Comparative Study of Reed-Solomon and Liberation Codes in RAID6 Systems:Implementation, Evaluation and Performance Analysis

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].

All digital television broadcast schemes require some form of forward error correction. Trellis coded modulation and Reed-Solomon (RS) codes are two of the most commonly used error correction strategies. While RS codes have been used for a fairly long time in various applications and are well understood, use of trellis coded modulation has been restricted mostly to telephone modems. We explain the principles of trellis coded modulation and how trellis codes can be chosen as part of the error correction strategy for digital television broadcasting. A variety of coding and interleaving schemes that can be used to obtain the desired performance for compressed TV signal transmission is discussed. The nature of the terrestrial channel and the system performance requirements for a high definition television (HDTV) system are described, including a coding strategy called concatenated coding. >

Forward error correction (FEC) is a key capability in modern satellite communications that provide the system designer with the needed flexibility to comply with the different applications' requirements. Reed-Solomon (RS) codes are well known for their ability to optimize between the system power, bandwidth, data rate, and the quality of service. This paper introduces an efficient decoding scheme for decoding the RS codes adhering to the Consultative Committee for Space Data Systems (CCSDS) standards based on Justesen's construction of concatenation. To maintain the standard output size, the proposed scheme first encodes every m - 1 bits using the single-parity-check (SPC) code, while the RS code encodes K SPC codewords into N symbols that are of the same size as CCSDS standard. Decoding on the inner SPC code is based on maximum-likelihood decoding Kaneko algorithm, while for the proposed coding scheme, the reduced test-pattern Chase algorithm is adapted for decoding the outer RS code. The simulation results show the coding gains of 1.4 and 7 dB compared with the algebraic decoding of RS codes over the AWGN and Rayleigh fading channels, respectively. Moreover, the adopted reduced test-pattern Chase algorithm for decoding the RS code achieves an overall complexity reduction of 40% compared with the conventional Chase decoding algorithm.

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 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.

In multi-hop wireless sensor networks (WSN), the data transmitted from sensor nodes is highly susceptible to error corruption introduced by noisy channels. It is important to use some error correction schemes in order to control the errors introduced and to reduce the number of automatic requests for retransmission. The sensor nodes are typically wireless nodes with limited storage and computational power. Thus the error control schemes should be energy efficient at sensor nodes level. Most of the reported works assume that both encoding and decoding are performed at every node. The idea of this paper is to perform encoding only at the first node and decoding will be done at the base station. We assume that the base station has enough power to run complex decoding algorithms. This paper provides a model for energy consumed and the probability of receiving a correct word for multi-hop WSN at base station when considering a number of coding schemes and hops. The emphasis is on using powerful codes with low complexity encoding. Several powerful codes like Reed-Solomon (RS), list decoded RS codes, multivariate interpolation decoded RS codes (MIDRS) and Hermitian codes are investigated for this scheme. All these codes have very simple encoding based on RS type encoders and we will show that this scheme consumes less power than reported schemes while maintaining a low probability of error.

In recent years, conveying information using molecules via diffusion has attracted significant interest in micro and nanoscale applications. Yet, very little research have been conducted on the performance analysis of large-scale molecular communication (MC) systems with swarm of interfering receivers. In this type of MC system, the collective signal strength at a desired receiver can be impaired by the interference caused by other MC transmitters. The paper presents the first tractable analytical framework for the collective signal strength at a partially absorbing receiver due to a desired transmitter under the impact of a swarm of interfering transmitters in a three-dimensional (3D) large-scale MC system using stochastic geometry. We propose Reed Solomon (RS) coding as an error correction technique to combat the effects of interference caused by the multiple transmitters and the delayed molecules from the previous bits. We derive analytical expressions for the bit error probability (BEP) to evaluate the level of reliability that can be achieved by the proposed scheme. The results obtained using Monte Carlo simulations, matched extactly with the anlytical results, justifying the accuracy of the derivations. Results reveal that the RS codes are capable of providing an acceptable level of improvement in terms of BEP. Furthermore, by carefully selecting the code length, n, and the message length, k, of the RS(n, k) code, we can fine tune the BEP improvement to match the system requirements.

Forward Error Correction code (FEC) is important for Wireless Communication and Cryptography applications to perform the Error Detection and Correction (EDC) mechanisms. FEC codes are classified into two types such as Convolutional code and Block code. Various FEC codes can be used in Wireless Communication likes Viterbi code, Reed Solomon code, Golay, BCH, Multidimensional parity, Hamming code and LDPC codes. In this paper, an analysis between Reed Solomon code, Bose, Chaudhuri, and Hocquenghem (BCH) code and LDPC codes are presented. A Low Density Parity Check (LDPC) code provides low area and high speed than all other codes. Simulation is carried out by ModelSim6.3C and Synthesis is carried out by Xilinx10.1.

We develop a systematic theory for the construction of quantum codes from classical Reed–Solomon over \({\mathbb {F}}_{p^k}\), where p is prime and \(k \in {\mathbb {Z}}^{+}\). Based on two classical \([n,n-d_m+1,d_m]\) Reed–Solomon codes over \({\mathbb {F}}_{p^k}\), we provide the construction of an \([[n,n+n_e-2(d_m-1),d_m]]\) entanglement-assisted Reed–Solomon code over qudits of dimension \(d=p^k\) that saturates the quantum Singleton bound and needs \(n_e\) entangled qudits, which involves obtaining the explicit form of the stabilizers for the code. Our contributions towards the entanglement-assisted Reed–Solomon code are multi-fold as follows: (a) Based on the parity check matrices of the classical Reed–Solomon codes, the explicit form of the stabilizers of the entanglement-assisted Reed–Solomon code are provided and the rate-optimal code is obtained. (b) Based on the entanglement-assisted Reed–Solomon code, a burst error correcting code for qudits of dimension p that corrects a burst of \(k(t_m-1)\) qudits or less, where \(t_m = \lfloor (d_m-1)/2\rfloor \) is provided. (c) Finally, we provide amenable circuits for encoding and error correction with quantum circuit complexity of \({\mathcal {O}}(n^2)\), useful for practical implementations.

In this paper, error resilient stereoscopic video streaming problem is addressed. Two different forward error correction (FEC) codes namely systematic LT and RS codes are utilized to protect the stereoscopic video data against transmission errors. Initially, the stereoscopic video is categorized in 3 layers with different priorities. Then, a packetization scheme is used to increase the efficiency of error protection. A comparative analysis of RS and LT codes are provided via simulations to observe the optimum packetization and UEP strategies.

Smart building is an effective solution to address the issue of energy consumption in today's cyber-physical systems (CPSs) connected world. As an important tool to collect information from a fleet of electric appliance that installed in the building scope, a wireless sensor network is widely employed for this purpose. In this paper, we propose to incorporate forward error correction into the media access control layer of IEEE 802.15.4 standard for packets transmission, as the originally used automatic repeat request mechanism is a timing and energy consumed process that should be avoided in a noisy wireless channel of a smart building environment. Based on the developed CPS simulation platform, the bits error rate and packets error rate of convolutional codes (CCs), Reed-Solomon (RS) codes, and their concatenated codes are intensively investigated, on condition of been applied to different packet length and code rates. We show that the CCs are superior to other codes in most cases. However, the RS and CCs concatenated codes are good candidates, and the RS codes with larger symbol length are preferred for a longer packet. In this regard, future research will concentrated on non-line-of-sight wireless channel, and take the pulse interference from other devices into account.

5G uses various types of access networks and has a high frequency spectrum. At present, there are no mobile handsets that support all spectrum ranges due to limitations of hardware and the expenses involved in the process. This challenge can be addressed with the help of Software Defined Radio (SDR). Universal Software Radio Peripheral (USRP) is a range of software-defined radios and most USRPs connect to a host computer through a high-speed link, which the host-based software uses to control the USRP hardware and transmit/receive data. In general, error occurs during data transmission through different communication channel. These errors can be corrected using Forward Error Correction (FEC) techniques. In this paper, an error correction code named as Reed-Solomon (RS) code, which is suitable for correcting burst error is presented. Further, a Simulation module in LabVIEW for RS Codes using MSK modulation scheme through AWGN channel is presented and its Bit Error Rate (BER) performance for codes such as RS (32,16), RS (127,111) and RS (255,239) are evaluated.

Highly reliable data transmission using Reed-Solomon(RS)/convolutional concatenated coding in conjunction with hybrid type-I automatic retransmission request (ARQ) is investigated. The study is based on an analysis of the tradeoff between error correction and error detection capabilities of RS codes for the direct-sequence code-division multiple-access (DS-CDMA) systems. Two RS decoding schemes are considered: error-only decoding (decoding without side information) and error-and-erasure (EE) decoding. In the EE scheme, the inner code is decoded with a modified Viterbi algorithm, which produces reliable information along with the decoded output. Numerical results illustrate the importance of properly selecting the design parameters for the purpose of maximizing spectral efficiency. The results also show that an E/sub b//N/sub 0/ gain of 0.1 to 0.5 dB can be obtained by using EE decoding, depending on the selected parameters.

The use of Reed-Solomon error correction (RSEC) in bar code symbologies has been around since the early 1990s, but there hasn’t been an in depth, publicly accessible study done on their resistance to character substitution errors since the original studies done in the early 1990s at Ohio University. This article reports on the test results from the scanning of more 23 million scans resulting in more than 2.39 billion characters across four different RSEC enabled bar code symbologies (Data Matrix, QR code, PDF417 and Aztec Code) with five different scanners. The results show that the RSEC enabled symbologies are capable of achieving at least a 1 in 797 million error rate, allowing for their use in instances where decoded data accuracy is imperative.

Polar codes have attracted much recent attention as one of the first codes with low computational complexity that provably achieve optimal rate-regions for a large class of information-theoretic problems. One significant drawback, however, is that for current constructions the probability of error decays sub-exponentially in the block-length more detailed designs improve the probability of error at the cost of significantly increased computational complexity. In this work we show how the the classical idea of code concatenation - using "short" polar codes as inner codes and a "high-rate" Reed-Solomon code as the outer code - results in substantially improved performance. In particular, code concatenation with a careful choice of parameters boosts the rate of decay of the probability of error to almost exponential in the block-length with essentially no loss in computational complexity. We demonstrate such performance improvements for three sets of information-theoretic problems - a classical point-to-point channel coding problem, a class of multiple-input multiple output channel coding problems, and some network source coding problems.