Generator Polynomial Model-Based Eye Diagram Estimation Method for Bose-Chaudhuri-Hocquenghem (BCH) Code and Reed-Solomon (RS) Code

The objective of this paper is to explore the use of error correction codes in digital image watermarking for purposes of copyright protection. A comparative study between two error correction codes, Reed Solomon Code and Bose Chaudhuri-Hocquenghem Code was conducted to identify a more suitable error correction code to be used for this application. A bitmap image watermark was encoded with each chosen Error Correction Code, respectively then embedded on a color JPEG image. The effectiveness of each code was determined by the BER value obtained when the host image was subjected to several image manipulation attacks.

In this chapter, we describe different forward error correction (FEC) schemes currently in use or suitable for use in optical communication systems. We start with the description of standard block codes. The state-of-the-art in optical communication systems standardized by the ITU employ concatenated Bose–Chaudhuri–Hocquenghem (BCH)/RS codes [1, 2]. The RS(255,239), in particular, has been used in a broad range of long-haul communication systems [1, 2], and it is commonly considered as the first-generation of FEC [3, 4]. The elementary FEC schemes (BCH, RS, or convolutional codes) may be combined to design more powerful FEC schemes, e.g., RS(255,239) + RS(255,233). Several classes of concatenation codes are listed in ITU-T G975.1. Different concatenation schemes, such as the concatenation of two RS codes or the concatenation of RS and convolutional codes, are commonly considered as second generation of FEC [3, 4].

Reed-Solomon (RS) codes are error-correcting codes used in many of today's communication systems. RS encoding and decoding are typically implemented using dedicated hardware elements used in ADSL modems and digital TV. In this paper we propose using software in lieu of a hardware-based RS de-coder. This is accomplished using the Berlekamp-Massey algorithm, implemented on a programmable DSP. This software-based RS decoder using Berlekamp-Massey is implemented on Motorola's MSC8101 StarCore DSP. In order to evaluate the algorithm we use the following criteria: computation cost; cycle count, critical paths in the decoding scheme, and error location in the codeword. Furthermore, we examine the effect of changing the RS code's k, t parameters and its primitive polynomial in real-time implementation. The Berlekamp-Massey algorithm operates over finite field arithmetic, whose steps, as applied on the StarCore DSP, are discussed in detail and specifically evaluated using the assembly code for the syndrome search. We conclude that the Berlekamp-Massey algorithm, used for RS decoding, should be implemented using long codewords.

A new method for compact storage and transmission of medical images with concealed patient information in noisy environment is evinced. Digital watermarking is the technique adapted here for interleaving patient information with medical images. The patient information, which comprises of text data and signal graph, is encrypted to prevent unauthorized access of data. The latest encryption algorithm (Rijndael) is used for encrypting the text information. Signal graphs (ECG, EEG EMG etc.) are compressed using DPCM technique. To enhance the robustness of the embedded information, the patient information is coded by error correcting codes (ECC) such as (7,4) Hamming, Bose-Chaudhuri-Hocquenghem (BCH) and Reed Solomon (RS) codes .The noisy scenario is simulated by adding salt and pepper (S&P) noise to the embedded image. For different signal to noise ratio (SNR) of the image, bit error rate (BER) and number of character altered (NOCA) for text data and percentage distortion (PDIST) for the signal graph are evaluated. The performance comparison based on the above parameters is conducted for three types of ECC. It is elicited that coded systems can perform better than the uncoded systems.

In this paper, a blind reconstruction method of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes is proposed, which aims to reconstruct the generator polynomials of BCH and RS codes from the intercepted noisy bitstream in non-cooperative environments. The previous blind reconstruction methods try to select correct (or noiseless) codewords or extract code information from the received bitstream. However, they need to have sufficiently correct codewords for successful reconstruction. Therefore, the existing methods are highly dependent on the channel conditions because a sufficient number of correct codewords should be available for achieving good reconstruction performance. To overcome such high dependency on channel conditions, the proposed blind reconstruction method performs error correction of single-error codewords through the proposed single-error correction method (SCM). Then, both the correct codewords and the corrected single-error codewords are utilized for blind reconstruction. Specifically, the correct codewords are selected based on the algebraic property of BCH and RS codes that codeword polynomials have consecutive roots. At the same time, the codewords presumably with a single error are selected and corrected based on the fact that the most of BCH and RS codeword polynomials with either a single error or double errors do not have consecutive roots. Based on this fact, SCM is devise, which selects and corrects the codewords with a single error with high probability. Finally, it is verified that the proposed blind reconstruction method shows much better reconstruction performance by using only one-tenth of the received codewords compared to the existing best methods even under harsh channel conditions.

Efficient error detection in Double Error Correction BCH codes for memory applications

RAID schemes have been in use for hard disk drives to provide improved performance or fault tolerance. RAID schemes, particularly RAID 2 and higher comprise of striping and adding parity. For solid state drives (SSDs), to combat die failures, schemes like RAID 5 and RAID 6 are effective. The idea is to have block level striping with distributed parity to provide data recovery in case of single or multiple die failures. In this work, we analyze the RAID 5 case used in SSDs when there is no die failure. For this case, the distributed parity information can be used to either combat higher raw bit error rate (RBER) or to improve the uncorrectable bit error rate (UBER). The algorithm we propose can be used with BCH or LDPC codes. With LDPC codes, it uses soft information on RAID 5 parity in order to recover from multiple error correcting code (ECC) fatals in a RAID stripe. The entire stripe data is read and then decoding is performed using the RAID parity to try recovery. We used simulations in order to characterize the RBER gains. First, modeling of the RBER distribution of the dies was performed using the log-normal distribution. Using this model, a performance evaluation for ECC schemes like LDPC codes and BCH codes was performed. We show that if LDPC codes are used, RAID 5 theoretically offers an RBER capability which is 3 times that of the error correcting code RBER capability. However, it is not possible to realize this RBER benefit completely since it has SSD performance downside. In order to meet the Quality of Service (QoS) specification, a realizable RBER gain is by a factor of 1.3 to 2 instead of 3. Further, these gains can only be realized if an LDPC code is used as an error correcting code, algebraic codes like BCH codes will provide limited gains. If no RBER benefit is permitted by a particular configuration of solid state drive, this scheme can be used to provide UBER gain by 3 orders of magnitude.

Error-Control Coding, Modulation and Equalization for All-Digital Advanced Television: State of the Art and Future Possibilities

This paper investigates the design of Reed Solomon (RS) encoder. Based on the message symbols, the RS encoder generates the code-word. By carrying out a polynomial division using Galois Field algebra, the parity symbols are calculated. Reed-Solomon codes are one of the most effective and effective non-binary error codes to detect and correct burst errors. This is the focus work for my dissertation to implement RS encoder and decoder that is a complex algorithm and it is used for the reliable memory operation in a system. The RS Encoder and decoder are design in structural modeling and develop the hardware. The sift and multiplier type divider is used for Encoder and Syndrome module design.

In the field of military wireless communication, digital signals will generate bit errors due to various interferences in the process of channel transmission. In order to improve the communication quality and ensure the correctness and reliability of the communication, the method of error control is usually used for error correction. Error control often adopts the technology of error correction code, which improves the reliability of information transmission by adding redundant information to the effective information code stream. RS code is a typical error correction code with strong error correction ability, which can correct both random errors and burst errors. On this basis, this paper studies the basic rules of RS code construction, designs a new type of codec system circuit for RS (35,27) code, and finally uses hardware description language to design the encoder and decoder, implementation and simulation verification. It has been verified that the RS encoding, and decoding system is stable and reliable, has excellent performance, strong error correction ability, strong scalability, and has a broad application space.

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.

Reed-Solomon (RS) codes are used as error-correcting codes in numerous digital communication and storage systems. Algebraic soft-decision decoding (ASD) of RS codes can achieve substantial coding gain with polynomial complexity. Among practical ASD algorithms, the iterative bit-level generalized minimum distance (BGMD) decoding can achieve similar or higher coding gain with lower complexity. The interpolation is a major step of ASD. The maximum achievable speed of this step is limited by the inherent serial nature of the interpolation algorithm. In this paper, a novel interpolation scheme that is capable of combining multiple interpolation iterations, as well as sharing interpolation results from previous decoding iterations, is developed for the iterative BGMD decoding. In addition, efficient VLSI architectures are proposed to implement the developed scheme. Based on the proposed architectures, an interpolator for a (255, 239) RS code is implemented on field programmable gate array (FPGA) devices. On a Xilinx Virtex-II device, our interpolator can achieve a throughput of 440 Mbps, which is 64% higher than the fastest previous design, with 51% less FPGA resource.

Due to elevated data rate transmission, fading immunity and bandwidth efficient capability, OFDM comes out as a key technology in contemporary wireless communication systems. Additionally, to access error free transmission, diverse channel codes for instance Convolution Codes (CC), Reed Solomon (RS) and BCH codes are being used in the previous work. In this work, an OFDM based M-PSK wireless system is demonstrated incorporating diverse channel encoding techniques over Rayleigh fading channel. The performance of the simulated test-bed is investigated via BER assessment as a function of SNR. Our simulation results reports a significant improvement in BER of the simulated system using BCH over CC and RS.

The Reed-Solomon (RS) codes, proposed in 1960 by Irving Reed and Gustav Solomon as a subset of error-correcting codes, have many current applications. The most significant of which are data recovery in storage systems, including hard drives, minidiscs, CDs, DVDs, Google's GFS, BigTable, and RAID 6, as well as in communication systems such as DSL, WiMAX, DVB, ATSC, and satellite communications. Additionally, RS codes are used as Bar codes in management and advertising systems, such as PDF-417, MaxiCode, Datamatrix, QR Code, and Aztec Code. Nowadays, RS codes over Galois Fields GF(2m) with base 2 are commonly used in these applications, with the GF(28) field being the most widely used. This allows all 256 values of a byte to be represented as a polynomial with 8 binary coefficients over GF(28). Considering RS codes as cyclic codes in GF(2m) fields, as well as the validity of mathematical dependencies in arbitrary field GF(pm), is a motivation to verify and generalize the idea of generating RS codes in a field with base other prime than 2. As a result, the paper derives the specific features of the construction of Reed-Solomon codes by considering them as a family of codes over any field GF(pm) whose base is a prime p other than 2. The paper also discusses the unique properties of basic arithmetic operations in the arbitrary field GF(pm), which arise from the non-uniqueness of the inverse elements a and -a in a field with base other than 2.

A fast decoding algorithm for double error correcting Reed Solomon (RS) codes is presented. A new approach to the RS code decoding is introduced to eliminate the Chien search, to reduce substantially the root search time and hence to increase the decoding speed. It is interesting that the error locating time almost constant as the number of bits per symbol is increased. The algorithm finds suitable for microprocessor-based as well as VLSI implementations. Various RS codes are considered. Complexity, memory space and decoding time are all minimized as compared to the normal Chien search approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>