The central idea is that the sender encodes the message in a redundant way, most often by using an error correction code or error correcting code (ECC).[4][5] The redundancy allows the receiver not only to detect errors that may occur anywhere in the message, but often to correct a limited number of errors. Therefore a reverse channel to request re-transmission may not be needed. The cost is a fixed, higher forward channel bandwidth.
The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code.[5]
FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in multicast.
Long-latency connections also benefit; in the case of satellites orbiting distant planets, retransmission due to errors would create a delay of several hours. FEC is also widely used in modems and in cellular networks.
FEC processing in a receiver may be applied to a digital bit stream or in the demodulation of a digitally modulated carrier. For the latter, FEC is an integral part of the initial analog-to-digital conversion in the receiver. The Viterbi decoder implements a soft-decision algorithm to demodulate digital data from an analog signal corrupted by noise. Many FEC decoders can also generate a bit-error rate (BER) signal which can be used as feedback to fine-tune the analog receiving electronics.
FEC information is added to mass storage (magnetic, optical and solid state/flash based) devices to enable recovery of corrupted data, and is used as ECCcomputer memory on systems that require special provisions for reliability.
The maximum proportion of errors or missing bits that can be corrected is determined by the design of the ECC, so different forward error correcting codes are suitable for different conditions. In general, a stronger code induces more redundancy that needs to be transmitted using the available bandwidth, which reduces the effective bit-rate while improving the received effective signal-to-noise ratio. The noisy-channel coding theorem of Claude Shannon can be used to compute the maximum achievable communication bandwidth for a given maximum acceptable error probability. This establishes bounds on the theoretical maximum information transfer rate of a channel with some given base noise level. However, the proof is not constructive, and hence gives no insight of how to build a capacity achieving code. After years of research, some advanced FEC systems like polar code[3] come very close to the theoretical maximum given by the Shannon channel capacity under the hypothesis of an infinite length frame.
Method
ECC is accomplished by adding redundancy to the transmitted information using an algorithm. A redundant bit may be a complicated function of many original information bits. The original information may or may not appear literally in the encoded output; codes that include the unmodified input in the output are systematic, while those that do not are non-systematic.
A simplistic example of ECC is to transmit each data bit 3 times, which is known as a (3,1) repetition code. Through a noisy channel, a receiver might see 8 versions of the output, see table below.
Triplet received
Interpreted as
000
0 (error-free)
001
0
010
0
100
0
111
1 (error-free)
110
1
101
1
011
1
This allows an error in any one of the three samples to be corrected by "majority vote", or "democratic voting". The correcting ability of this ECC is:
Up to 1 bit of triplet in error, or
up to 2 bits of triplet omitted (cases not shown in table).
Though simple to implement and widely used, this triple modular redundancy is a relatively inefficient ECC. Better ECC codes typically examine the last several tens or even the last several hundreds of previously received bits to determine how to decode the current small handful of bits (typically in groups of 2 to 8 bits).
Averaging noise to reduce errors
ECC could be said to work by "averaging noise"; since each data bit affects many transmitted symbols, the corruption of some symbols by noise usually allows the original user data to be extracted from the other, uncorrupted received symbols that also depend on the same user data.
Because of this "risk-pooling" effect, digital communication systems that use ECC tend to work well above a certain minimum signal-to-noise ratio and not at all below it.
This all-or-nothing tendency – the cliff effect – becomes more pronounced as stronger codes are used that more closely approach the theoretical Shannon limit.
Interleaving ECC coded data can reduce the all or nothing properties of transmitted ECC codes when the channel errors tend to occur in bursts. However, this method has limits; it is best used on narrowband data.
Most telecommunication systems use a fixed channel code designed to tolerate the expected worst-case bit error rate, and then fail to work at all if the bit error rate is ever worse. However, some systems adapt to the given channel error conditions: some instances of hybrid automatic repeat-request use a fixed ECC method as long as the ECC can handle the error rate, then switch to ARQ when the error rate gets too high; adaptive modulation and coding uses a variety of ECC rates, adding more error-correction bits per packet when there are higher error rates in the channel, or taking them out when they are not needed.
A block code (specifically a Hamming code) where redundant bits are added as a block to the end of the initial messageA continuous convolutional code where redundant bits are added continuously into the structure of the code word
Block codes work on fixed-size blocks (packets) of bits or symbols of predetermined size. Practical block codes can generally be hard-decoded in polynomial time to their block length.
Convolutional codes work on bit or symbol streams of arbitrary length. They are most often soft decoded with the Viterbi algorithm, though other algorithms are sometimes used. Viterbi decoding allows asymptotically optimal decoding efficiency with increasing constraint length of the convolutional code, but at the expense of exponentially increasing complexity. A convolutional code that is terminated is also a 'block code' in that it encodes a block of input data, but the block size of a convolutional code is generally arbitrary, while block codes have a fixed size dictated by their algebraic characteristics. Types of termination for convolutional codes include "tail-biting" and "bit-flushing".
Hamming ECC is commonly used to correct NAND flash memory errors.[6] This provides single-bit error correction and 2-bit error detection. Hamming codes are only suitable for more reliable single-level cell (SLC) NAND. Denser multi-level cell (MLC) NAND may use multi-bit correcting ECC such as BCH or Reed–Solomon.[7][8] NOR Flash typically does not use any error correction.[7]
Classical block codes are usually decoded using hard-decision algorithms,[9] which means that for every input and output signal a hard decision is made whether it corresponds to a one or a zero bit. In contrast, convolutional codes are typically decoded using soft-decision algorithms like the Viterbi, MAP or BCJR algorithms, which process (discretized) analog signals, and which allow for much higher error-correction performance than hard-decision decoding.
Nearly all classical block codes apply the algebraic properties of finite fields. Hence classical block codes are often referred to as algebraic codes.
In contrast to classical block codes that often specify an error-detecting or error-correcting ability, many modern block codes such as LDPC codes lack such guarantees. Instead, modern codes are evaluated in terms of their bit error rates.
Most forward error correction codes correct only bit-flips, but not bit-insertions or bit-deletions. In this setting, the Hamming distance is the appropriate way to measure the bit error rate. A few forward error correction codes are designed to correct bit-insertions and bit-deletions, such as Marker Codes and Watermark Codes. The Levenshtein distance is a more appropriate way to measure the bit error rate when using such codes. [10]
Code-rate and the tradeoff between reliability and data rate
The fundamental principle of ECC is to add redundant bits in order to help the decoder to find out the true message that was encoded by the transmitter. The code-rate of a given ECC system is defined as the ratio between the number of information bits and the total number of bits (i.e., information plus redundancy bits) in a given communication package. The code-rate is hence a real number. A low code-rate close to zero implies a strong code that uses many redundant bits to achieve a good performance, while a large code-rate close to 1 implies a weak code.
The redundant bits that protect the information have to be transferred using the same communication resources that they are trying to protect. This causes a fundamental tradeoff between reliability and data rate.[11] In one extreme, a strong code (with low code-rate) can induce an important increase in the receiver SNR (signal-to-noise-ratio) decreasing the bit error rate, at the cost of reducing the effective data rate. On the other extreme, not using any ECC (i.e., a code-rate equal to 1) uses the full channel for information transfer purposes, at the cost of leaving the bits without any additional protection.
One interesting question is the following: how efficient in terms of information transfer can an ECC be that has a negligible decoding error rate? This question was answered by Claude Shannon with his second theorem, which says that the channel capacity is the maximum bit rate achievable by any ECC whose error rate tends to zero:[12] His proof relies on Gaussian random coding, which is not suitable to real-world applications. The upper bound given by Shannon's work inspired a long journey in designing ECCs that can come close to the ultimate performance boundary. Various codes today can attain almost the Shannon limit. However, capacity achieving ECCs are usually extremely complex to implement.
The most popular ECCs have a trade-off between performance and computational complexity. Usually, their parameters give a range of possible code rates, which can be optimized depending on the scenario. Usually, this optimization is done in order to achieve a low decoding error probability while minimizing the impact to the data rate. Another criterion for optimizing the code rate is to balance low error rate and retransmissions number in order to the energy cost of the communication.[13]
Classical (algebraic) block codes and convolutional codes are frequently combined in concatenated coding schemes in which a short constraint-length Viterbi-decoded convolutional code does most of the work and a block code (usually Reed–Solomon) with larger symbol size and block length "mops up" any errors made by the convolutional decoder. Single pass decoding with this family of error correction codes can yield very low error rates, but for long range transmission conditions (like deep space) iterative decoding is recommended.
Concatenated codes have been standard practice in satellite and deep space communications since Voyager 2 first used the technique in its 1986 encounter with Uranus. The Galileo craft used iterative concatenated codes to compensate for the very high error rate conditions caused by having a failed antenna.
Low-density parity-check (LDPC) codes are a class of highly efficient linear block codes made from many single parity check (SPC) codes. They can provide performance very close to the channel capacity (the theoretical maximum) using an iterated soft-decision decoding approach, at linear time complexity in terms of their block length. Practical implementations rely heavily on decoding the constituent SPC codes in parallel.
LDPC codes were first introduced by Robert G. Gallager in his PhD thesis in 1960, but due to the computational effort in implementing encoder and decoder and the introduction of Reed–Solomon codes, they were mostly ignored until the 1990s.
LDPC codes are now used in many recent high-speed communication standards, such as DVB-S2 (Digital Video Broadcasting – Satellite – Second Generation), WiMAX (IEEE 802.16e standard for microwave communications), High-Speed Wireless LAN (IEEE 802.11n),[14]10GBase-T Ethernet (802.3an) and G.hn/G.9960 (ITU-T Standard for networking over power lines, phone lines and coaxial cable). Other LDPC codes are standardized for wireless communication standards within 3GPPMBMS (see fountain codes).
Turbo coding is an iterated soft-decoding scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the Shannon limit. Predating LDPC codes in terms of practical application, they now provide similar performance.
One of the earliest commercial applications of turbo coding was the CDMA2000 1x (TIA IS-2000) digital cellular technology developed by Qualcomm and sold by Verizon Wireless, Sprint, and other carriers. It is also used for the evolution of CDMA2000 1x specifically for Internet access, 1xEV-DO (TIA IS-856). Like 1x, EV-DO was developed by Qualcomm, and is sold by Verizon Wireless, Sprint, and other carriers (Verizon's marketing name for 1xEV-DO is Broadband Access, Sprint's consumer and business marketing names for 1xEV-DO are Power Vision and Mobile Broadband, respectively).
Sometimes it is only necessary to decode single bits of the message, or to check whether a given signal is a codeword, and do so without looking at the entire signal. This can make sense in a streaming setting, where codewords are too large to be classically decoded fast enough and where only a few bits of the message are of interest for now. Also such codes have become an important tool in computational complexity theory, e.g., for the design of probabilistically checkable proofs.
Locally decodable codes are error-correcting codes for which single bits of the message can be probabilistically recovered by only looking at a small (say constant) number of positions of a codeword, even after the codeword has been corrupted at some constant fraction of positions. Locally testable codes are error-correcting codes for which it can be checked probabilistically whether a signal is close to a codeword by only looking at a small number of positions of the signal.
Not all testing codes are locally decoding and testing of codes
Not all locally decodable codes (LDCs) are locally testable codes (LTCs)[15] neither locally correctable codes (LCCs),[16] q-query LCCs are bounded exponentially[17][18] while LDCs can have subexponential lengths.[19][20]
Interleaving
"Interleaver" redirects here. For the fiber-optic device, see optical interleaver.
A short illustration of interleaving idea
Interleaving is frequently used in digital communication and storage systems to improve the performance of forward error correcting codes. Many communication channels are not memoryless: errors typically occur in bursts rather than independently. If the number of errors within a code word exceeds the error-correcting code's capability, it fails to recover the original code word. Interleaving alleviates this problem by shuffling source symbols across several code words, thereby creating a more uniform distribution of errors.[21] Therefore, interleaving is widely used for burst error-correction.
The analysis of modern iterated codes, like turbo codes and LDPC codes, typically assumes an independent distribution of errors.[22] Systems using LDPC codes therefore typically employ additional interleaving across the symbols within a code word.[23]
For turbo codes, an interleaver is an integral component and its proper design is crucial for good performance.[21][24] The iterative decoding algorithm works best when there are not short cycles in the factor graph that represents the decoder; the interleaver is chosen to avoid short cycles.
Interleaver designs include:
rectangular (or uniform) interleavers (similar to the method using skip factors described above)
convolutional interleavers
random interleavers (where the interleaver is a known random permutation)
S-random interleaver (where the interleaver is a known random permutation with the constraint that no input symbols within distance S appear within a distance of S in the output).[25]
In multi-carrier communication systems, interleaving across carriers may be employed to provide frequency diversity, e.g., to mitigate frequency-selective fading or narrowband interference.[28]
Example
Transmission without interleaving:
Error-free message: aaaabbbbccccddddeeeeffffgggg Transmission with a burst error: aaaabbbbccc____deeeeffffgggg
Here, each group of the same letter represents a 4-bit one-bit error-correcting codeword. The codeword cccc is altered in one bit and can be corrected, but the codeword dddd is altered in three bits, so either it cannot be decoded at all or it might be decoded incorrectly.
With interleaving:
Error-free code words: aaaabbbbccccddddeeeeffffgggg Interleaved: abcdefgabcdefgabcdefgabcdefg Transmission with a burst error: abcdefgabcd____bcdefgabcdefg Received code words after deinterleaving: aa_abbbbccccdddde_eef_ffg_gg
In each of the codewords "aaaa", "eeee", "ffff", and "gggg", only one bit is altered, so one-bit error-correcting code will decode everything correctly.
Transmission without interleaving:
Original transmitted sentence: ThisIsAnExampleOfInterleaving Received sentence with a burst error: ThisIs______pleOfInterleaving
The term "AnExample" ends up mostly unintelligible and difficult to correct.
With interleaving:
Transmitted sentence: ThisIsAnExampleOfInterleaving... Error-free transmission: TIEpfeaghsxlIrv.iAaenli.snmOten. Received sentence with a burst error: TIEpfe______Irv.iAaenli.snmOten. Received sentence after deinterleaving: T_isI_AnE_amp_eOfInterle_vin_...
No word is completely lost and the missing letters can be recovered with minimal guesswork.
Disadvantages of interleaving
Use of interleaving techniques increases total delay. This is because the entire interleaved block must be received before the packets can be decoded.[29] Also interleavers hide the structure of errors; without an interleaver, more advanced decoding algorithms can take advantage of the error structure and achieve more reliable communication than a simpler decoder combined with an interleaver[citation needed]. An example of such an algorithm is based on neural network[30] structures.
Software for error-correcting codes
Simulating the behaviour of error-correcting codes (ECCs) in software is a common practice to design, validate and improve ECCs. The upcoming wireless 5G standard raises a new range of applications for the software ECCs: the Cloud Radio Access Networks (C-RAN) in a Software-defined radio (SDR) context. The idea is to directly use software ECCs in the communications. For instance in the 5G, the software ECCs could be located in the cloud and the antennas connected to this computing resources: improving this way the flexibility of the communication network and eventually increasing the energy efficiency of the system.
In this context, there are various available Open-source software listed below (non exhaustive).
AFF3CT(A Fast Forward Error Correction Toolbox): a full communication chain in C++ (many supported codes like Turbo, LDPC, Polar codes, etc.), very fast and specialized on channel coding (can be used as a program for simulations or as a library for the SDR).
IT++: a C++ library of classes and functions for linear algebra, numerical optimization, signal processing, communications, and statistics.
OpenAir: implementation (in C) of the 3GPP specifications concerning the Evolved Packet Core Networks.
In information theory and coding theory with applications in computer science and telecommunication, error detection and correction (EDAC) or error control are techniques that enable reliable delivery of digital data over unreliable communication channels. Many communication channels are subject to channel noise, and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases.
In telecommunication, a convolutional code is a type of error-correcting code that generates parity symbols via the sliding application of a boolean polynomial function to a data stream. The sliding application represents the 'convolution' of the encoder over the data, which gives rise to the term 'convolutional coding'. The sliding nature of the convolutional codes facilitates trellis decoding using a time-invariant trellis. Time invariant trellis decoding allows convolutional codes to be maximum-likelihood soft-decision decoded with reasonable complexity.
In computer science and telecommunication, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors, or correct one-bit errors without detection of uncorrected errors. By contrast, the simple parity code cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are perfect codes, that is, they achieve the highest possible rate for codes with their block length and minimum distance of three. Richard W. Hamming invented Hamming codes in 1950 as a way of automatically correcting errors introduced by punched card readers. In his original paper, Hamming elaborated his general idea, but specifically focused on the Hamming(7,4) code which adds three parity bits to four bits of data.
Reed–Solomon codes are a group of error-correcting codes that were introduced by Irving S. Reed and Gustave Solomon in 1960. They have many applications, including consumer technologies such as MiniDiscs, CDs, DVDs, Blu-ray discs, QR codes, Data Matrix, data transmission technologies such as DSL and WiMAX, broadcast systems such as satellite communications, DVB and ATSC, and storage systems such as RAID 6.
A satellite modem or satmodem is a modem used to establish data transfers using a communications satellite as a relay. A satellite modem's main function is to transform an input bitstream to a radio signal and vice versa.
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied by various scientific disciplines—such as information theory, electrical engineering, mathematics, linguistics, and computer science—for the purpose of designing efficient and reliable data transmission methods. This typically involves the removal of redundancy and the correction or detection of errors in the transmitted data.
In information theory, turbo codes are a class of high-performance forward error correction (FEC) codes developed around 1990–91, but first published in 1993. They were the first practical codes to closely approach the maximum channel capacity or Shannon limit, a theoretical maximum for the code rate at which reliable communication is still possible given a specific noise level. Turbo codes are used in 3G/4G mobile communications and in satellite communications as well as other applications where designers seek to achieve reliable information transfer over bandwidth- or latency-constrained communication links in the presence of data-corrupting noise. Turbo codes compete with low-density parity-check (LDPC) codes, which provide similar performance.
In information theory, a low-density parity-check (LDPC) code is a linear error correcting code, a method of transmitting a message over a noisy transmission channel. An LDPC code is constructed using a sparse Tanner graph. LDPC codes are capacity-approaching codes, which means that practical constructions exist that allow the noise threshold to be set very close to the theoretical maximum for a symmetric memoryless channel. The noise threshold defines an upper bound for the channel noise, up to which the probability of lost information can be made as small as desired. Using iterative belief propagation techniques, LDPC codes can be decoded in time linear in their block length.
In coding theory, block codes are a large and important family of error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract definition of block codes is conceptually useful because it allows coding theorists, mathematicians, and computer scientists to study the limitations of all block codes in a unified way. Such limitations often take the form of bounds that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors.
In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although turbo codes can be seen as a hybrid of these two types. Linear codes allow for more efficient encoding and decoding algorithms than other codes.
Hybrid automatic repeat request is a combination of high-rate forward error correction (FEC) and automatic repeat request (ARQ) error-control. In standard ARQ, redundant bits are added to data to be transmitted using an error-detecting (ED) code such as a cyclic redundancy check (CRC). Receivers detecting a corrupted message will request a new message from the sender. In Hybrid ARQ, the original data is encoded with an FEC code, and the parity bits are either immediately sent along with the message or only transmitted upon request when a receiver detects an erroneous message. The ED code may be omitted when a code is used that can perform both forward error correction (FEC) in addition to error detection, such as a Reed–Solomon code. The FEC code is chosen to correct an expected subset of all errors that may occur, while the ARQ method is used as a fall-back to correct errors that are uncorrectable using only the redundancy sent in the initial transmission. As a result, hybrid ARQ performs better than ordinary ARQ in poor signal conditions, but in its simplest form this comes at the expense of significantly lower throughput in good signal conditions. There is typically a signal quality cross-over point below which simple hybrid ARQ is better, and above which basic ARQ is better.
In computer science, repeat-accumulate codes are a low complexity class of error-correcting codes. They were devised so that their ensemble weight distributions are easy to derive. RA codes were introduced by Divsalar et al.
The error floor is a phenomenon encountered in modern iterated sparse graph-based error correcting codes like LDPC codes and turbo codes. When the bit error ratio (BER) is plotted for conventional codes like Reed–Solomon codes under algebraic decoding or for convolutional codes under Viterbi decoding, the BER steadily decreases in the form of a curve as the SNR condition becomes better. For LDPC codes and turbo codes there is a point after which the curve does not fall as quickly as before, in other words, there is a region in which performance flattens. This region is called the error floor region. The region just before the sudden drop in performance is called the waterfall region.
In coding theory, concatenated codes form a class of error-correcting codes that are derived by combining an inner code and an outer code. They were conceived in 1966 by Dave Forney as a solution to the problem of finding a code that has both exponentially decreasing error probability with increasing block length and polynomial-time decoding complexity. Concatenated codes became widely used in space communications in the 1970s.
In coding theory, list decoding is an alternative to unique decoding of error-correcting codes for large error rates. The notion was proposed by Elias in the 1950s. The main idea behind list decoding is that the decoding algorithm instead of outputting a single possible message outputs a list of possibilities one of which is correct. This allows for handling a greater number of errors than that allowed by unique decoding.
A locally decodable code (LDC) is an error-correcting code that allows a single bit of the original message to be decoded with high probability by only examining a small number of bits of a possibly corrupted codeword. This property could be useful, say, in a context where information is being transmitted over a noisy channel, and only a small subset of the data is required at a particular time and there is no need to decode the entire message at once. Note that locally decodable codes are not a subset of locally testable codes, though there is some overlap between the two.
In digital communications, a turbo equalizer is a type of receiver used to receive a message corrupted by a communication channel with intersymbol interference (ISI). It approaches the performance of a maximum a posteriori (MAP) receiver via iterative message passing between a soft-in soft-out (SISO) equalizer and a SISO decoder. It is related to turbo codes in that a turbo equalizer may be considered a type of iterative decoder if the channel is viewed as a non-redundant convolutional code. The turbo equalizer is different from classic a turbo-like code, however, in that the 'channel code' adds no redundancy and therefore can only be used to remove non-gaussian noise.
In information theory, a polar code is a linear block error-correcting code. The code construction is based on a multiple recursive concatenation of a short kernel code which transforms the physical channel into virtual outer channels. When the number of recursions becomes large, the virtual channels tend to either have high reliability or low reliability, and the data bits are allocated to the most reliable channels. It is the first code with an explicit construction to provably achieve the channel capacity for symmetric binary-input, discrete, memoryless channels (B-DMC) with polynomial dependence on the gap to capacity. Polar codes were developed by Erdal Arikan, a professor of electrical engineering at Bilkent University.
In coding theory, burst error-correcting codes employ methods of correcting burst errors, which are errors that occur in many consecutive bits rather than occurring in bits independently of each other.
Serial concatenated convolutional codes (SCCC) are a class of forward error correction (FEC) codes highly suitable for turbo (iterative) decoding. Data to be transmitted over a noisy channel may first be encoded using an SCCC. Upon reception, the coding may be used to remove any errors introduced during transmission. The decoding is performed by repeated decoding and [de]interleaving of the received symbols.
References
↑ Charles Wang; Dean Sklar; Diana Johnson (Winter 2001–2002). "Forward Error-Correction Coding". Crosslink. 3 (1). The Aerospace Corporation. Archived from the original on 14 March 2012. Retrieved 5 March 2006.
↑ Charles Wang; Dean Sklar; Diana Johnson (Winter 2001–2002). "Forward Error-Correction Coding". Crosslink. 3 (1). The Aerospace Corporation. Archived from the original on 14 March 2012. Retrieved 5 March 2006. How Forward Error-Correcting Codes Work]
1 2 "What Types of ECC Should Be Used on Flash Memory?"(Application note). Spansion. 2011. Both Reed–Solomon algorithm and BCH algorithm are common ECC choices for MLC NAND flash. ... Hamming based block codes are the most commonly used ECC for SLC.... both Reed–Solomon and BCH are able to handle multiple errors and are widely used on MLC flash.
↑ Luby, Michael; Mitzenmacher, M.; Shokrollahi, A.; Spielman, D.; Stemann, V. (1997). "Practical Loss-Resilient Codes". Proc. 29th Annual Association for Computing Machinery (ACM) Symposium on Theory of Computation.
↑ "Digital Video Broadcast (DVB); Second generation framing structure, channel coding and modulation systems for Broadcasting, Interactive Services, News Gathering and other satellite broadband applications (DVB-S2)". En 302 307 (V1.2.1). ETSI. April 2009.
↑ Andrews, K. S.; Divsalar, D.; Dolinar, S.; Hamkins, J.; Jones, C. R.; Pollara, F. (November 2007). "The Development of Turbo and LDPC Codes for Deep-Space Applications". Proceedings of the IEEE. 95 (11): 2142–2156. doi:10.1109/JPROC.2007.905132. S2CID9289140.
↑ Dolinar, S.; Divsalar, D. (15 August 1995). "Weight Distributions for Turbo Codes Using Random and Nonrandom Permutations". TDA Progress Report. 122: 42–122. Bibcode:1995TDAPR.122...56D. CiteSeerX10.1.1.105.6640.
↑ "Digital Video Broadcast (DVB); Frame structure, channel coding and modulation for a second generation digital terrestrial television broadcasting system (DVB-T2)". En 302 755 (V1.1.1). ETSI. September 2009.
Wicker, Stephen B. (1995). Error Control Systems for Digital Communication and Storage. Englewood Cliffs, New Jersey, USA: Prentice-Hall. ISBN0-13-200809-2.
Wilson, Stephen G. (1996). Digital Modulation and Coding. Englewood Cliffs, New Jersey, USA: Prentice-Hall. ISBN0-13-210071-1.
Sphere Packings, Lattices and Groups, By J. H. Conway, Neil James Alexander Sloane, Springer Science & Business Media, 2013-03-09 – Mathematics – 682 pages.
This page is based on this Wikipedia article Text is available under the CC BY-SA 4.0 license; additional terms may apply. Images, videos and audio are available under their respective licenses.
