean algorithm. An optimized Berlekamp Massey algorithm is also given for the case ofa field of characteristic 2. The Berlekamp Massey algorithm has a quadratic complexity and is used, for example, to solve systems of linear equations in which the matrix of the system is the Toeplitz matrix. In particular, such systems of equations appear in algorithms for the syndrome decoding of BCH codes, Reed Solomon codes, generalized Reed Solomon codes, and Goppa codes. Algorithms for decoding the listed codes based on the Berlekamp Massey algorithm are given.

On the construction of irreducible and primitive polynomials from [formula omitted] to [formula omitted

In this paper, we present two constructions of irreducible (primitive) polynomials over Fq of degree rm from irreducible (primitive) polynomial over Fqm of degree r, and we show that these two constructions coincide. The first construction is based on the Frobenius automorphism of Fqm over Fq. The second one comes from a generalization of a construction of primitive polynomials over Fq which uses the companion matrix. From this generalization, given an irreducible (resp. primitive) polynomial over Fqm of degree r, we generate multiple (resp. all) irreducible polynomials over Fq of degree rm. As an application, a characterization of the generator polynomial of a BCH code over Fq is given. Then, we show how two BCH codes over Fq and Fqm, respectively, and their generator polynomials are related.

Degenerate Quantum LDPC Codes With Good Finite Length Performance

We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. It is shown that in some cases such codes have better performance than hypergraph product codes. Finally, we demonstrate that the performance of the proposed BP-OSD decoder for some of the constructed codes is better than for a relatively large surface code decoded by a near-optimal decoder.

The modal Θ-valent extensions of BCH codes

In this paper, the purpose is to define a modal Θ-valent extension of a BCH code over the finite modal Θ-valent field and to propose a decoding process of this new class of codes on the modal Θ-valent view.

Some t-designs from BCH codes

Some t-designs from BCH codes

Parameters and characterizations of hulls of some projective narrow-sense BCH codes

Parameters and characterizations of hulls of some projective narrow-sense BCH codes

The Hermitian dual-containing LCD BCH codes and related quantum codes

The Hermitian dual-containing LCD BCH codes and related quantum codes

Fast Blind Recognition of BCH Code Based on Spectral Analysis and Probability Statistics

This letter presents a fast blind recognition method for Bose, Chaudhuri, and Hocquenghem (BCH) code from noisy intercepted bit-streams. This approach employs the fact that the t-error-correcting BCH code has 2t continuous roots, and the roots correspond with the zero spectral. Specifically, the Galois field Fourier transform is simplified (S-GFFT) by using the property of conjugate roots. Then, the distribution characteristics of continuous zero spectra are analyzed. If the continuous zero spectra satisfy the threshold condition which is derived by the probability of zero spectrum component, the corresponding estimated length is the code length, and the relevant positions correspond to the continuous roots of the generated polynomial. Finally, the generator polynomial can be reconstructed based on algebras theory. Compared to the existing solutions, the computational amount of S-GFFT at most accounts for 60% of GFFT, and the proposed scheme manages to recognize the code parameters without traversing the whole candidate datasets.

On Hulls of Some Primitive BCH Codes and Self-Orthogonal Codes

Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}_{(r,r^{m}-1,\delta,b)}$ </tex-math></inline-formula> be the primitive BCH code over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{r}$ </tex-math></inline-formula> of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r^{m}-1$ </tex-math></inline-formula> with designed distance <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{r}$ </tex-math></inline-formula> is the finite field of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> . In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}_{(r,r^{2}-1,\delta,1)}$ </tex-math></inline-formula> and its hull in both Hermitian and Euclidean cases for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \le \delta \le r^{2}-1$ </tex-math></inline-formula> . We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.

Efficient image tampering localization using semi-fragile watermarking and error control codes

In this paper, we propose an image tampering localization algorithm using semi-fragile watermarking and Error-Locating codes in the DWT domain. By introducing different families of codes, we show the benefit in terms of image tampering localization and complexity of using control code error localization as an authentication function. Indeed, we first experimentally show that error localization block codes is as precise as using classical error correcting codes (Reed-Solomon and BCH codes) to locate image tampering. However, their corresponding decoding algorithms complexity is at least quadratic which make them impractical for some real time applications. To solve this problem, we introduce error-control codes called Error-Locating codes where error localization is reduced to a single syndrome computation performed with low number of binary operations (detailed later in the paper). We provide comparisons of image quality and tampering localization performances using error-detection, error-localization and error-correction approaches with different error control codes.

A correlation-breaking interleaving of polar codes in concatenated systems

It is known that the bit errors of polar codes with successive cancellation (SC) decoding are coupled. However, existing concatenation schemes of polar codes with other error correction codes rarely take this coupling effect into consideration. To achieve a better error performance of concatenated systems with polar codes as inner codes, one can divide all bits in an outer block into different polar blocks to completely de-correlate the possible coupled errors in the transmitter side. We call this interleaving a blind interleaving (BI) which serves as a benchmark. Two BI schemes, termed BI-DP and BI-CDP, are proposed in the paper. To better balance performance, memory size, and the decoding delay from the de-interleaving, a novel interleaving scheme, named the correlation-breaking interleaving (CBI), is proposed. The CBI breaks the correlated information bits based on the error correlation pattern proposed and proven in this paper. The proposed CBI scheme is general in the sense that any error correction code can serve as the outer code. In this paper, Low-Density Parity-Check (LDPC) codes and BCH codes are used as two examples of the outer codes of the interleaving scheme. The CBI scheme 1) can keep the simple SC polar decoding while achieving a better error performance than the state-of-the-art (SOA) direct concatenation of polar codes with LDPC codes and BCH codes; 2) achieves a comparable error performance as the BI-DP scheme with a smaller memory size and a shorter decoding delay. Numerical results are provided to verify the performance of the BI schemes and the CBI scheme.

Optimized Polar Codes as Forward Error Correction Coding for Digital Video Broadcasting Systems

Polar codes are featured by their low encoding/decoding complexity for symmetric binary input-discrete memoryless channels. Recently, flexible generic Successive Cancellation List (SCL) decoders for polar codes were proposed to provide different throughput, latency, and decoding performances. In this paper, we propose to use polar codes with flexible fast-adaptive SCL decoders in Digital Video Broadcasting (DVB) systems to meet the growing demand for more bitrates. In addition, they can provide more interactive services with less latency and more throughput. First, we start with the construction of polar codes and propose a new mathematical relation to get the optimized design point for the polar code. We prove that our optimized design point is too close to the one that achieves minimum Bit Error Rate (BER). Then, we compare the performance of polar and Low-Density Parity Check (LDPC) codes in terms of BER, encoder/decoder latencies, and throughput. The results show that both channel coding techniques have comparable BER. However, polar codes are superior to LDPC in terms of decoding latency, and system throughput. Finally, we present the possible performance enhancement of DVB systems in terms of decoding latency and complexity when using optimized polar codes as a Forward Error Correction (FEC) technique instead of Bose Chaudhuri Hocquenghem (BCH) and LDPC codes that are currently adopted in DVB standards.

Soft decoding of short/medium length codes using ordered statistics for quantum key distribution

Quantum key distribution (QKD) is a cryptographic communication protocol that utilizes quantum mechanical properties for provable absolute security against an eavesdropper. The communication is carried between two terminals using random photon polarization states represented through quantum states. Both these terminals are interconnected through disjoint quantum and classical channels. Information reconciliation using delay controlled joint decoding is performed at the receiving terminal. Its performance is characterized using data and error rates. Achieving low error rates is particularly challenging for schemes based on error correcting codes with short code lengths. This article addresses the decoding process using ordered statistics decoding for information reconciliation of both short and medium length Bose–Chaudhuri–Hocquenghem codes over a QKD link. The link’s quantum channel is modeled as a binary symmetric quantum depolarization channel, whereas the classical channel is configured with additive white Gaussian noise. Our results demonstrate the achievement of low bit error rates, and reduced decoding complexity when compared to other capacity achieving codes of similar length and configuration.

Design of LDBCH Codes for Ultra Reliable Low Latency Communications

In ultra reliable low latency communications, the demanding requirements on high reliability and low latency make it a challenge to design capable channel codes. In this work, a candidate named low-density BCH-constraint (LDBCH) code is introduced, which generalizes low-density parity-check codes by replacing the single parity check code constraints with the BCH code constraints. Specifically, a design methodology is introduced to construct efficient LDBCH codes of short information lengths and low coding rates. Moreover, the message passing decoding of LDBCH is enhanced by max-log approximation and weighted scaling when updating the BCH check nodes. Numerical results show that, with tolerable increase of decoding complexity, LDBCH codes with the proposed decoding enhancement outperforms CA-Polar and 5G LDPC by about 0.3 dB and 0.5 dB in SNR, respectively, for the information length of 104 and the block error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">- 6</sup> over an AWGN channel with QPSK modulation.

Performance of Transmit Antenna Selection in Multiple Input Multiple Output-Orthogonal Space Time Block Code (MIMO-OSTBC) System Joint with Bose-Chaudhuri-Hocquenghem (BCH)-Turbo Code (TC) in Rayleigh Fading Channel

The Transmit antenna selection (TAS) technique is essential to increase the efficiency of spatial modulation (SM) systems. This TAS is an effective technique for reducing the Multiple Input Multiple Output (MIMO) systems, which can increase the computational difficulty and Bit error rate (BER) by various TAS algorithms. But these selection methods cannot provide code gain, so it is essential to join the external code in TAS to obtain code gain advantages in BER. In some existing work, the improved BER has been perceived by joining Forward Error Correction Code (FEC) and Space Time Block Code (STBC) for MIMO systems provided greater code gain. A multiple TAS-OSTBC technique with new integration of Bose–Chaudhuri–Hocquenghem (BCH)-Turbo code (TC) is proposed in this paper. The TAS-OSTBC system connects with the external BCH code in sequence of internal Turbo code. This combination can provide increasing code gain and the effective advantages of the TAS-OSTBC system. To perform the system analysis Rayleigh channel is utilized. In the case with multiple TAS-OSTBC systems, better performance can provide by this new combination of the BCH-Turbo compared to the conventional Turbo code for the Rayleigh fading.

MIMO-Orthogonal space time block code system joint with BCH-TC in Rayleigh fading channel for performance of transmit antenna selection

Purpose To enhance the performance transmit antenna selection (TAS) of spatial modulation (SM), systems technique needs to be essential. This TAS is an effective technique for reducing the multiple input multiple output (MIMO) systems computational difficulty, and bit error rate (BER) can increase remarkably by various TAS algorithms. But these selection methods cannot provide code gain, so it is essential to join the TAS with external code to obtain cy -ode gain advantages in BER. Design/methodology/approach In this paper, Bose–Chaudhuri–Hocquenghem (BCH)-Turbo code TC is combined with the orthogonal space time block code system. Findings In some existing work, the improved BER has been perceived by joining forward error correction code and space time block code (STBC) for MIMO systems provided greater code gain. The proposed work can provide increasing code gain and the effective advantages of the TAS-OSTBC system. Originality/value To perform the system analysis, Rayleigh channel is used. In the case with multiple TAS-OSTBC systems, better performance can provide by this new joint of the BCH-Turbo compared to the conventional Turbo code for the Rayleigh fading.

Construction of matrix product codes where defining matrix is generator matrix of some BCH code or RS code, using SageMath functions

For a given finitely many codes, Matrix Product Code (MPC) can generate a code with better dimension and minimum distance. In this paper, SageMath functions are defined and using these functions, matrix product codes (MPC) are constructed with defining matrix as the generator matrix of some BHC-code or RS-code.

Improving bit error-rate based on adaptive Bose-Chaudhuri Hocquenghem concatenated with convolutional codes

Several algorithms have been proposed to avoid the error floor region, such as the concatenation codes that requires high computational demands in addition to high complexity. This paper proposes a technique based on using cascaded BCH and convolutional codes that leads to better error correction performance. Moreover, an adaptive method based on sensing the channel's noise to determine the number of the parity bits that will be added to the used BCH that reduces the consumed bandwidth and the transmitted parity bits is presented. A further enhancement is fulfilled by using parallel processing branches, resulting in reducing the consumed time and speed up the performance. The results show that the proposed code presents a better performance. A high reduction in the number of cycles that will be used in the encoding and decoding compared with the classical method and finally a flexible parity bits method based on the signal-to-noise ratio of the channel that reduced the parity bits which leads to reduce the consumed bandwidth. The MATLAB simulation and the field programmable gate array (FPGA) implementation will be provided in this paper to validate the proposed concept.

Sum-Rank BCH Codes and Cyclic-Skew-Cyclic Codes

In this work, cyclic-skew-cyclic codes and sum-rank BCH codes are introduced. Cyclic-skew-cyclic codes are characterized as left ideals of a suitable non-commutative finite ring, constructed using skew polynomials on top of polynomials (or vice versa). Single generators of such left ideals are found, and they are used to construct generator matrices of the corresponding codes. The notion of defining set is introduced, using pairs of roots of skew polynomials on top of poynomials. A lower bound (called sum-rank BCH bound) on the minimum sum-rank distance is given for cyclic-skew-cyclic codes whose defining set contains certain consecutive pairs. Sum-rank BCH codes, with prescribed minimum sum-rank distance, are then defined as the largest cyclic-skew-cyclic codes whose defining set contains such consecutive pairs. The defining set of a sum-rank BCH code is described, and a lower bound on its dimension is obtained. Thanks to it, tables are provided showing that sum-rank BCH codes beat previously known codes for the sum-rank metric for binary 2 ×2 matrices (i.e., codes whose codewords are lists of 2 ×2 binary matrices, for a wide range of list lengths that correspond to the code length). Finally, a decoder for sum-rank BCH codes up to half their prescribed distance is obtained.

Comparison of BCH Code Performance in LTE-Advanced Cooperative System in Suburban, Urban and Rural Environment Based on WINNER II Channel Model

Multi-path fading is a severe issue that causes fluctuations in the amplitudes of the received signals. For combating this issue, several researchers have used cooperative communication in the Long-Term Evolution (LTE)-advanced system. To further decrease the degradative effects of transmitted signals, various error correction techniques can be used to recover the erroneous components, especially when the channel has been deep fading. The paper investigates the general performance of the Convolutional Codes (CC) in the LTE-advanced cooperative system and compare it with the Bose-Chaudhuri-Hocquenghem (BCH) codes. The comparison is based on WINNER II project as the channel model between the nodes in the two coding processes. To validate the advantages of BCH code, the process were tested on three different environments; a suburban macro-cell, urban macro-cell, and the rural micro-cell with prior study to understand the appropriate relay height in every environment. Path loss models for each environment were derived and proposed as part of the proposed simulation framework. Conducted simulation studies based on bit error rate and direct connection showed that the use of the CC and the BCH codes in a relay technologies could significantly increase the system performance. It was noted that the performance of the system which used BCH nodes was better than the system using CC nodes in an LTE-advanced cooperative system.