Error Correction Coding
About
In mathematics, computer science, telecommunication, and information theory, error detection and correction has great practical importance in maintaining data (information) integrity across noisy channels and less-than-reliable storage media.
General definitions of terms
Definitions of error detection and error correction:
- Error detection is the ability to detect the presence of errors caused by noise or other impairments during transmission from the transmitter to the receiver.
- Error correction is the additional ability to reconstruct the original, error-free data.
There are two basic ways to design the channel code and protocol for an error correcting system:
- Automatic repeat-request (ARQ): The transmitter sends the data and also an error detection code, which the receiver uses to check for errors, and requests retransmission of erroneous data. In many cases, the request is implicit; the receiver sends an acknowledgement (ACK) of correctly received data, and the transmitter re-sends anything not acknowledged within a reasonable period of time.
- Forward error correction (FEC): The transmitter encodes the data with an error-correcting code (ECC) and sends the coded message. The receiver never sends any messages back to the transmitter. The receiver decodes what it receives into the «most likely» data. The codes are designed so that it would take an «unreasonable» amount of noise to trick the receiver into misinterpreting the data.
It is possible to combine the two, so that minor errors are corrected without retransmission, and major errors are detected and a retransmission requested. The combination is called hybrid automatic repeat-request.
Error detection schemes
In telecommunication, a redundancy check is extra data added to a message for the purposes of error detection.
Several schemes exist to achieve error detection, and generally they are quite simple. All error detection codes (which include all error-detection-and-correction codes) transmit more bits than were in the original data. Most codes are «systematic«: the transmitter sends a fixed number of original data bits, followed by fixed number of check bits (usually referred to as redundancy in the literature) which are derived from the data bits by some deterministic algorithm. The receiver applies the same algorithm to the received data bits and compares its output to the received check bits; if the values do not match, an error has occurred at some point during the transmission. In a system that uses a «non-systematic» code, such as some raptor codes, data bits are transformed into at least as many code bits, and the transmitter sends only the code bits.
Repetition schemes
- Main article: repetition code
Variations on this theme exist. Given a stream of data that is to be sent, the data is broken up into blocks of bits, and in sending, each block is sent some predetermined number of times. For example, if we want to send «1011», we may repeat this block three times each.
Suppose we send «1011 1011 1011», and this is received as «1010 1011 1011». As one group is not the same as the other two, we can determine that an error has occurred. This scheme is not very efficient, and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g. «1010 1010 1010» in the example above will be detected as correct in this scheme).
The scheme however is extremely simple, and is in fact used in some transmissions of numbers stations.Template:Fact
Parity schemes
- Main article: Parity bit
A parity bit is an error detection mechanism that can only detect an odd number of errors.
The stream of data is broken up into blocks of bits, and the number of 1 bits is counted. Then, a «parity bit» is set (or cleared) if the number of one bits is odd (or even). (This scheme is called even parity; odd parity can also be used.) If the tested blocks overlap, then the parity bits can be used to isolate the error, and even correct it if the error affects a single bit: this is the principle behind the Hamming code.
There is a limitation to parity schemes. A parity bit is only guaranteed to detect an odd number of bit errors (one, three, five, and so on). If an even number of bits (two, four, six and so on) are flipped, the parity bit appears to be correct, even though the data is corrupt.
Checksum
- Main article: Checksum
A checksum of a message is an arithmetic sum of message code words of a certain word length, for example byte values, and their carry value. The sum is negated by means of ones-complement, and stored or transferred as an extra code word extending the message.
On the receiver side, a new checksum may be calculated from the extended message. If the new checksum is not 0, an error has been detected.
Checksum schemes include parity bits, check digits, and longitudinal redundancy check.
Cyclic redundancy checks
- Main article: Cyclic redundancy check
More complex error detection (and correction) methods make use of the properties of finite fields and polynomials over such fields.
The cyclic redundancy check considers a block of data as the coefficients to a polynomial and then divides by a fixed, predetermined polynomial. The coefficients of the result of the division is taken as the redundant data bits, the CRC.
On reception, one can recompute the CRC from the payload bits and compare this with the CRC that was received. A mismatch indicates that an error occurred.
Hamming distance based checks
If we want to detect d bit errors in an n bit word we can map every n bit word into a bigger n+d+1 bit word so that the minimum Hamming distance between each valid mapping is d+1. This way, if one receives a n+d+1 word that doesn’t match any word in the mapping (with a Hamming distance x <= d+1 from any word in the mapping) it can successfully detect it as an erroneous word. Even more, d or fewer errors will never transform a valid word into another, because the Hamming distance between each valid word is at least d+1, and such errors only lead to invalid words that are detected correctly.
Given a stream of m*n bits, we can detect x <= d bit errors successfully using the above method on every n bit word. In fact, we can detect a maximum of m*d errors if every n word is transmitted with maximum d errors.
Hash function
Any hash function can be used as a integrity check.
Horizontal and vertical redundancy check
Other types of redundancy check include horizontal redundancy check, vertical redundancy check and «double», «dual» or «diagonal» parity (used in RAID-DP).
Automatic repeat request
- Main article: Automatic repeat-request
Automatic Repeat-reQuest (ARQ) is an error control method for data transmission which makes use of error detection codes, acknowledgment and/or negative acknowledgement messages and timeouts to achieve reliable data transmission. An acknowledgment is a message sent by the receiver to the transmitter to indicate that it has correctly received a data frame.
Usually, when the transmitter does not receive the acknowledgment before the timeout occurs (i.e. within a reasonable amount of time after sending the data frame), it retransmits the frame until it is either correctly received or the error persists beyond a predetermined number of retransmissions.
A few types of ARQ protocols are Stop-and-wait ARQ, Go-Back-N ARQ and Selective Repeat ARQ.
Hybrid ARQ is a combination of ARQ and forward error correction.
Error-correcting code
- Main article: Forward error correction
An error-correcting code (ECC) or forward error correction (FEC) code is redundant data that is added to the message on the sender side. If the number of errors is within the capability of the code being used, the receiver can use the extra information to discover the locations of the errors and correct them. Since the receiver does not have to ask the sender for retransmission of the data, a back-channel is not necessary in forward error correction, so it is suitable for simplex communication such as broadcasting. Error-correcting codes are used in computer data storage, for example CDs, DVDs and in dynamic RAM. It is also used in digital transmission, especially wireless communication, since wireless communication without FEC often would suffer from packet-error rates close to 100%, and conventional automatic repeat request error control would yield a very low goodput.
Two main categories are convolutional codes and block codes.
Convolutional codes are processed on a bit-by-bit basis, and only causes a processing delay corresponding to a few bit periods. In convolutional coding, a Viterbi decoder is typically used on the receiver side.
Block codes are processed on a block-by-block basis. Early examples of block codes are repetition codes, Hamming codes and multidimensional parity-check codes. More efficient codes, often used in modern systems, are Reed-Solomon codes, turbo codes, BCH codes, Reed-Muller codes, Binary Golay codes, and low-density parity-check codes (LDPC). For example, the code rate of the Reed Solomon block code denoted RS(204,188) processes blocks of 188 bytes of useful information at a time, and appends 204 — 188 = 16 redundant bytes to each block. It can handle 8 incorrect bytes per data block.
Shannon’s theorem is an important theorem in forward error correction which describes a theoretical upper bound for the attainable spectral efficiency in bit/s/Hz of a channel coding scheme (an error-correcting scheme, typically combined width a digital modulation scheme) for a specific signal-to-noise ratio (SNR). For example, if the SNR is 0 dB, the spectral efficiency can not be higher than 1 bit/s/Hz, resulting in that the information rate in bit/s (the useful bitrate, excluding redundant ECC) can not be higher than the bandwidth in hertz.
The effectiveness of the coding scheme may also be measured in terms of code rate k/n, which is the ratio between k information bits and n transmitted data bits. Finally the effectiveness may be expressed as coding gain in decibel, which is the allowable reduction in signal-to-noise ratio whilst still attaining the same bit error rate and data rate as the uncoded system.
Error-correcting memory
- Main article: Dynamic random access memory#Errors and error correction
Because soft errors are extremely common in the DRAM of computers used in satellites and space probes, such memory is structured as ECC memory (also called «EDAC protected memory»).
Such memory controllers traditionally use a Hamming code, although some use triple modular redundancy.
Even though a single cosmic ray can upset many physically neighboring bits in a DRAM, such memory systems are designed so that neighboring bits belong to different words, so that a single event upset (SEU) causes only a single error in any particular word, and so can be corrected by a single-bit error correcting code.
As long as no more than a single bit in any particular word is affected by an error between accesses, such a memory system presents the illusion of an error-free memory.[1]
Applications
Applications that require low latency (such as telephone conversations) cannot use Automatic Repeat reQuest (ARQ); they must use Forward Error Correction (FEC). By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be any good.
Applications where the transmitter immediately forgets the information as soon as it is sent (such as most television cameras) cannot use ARQ; they must use FEC because when an error occurs, the original data is no longer available. (This is also why FEC is used in data storage systems such as RAID and distributed data store).
Applications that use ARQ must have a return channel. Applications that have no return channel cannot use ARQ.
Applications that require extremely low error rates (such as digital money transfers) must use ARQ.
The Internet
In a typical TCP/IP stack, error detection is performed at multiple levels:
- Each Ethernet frame carries a CRC-32 checksum. The receiver discards frames if their checksums do not match.
- The IPv4 header contains a header checksum of the contents of the header (excluding the checksum field). Packets with checksums that don’t match are discarded.
- The checksum was omitted from the IPv6 header, because most current link layer protocols have error detection.
- UDP has an optional checksum. Packets with wrong checksums are discarded.
- TCP has a checksum of the payload, TCP header (excluding the checksum field) and source- and destination addresses of the IP header. Packets found to have incorrect checksums are discarded and eventually get retransmitted when the sender receives a triple-ack or a timeout occurs.
Deep-space telecommunications
NASA has used many different error correcting codes. For missions between 1969 and 1977 the Mariner spacecraft used a Reed-Muller code. The noise these spacecraft were subject to was well approximated by a «bell-curve» (normal distribution), so the Reed-Muller codes were well suited to the situation.
The Voyager 1 & Voyager 2 spacecraft transmitted color pictures of Jupiter and Saturn in 1979 and 1980.
- Color image transmission required 3 times the amount of data, so the Golay (24,12,8) code was used.Template:Fact[2]
- This Golay code is only 3-error correcting, but it could be transmitted at a much higher data rate.
- Voyager 2 went on to Uranus and Neptune and the code was switched to a concatenated Reed-Solomon code—Convolutional code for its substantially more powerful error correcting capabilities.
- Current DSN error correction is done with dedicated hardware.
- For some NASA deep space craft such as those in the Voyager program, Cassini-Huygens (Saturn), New Horizons (Pluto) and Deep Space 1—the use of hardware ECC may not be feasible for the full duration of the mission.
The different kinds of deep space and orbital missions that are conducted suggest that trying to find a «one size fits all» error correction system will be an ongoing problem for some time to come.
- For missions close to the earth the nature of the «noise» is different from that on a spacecraft headed towards the outer planets.
- In particular, if a transmitter on a spacecraft far from earth is operating at a low power, the problem of correcting for noise gets larger with distance from the earth.
Satellite broadcasting (DVB)
The demand for satellite transponder bandwidth continues to grow, fueled by the desire to deliver television (including new channels and High Definition TV) and IP data. Transponder availability and bandwidth constraints have limited this growth, because transponder capacity is determined by the selected modulation scheme and Forward error correction (FEC) rate.
Overview
- QPSK coupled with traditional Reed Solomon and Viterbi codes have been used for nearly 20 years for the delivery of digital satellite TV.
- Higher order modulation schemes such as 8PSK, 16QAM and 32QAM have enabled the satellite industry to increase transponder efficiency by several orders of magnitude.
- This increase in the information rate in a transponder comes at the expense of an increase in the carrier power to meet the threshold requirement for existing antennas.
- Tests conducted using the latest chipsets demonstrate that the performance achieved by using Turbo Codes may be even lower than the 0.8 dB figure assumed in early designs.
Data storage
Error detection and correction codes are often used to improve the reliability of data storage media.
A «parity track» was present on the first magnetic tape data storage in 1951. The «Optimal Rectangular Code» used in group code recording tapes not only detects but also corrects single-bit errors.
Some file formats, particularly archive formats, include a checksum (most often CRC32) to detect corruption and truncation and can employ redundancy and/or parity files to recover portions of corrupted data.
Reed Solomon codes are used in compact discs to correct errors caused by scratches.
Modern hard drives use CRC codes to detect and Reed-Solomon codes to correct minor errors in sector reads, and to recover data from sectors that have «gone bad» and store that data in the spare sectors.[3]
RAID systems use a variety of error correction techniques, to correct errors when a hard drive completely fails.
Information theory and error detection and correction
Information theory tells us that whatever the probability of error in transmission or storage, it is possible to construct error-correcting codes in which the likelihood of failure is arbitrarily low, although this requires adding increasing amounts of redundant data to the original, which might not be practical when the error probability is very high. Shannon’s theorem sets an upper bound to the error correction rate that can be achieved (and thus the level of noise that can be tolerated) using a fixed amount of redundancy, but does not tell us how to construct such an optimal code.
Error-correcting codes can be divided into block codes and convolutional codes. Other block error-correcting codes, such as Reed-Solomon codes, transform a chunk of bits into a (longer) chunk of bits in such a way that errors up to some threshold in each block can be detected and corrected.
However, in practice errors often occur in bursts rather than at random. This is often compensated for by shuffling (interleaving) the bits in the message after coding. Then any burst of bit-errors is broken up into a set of scattered single-bit errors when the bits of the message are unshuffled (de-interleaved) before being decoded.
List of error-correction, error-detection methods
This list contains methods of error correction (Reed-Solomon, for example is a method) and practical techniques for error correction (like the Check digit, a practical method).
- Berger code
- Chipkill, an application of ECC techniques to volatile system memory.
- Constant-weight code
- Convolutional codes are usually decoded with iterative Viterbi decoding techniques
- Differential space–time codes, related to space–time block codes.
- Dual modular redundancy, subset of N-modular redundancy, related to triple modular redundancy
- Erasure codes are a superset of Fountain codes
- Forward error correction
- Group code
- Golay code, the Binary Golay codes are the most commonly used Golay codes
- Goppa code that is used to create the McEliece cryptosystem
- Hadamard code
- Hagelbarger code
- Hamming code
- Lexicographic code
- Longitudinal redundancy check
- Low-density parity-check code
- LT codes are near optimal rateless erasure correcting codes.
- m of n codes
- Online codes are an example of rateless erasure codes.
- Parity bit
- Raptor codes are a class of fountain codes.
- Reed-Solomon error correction
- Reed-Muller code
- Repeat-accumulate code
- Sparse graph code
- Space–time code
- Space–time trellis code
- Tornado codes are optimal Fountain codes
- Triple modular redundancy
- Turbo code
- Viterbi algorithm
Practical uses of Error Correction methods
- Concatenated error correction codes, the Compact Disc and Voyager Program spacecraft use concatenated error correction technologies
- Check digit, commonly used on UPC barcodes
- Luhn algorithm, the most commonly used base 10 checksum that can perform limited error detection but not error correction
- Luhn mod N algorithm, the above algorithm but implementable in a non base 10 form
- Verhoeff algorithm, a modular based form not related to the Luhn algorithms that can detect most forms of transposition errors in financial cryptographic applications.
References
- ↑
- ↑ http://www-math.cudenver.edu/~wcherowi/courses/m6409/mariner9talk.pdf
- ↑ My Hard Drive Died | Scott A. Moulton
Links
Error Correction Coding
About
In mathematics, computer science, telecommunication, and information theory, error detection and correction has great practical importance in maintaining data (information) integrity across noisy channels and less-than-reliable storage media.
General definitions of terms
Definitions of error detection and error correction:
- Error detection is the ability to detect the presence of errors caused by noise or other impairments during transmission from the transmitter to the receiver.
- Error correction is the additional ability to reconstruct the original, error-free data.
There are two basic ways to design the channel code and protocol for an error correcting system:
- Automatic repeat-request (ARQ): The transmitter sends the data and also an error detection code, which the receiver uses to check for errors, and requests retransmission of erroneous data. In many cases, the request is implicit; the receiver sends an acknowledgement (ACK) of correctly received data, and the transmitter re-sends anything not acknowledged within a reasonable period of time.
- Forward error correction (FEC): The transmitter encodes the data with an error-correcting code (ECC) and sends the coded message. The receiver never sends any messages back to the transmitter. The receiver decodes what it receives into the «most likely» data. The codes are designed so that it would take an «unreasonable» amount of noise to trick the receiver into misinterpreting the data.
It is possible to combine the two, so that minor errors are corrected without retransmission, and major errors are detected and a retransmission requested. The combination is called hybrid automatic repeat-request.
Error detection schemes
In telecommunication, a redundancy check is extra data added to a message for the purposes of error detection.
Several schemes exist to achieve error detection, and generally they are quite simple. All error detection codes (which include all error-detection-and-correction codes) transmit more bits than were in the original data. Most codes are «systematic«: the transmitter sends a fixed number of original data bits, followed by fixed number of check bits (usually referred to as redundancy in the literature) which are derived from the data bits by some deterministic algorithm. The receiver applies the same algorithm to the received data bits and compares its output to the received check bits; if the values do not match, an error has occurred at some point during the transmission. In a system that uses a «non-systematic» code, such as some raptor codes, data bits are transformed into at least as many code bits, and the transmitter sends only the code bits.
Repetition schemes
- Main article: repetition code
Variations on this theme exist. Given a stream of data that is to be sent, the data is broken up into blocks of bits, and in sending, each block is sent some predetermined number of times. For example, if we want to send «1011», we may repeat this block three times each.
Suppose we send «1011 1011 1011», and this is received as «1010 1011 1011». As one group is not the same as the other two, we can determine that an error has occurred. This scheme is not very efficient, and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g. «1010 1010 1010» in the example above will be detected as correct in this scheme).
The scheme however is extremely simple, and is in fact used in some transmissions of numbers stations.Template:Fact
Parity schemes
- Main article: Parity bit
A parity bit is an error detection mechanism that can only detect an odd number of errors.
The stream of data is broken up into blocks of bits, and the number of 1 bits is counted. Then, a «parity bit» is set (or cleared) if the number of one bits is odd (or even). (This scheme is called even parity; odd parity can also be used.) If the tested blocks overlap, then the parity bits can be used to isolate the error, and even correct it if the error affects a single bit: this is the principle behind the Hamming code.
There is a limitation to parity schemes. A parity bit is only guaranteed to detect an odd number of bit errors (one, three, five, and so on). If an even number of bits (two, four, six and so on) are flipped, the parity bit appears to be correct, even though the data is corrupt.
Checksum
- Main article: Checksum
A checksum of a message is an arithmetic sum of message code words of a certain word length, for example byte values, and their carry value. The sum is negated by means of ones-complement, and stored or transferred as an extra code word extending the message.
On the receiver side, a new checksum may be calculated from the extended message. If the new checksum is not 0, an error has been detected.
Checksum schemes include parity bits, check digits, and longitudinal redundancy check.
Cyclic redundancy checks
- Main article: Cyclic redundancy check
More complex error detection (and correction) methods make use of the properties of finite fields and polynomials over such fields.
The cyclic redundancy check considers a block of data as the coefficients to a polynomial and then divides by a fixed, predetermined polynomial. The coefficients of the result of the division is taken as the redundant data bits, the CRC.
On reception, one can recompute the CRC from the payload bits and compare this with the CRC that was received. A mismatch indicates that an error occurred.
Hamming distance based checks
If we want to detect d bit errors in an n bit word we can map every n bit word into a bigger n+d+1 bit word so that the minimum Hamming distance between each valid mapping is d+1. This way, if one receives a n+d+1 word that doesn’t match any word in the mapping (with a Hamming distance x <= d+1 from any word in the mapping) it can successfully detect it as an erroneous word. Even more, d or fewer errors will never transform a valid word into another, because the Hamming distance between each valid word is at least d+1, and such errors only lead to invalid words that are detected correctly.
Given a stream of m*n bits, we can detect x <= d bit errors successfully using the above method on every n bit word. In fact, we can detect a maximum of m*d errors if every n word is transmitted with maximum d errors.
Hash function
Any hash function can be used as a integrity check.
Horizontal and vertical redundancy check
Other types of redundancy check include horizontal redundancy check, vertical redundancy check and «double», «dual» or «diagonal» parity (used in RAID-DP).
Automatic repeat request
- Main article: Automatic repeat-request
Automatic Repeat-reQuest (ARQ) is an error control method for data transmission which makes use of error detection codes, acknowledgment and/or negative acknowledgement messages and timeouts to achieve reliable data transmission. An acknowledgment is a message sent by the receiver to the transmitter to indicate that it has correctly received a data frame.
Usually, when the transmitter does not receive the acknowledgment before the timeout occurs (i.e. within a reasonable amount of time after sending the data frame), it retransmits the frame until it is either correctly received or the error persists beyond a predetermined number of retransmissions.
A few types of ARQ protocols are Stop-and-wait ARQ, Go-Back-N ARQ and Selective Repeat ARQ.
Hybrid ARQ is a combination of ARQ and forward error correction.
Error-correcting code
- Main article: Forward error correction
An error-correcting code (ECC) or forward error correction (FEC) code is redundant data that is added to the message on the sender side. If the number of errors is within the capability of the code being used, the receiver can use the extra information to discover the locations of the errors and correct them. Since the receiver does not have to ask the sender for retransmission of the data, a back-channel is not necessary in forward error correction, so it is suitable for simplex communication such as broadcasting. Error-correcting codes are used in computer data storage, for example CDs, DVDs and in dynamic RAM. It is also used in digital transmission, especially wireless communication, since wireless communication without FEC often would suffer from packet-error rates close to 100%, and conventional automatic repeat request error control would yield a very low goodput.
Two main categories are convolutional codes and block codes.
Convolutional codes are processed on a bit-by-bit basis, and only causes a processing delay corresponding to a few bit periods. In convolutional coding, a Viterbi decoder is typically used on the receiver side.
Block codes are processed on a block-by-block basis. Early examples of block codes are repetition codes, Hamming codes and multidimensional parity-check codes. More efficient codes, often used in modern systems, are Reed-Solomon codes, turbo codes, BCH codes, Reed-Muller codes, Binary Golay codes, and low-density parity-check codes (LDPC). For example, the code rate of the Reed Solomon block code denoted RS(204,188) processes blocks of 188 bytes of useful information at a time, and appends 204 — 188 = 16 redundant bytes to each block. It can handle 8 incorrect bytes per data block.
Shannon’s theorem is an important theorem in forward error correction which describes a theoretical upper bound for the attainable spectral efficiency in bit/s/Hz of a channel coding scheme (an error-correcting scheme, typically combined width a digital modulation scheme) for a specific signal-to-noise ratio (SNR). For example, if the SNR is 0 dB, the spectral efficiency can not be higher than 1 bit/s/Hz, resulting in that the information rate in bit/s (the useful bitrate, excluding redundant ECC) can not be higher than the bandwidth in hertz.
The effectiveness of the coding scheme may also be measured in terms of code rate k/n, which is the ratio between k information bits and n transmitted data bits. Finally the effectiveness may be expressed as coding gain in decibel, which is the allowable reduction in signal-to-noise ratio whilst still attaining the same bit error rate and data rate as the uncoded system.
Error-correcting memory
- Main article: Dynamic random access memory#Errors and error correction
Because soft errors are extremely common in the DRAM of computers used in satellites and space probes, such memory is structured as ECC memory (also called «EDAC protected memory»).
Such memory controllers traditionally use a Hamming code, although some use triple modular redundancy.
Even though a single cosmic ray can upset many physically neighboring bits in a DRAM, such memory systems are designed so that neighboring bits belong to different words, so that a single event upset (SEU) causes only a single error in any particular word, and so can be corrected by a single-bit error correcting code.
As long as no more than a single bit in any particular word is affected by an error between accesses, such a memory system presents the illusion of an error-free memory.[1]
Applications
Applications that require low latency (such as telephone conversations) cannot use Automatic Repeat reQuest (ARQ); they must use Forward Error Correction (FEC). By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be any good.
Applications where the transmitter immediately forgets the information as soon as it is sent (such as most television cameras) cannot use ARQ; they must use FEC because when an error occurs, the original data is no longer available. (This is also why FEC is used in data storage systems such as RAID and distributed data store).
Applications that use ARQ must have a return channel. Applications that have no return channel cannot use ARQ.
Applications that require extremely low error rates (such as digital money transfers) must use ARQ.
The Internet
In a typical TCP/IP stack, error detection is performed at multiple levels:
- Each Ethernet frame carries a CRC-32 checksum. The receiver discards frames if their checksums do not match.
- The IPv4 header contains a header checksum of the contents of the header (excluding the checksum field). Packets with checksums that don’t match are discarded.
- The checksum was omitted from the IPv6 header, because most current link layer protocols have error detection.
- UDP has an optional checksum. Packets with wrong checksums are discarded.
- TCP has a checksum of the payload, TCP header (excluding the checksum field) and source- and destination addresses of the IP header. Packets found to have incorrect checksums are discarded and eventually get retransmitted when the sender receives a triple-ack or a timeout occurs.
Deep-space telecommunications
NASA has used many different error correcting codes. For missions between 1969 and 1977 the Mariner spacecraft used a Reed-Muller code. The noise these spacecraft were subject to was well approximated by a «bell-curve» (normal distribution), so the Reed-Muller codes were well suited to the situation.
The Voyager 1 & Voyager 2 spacecraft transmitted color pictures of Jupiter and Saturn in 1979 and 1980.
- Color image transmission required 3 times the amount of data, so the Golay (24,12,8) code was used.Template:Fact[2]
- This Golay code is only 3-error correcting, but it could be transmitted at a much higher data rate.
- Voyager 2 went on to Uranus and Neptune and the code was switched to a concatenated Reed-Solomon code—Convolutional code for its substantially more powerful error correcting capabilities.
- Current DSN error correction is done with dedicated hardware.
- For some NASA deep space craft such as those in the Voyager program, Cassini-Huygens (Saturn), New Horizons (Pluto) and Deep Space 1—the use of hardware ECC may not be feasible for the full duration of the mission.
The different kinds of deep space and orbital missions that are conducted suggest that trying to find a «one size fits all» error correction system will be an ongoing problem for some time to come.
- For missions close to the earth the nature of the «noise» is different from that on a spacecraft headed towards the outer planets.
- In particular, if a transmitter on a spacecraft far from earth is operating at a low power, the problem of correcting for noise gets larger with distance from the earth.
Satellite broadcasting (DVB)
The demand for satellite transponder bandwidth continues to grow, fueled by the desire to deliver television (including new channels and High Definition TV) and IP data. Transponder availability and bandwidth constraints have limited this growth, because transponder capacity is determined by the selected modulation scheme and Forward error correction (FEC) rate.
Overview
- QPSK coupled with traditional Reed Solomon and Viterbi codes have been used for nearly 20 years for the delivery of digital satellite TV.
- Higher order modulation schemes such as 8PSK, 16QAM and 32QAM have enabled the satellite industry to increase transponder efficiency by several orders of magnitude.
- This increase in the information rate in a transponder comes at the expense of an increase in the carrier power to meet the threshold requirement for existing antennas.
- Tests conducted using the latest chipsets demonstrate that the performance achieved by using Turbo Codes may be even lower than the 0.8 dB figure assumed in early designs.
Data storage
Error detection and correction codes are often used to improve the reliability of data storage media.
A «parity track» was present on the first magnetic tape data storage in 1951. The «Optimal Rectangular Code» used in group code recording tapes not only detects but also corrects single-bit errors.
Some file formats, particularly archive formats, include a checksum (most often CRC32) to detect corruption and truncation and can employ redundancy and/or parity files to recover portions of corrupted data.
Reed Solomon codes are used in compact discs to correct errors caused by scratches.
Modern hard drives use CRC codes to detect and Reed-Solomon codes to correct minor errors in sector reads, and to recover data from sectors that have «gone bad» and store that data in the spare sectors.[3]
RAID systems use a variety of error correction techniques, to correct errors when a hard drive completely fails.
Information theory and error detection and correction
Information theory tells us that whatever the probability of error in transmission or storage, it is possible to construct error-correcting codes in which the likelihood of failure is arbitrarily low, although this requires adding increasing amounts of redundant data to the original, which might not be practical when the error probability is very high. Shannon’s theorem sets an upper bound to the error correction rate that can be achieved (and thus the level of noise that can be tolerated) using a fixed amount of redundancy, but does not tell us how to construct such an optimal code.
Error-correcting codes can be divided into block codes and convolutional codes. Other block error-correcting codes, such as Reed-Solomon codes, transform a chunk of bits into a (longer) chunk of bits in such a way that errors up to some threshold in each block can be detected and corrected.
However, in practice errors often occur in bursts rather than at random. This is often compensated for by shuffling (interleaving) the bits in the message after coding. Then any burst of bit-errors is broken up into a set of scattered single-bit errors when the bits of the message are unshuffled (de-interleaved) before being decoded.
List of error-correction, error-detection methods
This list contains methods of error correction (Reed-Solomon, for example is a method) and practical techniques for error correction (like the Check digit, a practical method).
- Berger code
- Chipkill, an application of ECC techniques to volatile system memory.
- Constant-weight code
- Convolutional codes are usually decoded with iterative Viterbi decoding techniques
- Differential space–time codes, related to space–time block codes.
- Dual modular redundancy, subset of N-modular redundancy, related to triple modular redundancy
- Erasure codes are a superset of Fountain codes
- Forward error correction
- Group code
- Golay code, the Binary Golay codes are the most commonly used Golay codes
- Goppa code that is used to create the McEliece cryptosystem
- Hadamard code
- Hagelbarger code
- Hamming code
- Lexicographic code
- Longitudinal redundancy check
- Low-density parity-check code
- LT codes are near optimal rateless erasure correcting codes.
- m of n codes
- Online codes are an example of rateless erasure codes.
- Parity bit
- Raptor codes are a class of fountain codes.
- Reed-Solomon error correction
- Reed-Muller code
- Repeat-accumulate code
- Sparse graph code
- Space–time code
- Space–time trellis code
- Tornado codes are optimal Fountain codes
- Triple modular redundancy
- Turbo code
- Viterbi algorithm
Practical uses of Error Correction methods
- Concatenated error correction codes, the Compact Disc and Voyager Program spacecraft use concatenated error correction technologies
- Check digit, commonly used on UPC barcodes
- Luhn algorithm, the most commonly used base 10 checksum that can perform limited error detection but not error correction
- Luhn mod N algorithm, the above algorithm but implementable in a non base 10 form
- Verhoeff algorithm, a modular based form not related to the Luhn algorithms that can detect most forms of transposition errors in financial cryptographic applications.
References
- ↑
- ↑ http://www-math.cudenver.edu/~wcherowi/courses/m6409/mariner9talk.pdf
- ↑ My Hard Drive Died | Scott A. Moulton
Links
Аннотация: Контроль по четности, CRC, алгоритм Хэмминга. Введение в коды Рида-Соломона: принципы, архитектура и реализация. Метод коррекции ошибок FEC (Forward Error Correction).
Каналы передачи данных ненадежны (шумы, наводки и т.д.), да и само оборудование обработки информации работает со сбоями. По этой причине важную роль приобретают механизмы детектирования ошибок. Ведь если ошибка обнаружена, можно осуществить повторную передачу данных и решить проблему. Если исходный код по своей длине равен полученному коду, обнаружить ошибку передачи не предоставляется возможным. Можно, конечно, передать код дважды и сравнить, но это уже двойная избыточность.
Простейшим способом обнаружения ошибок является контроль по четности. Обычно контролируется передача блока данных ( М бит). Этому блоку ставится в соответствие кодовое слово длиной N бит, причем N>M. Избыточность кода характеризуется величиной 1-M/N. Вероятность обнаружения ошибки определяется отношением M/N (чем меньше это отношение, тем выше вероятность обнаружения ошибки, но и выше избыточность).
При передаче информации она кодируется таким образом, чтобы с одной стороны характеризовать ее минимальным числом символов, а с другой – минимизировать вероятность ошибки при декодировании получателем. Для выбора типа кодирования важную роль играет так называемое расстояние Хэмминга.
Пусть А и Б — две двоичные кодовые последовательности равной длины. Расстояние Хэмминга между двумя этими кодовыми последовательностями равно числу символов, которыми они отличаются. Например, расстояние Хэмминга между кодами 00111 и 10101 равно 2.
Можно показать, что для детектирования ошибок в n битах схема кодирования требует применения кодовых слов с расстоянием Хэмминга не менее N + 1. Можно также показать, что для исправления ошибок в N битах необходима схема кодирования с расстоянием Хэмминга между кодами не менее 2N + 1. Таким образом, конструируя код, мы пытаемся обеспечить расстояние Хэмминга между возможными кодовыми последовательностями большее, чем оно может возникнуть из-за ошибок.
Широко распространены коды с одиночным битом четности. В этих кодах к каждым М бит добавляется 1 бит, значение которого определяется четностью (или нечетностью) суммы этих М бит. Так, например, для двухбитовых кодов 00, 01, 10, 11 кодами с контролем четности будут 000, 011, 101 и 110. Если в процессе передачи один бит будет передан неверно, четность кода из М+1 бита изменится.
Предположим, что частота ошибок ( BER – Bit Error Rate) равна р = 10-4. В этом случае вероятность передачи 8 бит с ошибкой составит 1 – (1 – p)8 = 7,9 х 10-4. Добавление бита четности позволяет детектировать любую ошибку в одном из переданных битах. Здесь вероятность ошибки в одном из 9 битов равна 9p(1 – p)8. Вероятность же реализации необнаруженной ошибки составит 1 – (1 – p)9 – 9p(1 – p)8 = 3,6 x 10-7. Таким образом, добавление бита четности уменьшает вероятность необнаруженной ошибки почти в 1000 раз. Использование одного бита четности типично для асинхронного метода передачи. В синхронных каналах чаще используется вычисление и передача битов четности как
для строк, так и для столбцов передаваемого массива данных. Такая схема позволяет не только регистрировать, но и исправлять ошибки в одном из битов переданного блока.
Контроль по четности достаточно эффективен для выявления одиночных и множественных ошибок в условиях, когда они являются независимыми. При возникновении ошибок в кластерах бит метод контроля четности неэффективен, и тогда предпочтительнее метод вычисления циклических сумм ( CRC — Cyclic Redundancy Check). В этом методе передаваемый кадр делится на специально подобранный образующий полином. Дополнение остатка от деления и является контрольной суммой.
В Ethernet вычисление CRC производится аппаратно. На
рис.
4.1 показан пример реализации аппаратного расчета CRC для образующего полинома R(x) = 1 + x2 + x3 + x5 + x7. В этой схеме входной код приходит слева.
Рис.
4.1.
Схема реализации расчета CRC
Эффективность CRC для обнаружения ошибок на многие порядки выше простого контроля четности. В настоящее время стандартизовано несколько типов образующих полиномов. Для оценочных целей можно считать, что вероятность невыявления ошибки в случае использования CRC, если ошибка на самом деле имеет место, равна (1/2)r, где r — степень образующего полинома.
| CRC -12 | x12 + x11 + x3 + x2 + x1 + 1 |
| CRC -16 | x16 + x15 + x2 + 1 |
| CRC -CCITT | x16 + x12 + x5 + 1 |
4.1. Алгоритмы коррекции ошибок
Исправлять ошибки труднее, чем их детектировать или предотвращать. Процедура коррекции ошибок предполагает два совмещеных процесса: обнаружение ошибки и определение места (идентификации сообщения и позиции в сообщении). После решения этих двух задач исправление тривиально — надо инвертировать значение ошибочного бита. В наземных каналах связи, где вероятность ошибки невелика, обычно используется метод детектирования ошибок и повторной пересылки фрагмента, содержащего дефект. Для спутниковых каналов с типичными для них большими задержками системы коррекции ошибок становятся привлекательными. Здесь используют коды Хэмминга или коды свертки.
Код Хэмминга представляет собой блочный код, который позволяет выявить и исправить ошибочно переданный бит в пределах переданного блока. Обычно код Хэмминга характеризуется двумя целыми числами, например, (11,7), используемыми при передаче 7-битных ASCII-кодов. Такая запись говорит, что при передаче 7-битного кода используется 4 контрольных бита (7 + 4 = 11). При этом предполагается, что имела место ошибка в одном бите и что ошибка в двух или более битах существенно менее вероятна. С учетом этого исправление ошибки осуществляется с определенной вероятностью. Например, пусть возможны следующие правильные коды (все они, кроме первого и последнего, отстоят друг от друга на расстояние Хэмминга 4):
00000000
11110000
00001111
11111111
При получении кода 00000111 нетрудно предположить, что правильное значение полученного кода равно 00001111. Другие коды отстоят от полученного на большее расстояние Хэмминга.
Рассмотрим пример передачи кода буквы s = 0x073 = 1110011 с использованием кода Хэмминга (11,7). Таблица 4.2.
| Позиция бита | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Значение бита | 1 | 1 | 1 | * | 0 | 0 | 1 | * | 1 | * | * |
Символами * помечены четыре позиции, где должны размещаться контрольные биты. Эти позиции определяются целой степенью 2 (1, 2, 4, 8 и т.д.). Контрольная сумма формируется путем выполнения операции XoR (исключающее ИЛИ) над кодами позиций ненулевых битов. В данном случае это 11, 10, 9, 5 и 3. Вычислим контрольную сумму:
| 11= | 1011 |
| 10= | 1010 |
| 09= | 1001 |
| 05= | 0101 |
| 03= | 0011 |
 |
1110 |
Таким образом, приемник получит код
| Позиция бита | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Значение бита | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 |
Просуммируем снова коды позиций ненулевых битов и получим нуль;
| 11= | 1011 |
| 10= | 1010 |
| 09= | 1001 |
| 08= | 1000 |
| 05= | 0101 |
| 04= | 0100 |
| 03= | 0011 |
| 02= | 0010 |
|
0000 |
Ну а теперь рассмотрим два случая ошибок в одном из битов посылки, например в бите 7 (1 вместо 0) и в бите 5 (0 вместо 1). Просуммируем коды позиций ненулевых битов еще раз:
|
|
В обоих случаях контрольная сумма равна позиции бита, переданного с ошибкой. Теперь для исправления ошибки достаточно инвертировать бит, номер которого указан в контрольной сумме. Понятно, что если ошибка произойдет при передаче более чем одного бита, код Хэмминга при данной избыточности окажется бесполезен.
В общем случае код имеет N = M + C бит и предполагается, что не более чем один бит в коде может иметь ошибку. Тогда возможно N+1 состояние кода (правильное состояние и n ошибочных). Пусть М = 4, а N = 7, тогда слово-сообщение будет иметь вид: M4, M3, M2, C3, M1, C2, C1. Теперь попытаемся вычислить значения С1, С2, С3. Для этого используются уравнения, где все операции представляют собой сложение по модулю 2:
С1 = М1 + М2 + М4 С2 = М1 + М3 + М4 С3 = М2 + М3 + М4
Для определения того, доставлено ли сообщение без ошибок, вычисляем следующие выражения (сложение по модулю 2):
С11 = С1 + М4 + М2 + М1 С12 = С2 + М4 + М3 + М1 С13 = С3 + М4 + М3 + М2
Результат вычисления интерпретируется следующим образом:
| C11 | C12 | C13 | Значение |
|---|---|---|---|
| 1 | 2 | 4 | Позиция бит |
| 0 | 0 | 0 | Ошибок нет |
| 0 | 0 | 1 | Бит С3 неверен |
| 0 | 1 | 0 | Бит С2 неверен |
| 0 | 1 | 1 | Бит M3 неверен |
| 1 | 0 | 0 | Бит С1 неверен |
| 1 | 0 | 1 | Бит M2 неверен |
| 1 | 1 | 0 | Бит M1 неверен |
| 1 | 1 | 1 | Бит M4 неверен |
Описанная схема легко переносится на любое число n и М.
Число возможных кодовых комбинаций М помехоустойчивого кода делится на n классов, где N — число разрешенных кодов. Разделение на классы осуществляется так, чтобы в каждый класс вошел один разрешенный код и ближайшие к нему (по расстоянию Хэмминга ) запрещенные коды. В процессе приема данных определяется, к какому классу принадлежит пришедший код. Если код принят с ошибкой, он заменяется ближайшим разрешенным кодом. При этом предполагается, что кратность ошибки не более qm.
В теории кодирования существуют следующие оценки максимального числа N n -разрядных кодов с расстоянием D.
| d=1 | n=2n |
| d=2 | n=2n-1 |
| d=3 | N 2n/(1 + n) |
| d = 2q + 1 | (для кода Хэмминга это неравенство превращается в равенство) |
В случае кода Хэмминга первые k разрядов используются в качестве информационных, причем
K = n – log(n + 1), откуда следует (логарифм по основанию 2), что k может принимать значения 0, 1, 4, 11, 26, 57 и т.д., это и определяет соответствующие коды Хэмминга (3,1); (7,4); (15,11); (31,26); (63,57) и т.д.
Обобщением кодов Хэмминга являются циклические коды BCH (Bose-Chadhuri-hocquenghem). Эти коды имеют широкий выбор длин и возможностей исправления ошибок.
Одной из старейших схем коррекции ошибок является двух-и трехмерная позиционная схема (
рис.
4.2). Для каждого байта вычисляется бит четности (бит <Ч>, направление Х). Для каждого столбца также вычисляется бит четности (направление Y. Производится вычисление битов четности для комбинаций битов с координатами (X,Y) (направление Z, слои с 1 до N ). Если при транспортировке будет искажен один бит, он может быть найден и исправлен по неверным битам четности X и Y. Если же произошло две ошибки в одной из плоскостей, битов четности данной плоскости недостаточно. Здесь поможет плоскость битов четности N+1.
Таким образом, на 512 передаваемых байтов данных пересылается около 200 бит четности.
Рис.
4.2.
Позиционная схема коррекции ошибок
См.
в лекциях Эллины
11. Контроль четности. Двумерный контроль четности
Возможно, простейшая форма обнаружения
ошибок заключается в использовании
одного бита четности. Предположим, что
на рис. 5.4 передаваемые данные D имеют
длину d разрядов. При проверке на четность
отправитель просто добавляет к данным
один бит, значение которого вычисляется
как сумма всех d разрядов данных по
модулю 2. В этом случае количество единиц
в получающемся в результате числе всегда
будет четным. Применяются также схемы,
в которых контрольный бит инвертируется,
в результате чего количество единиц в
получающемся в результате числе всегда
будет нечетным. На рис. 5.5 изображена
схема проверки на четность, а единственный
бит четности хранится в отдельном поле.
Действия, выполняемые получателем
при использовании такой схемы, также
очень просты. Получатель должен всего
лишь сосчитать количество единиц в
полученных d + 1 разрядах. Если при проверке
на четность получатель обнаруживает,
что в принятых им данных нечетное
количество единичных разрядов, он
понимает, что произошла ошибка, по
меньшей мере, в одном разряде. В общем
случае это означает, что в полученных
данных инвертировано нечетное количество
разрядов (произошла ошибка нечетной
кратности).
Что произойдет, если в полученном
пакете данных произойдет четное
количество однобитовых ошибок? В этом
случае получатель не сможет обнаружить
ошибку. Если вероятность ошибки в одном
разряде мала и можно предположить, что
ошибки в отдельных разрядах возникают
независимо друг от друга, тогда вероятность
нескольких ошибок в одном пакете крайне
мала. В таком случае единственного бита
четности может быть достаточно. Однако
практические наблюдения показали, что
в действительности ошибки не являются
независимыми, а часто группируются в
пакеты ошибок. В случае пакетных ошибок
вероятность того, что получатель не
обнаружит ошибку в пакете, может
приблизиться к величине 50 %. Очевидно,
в такой ситуации требуется более надежная
схема обнаружения ошибок! Но прежде чем
перейти к изучению схем обнаружения
ошибок, применяемых на практике,
рассмотрим простую схему, которая
обобщает предыдущую схему одноразрядного
контроля четности и помогает понять
принцип работы методов исправления
ошибок.
На рис. 5.6 показано двухмерное обобщение
одноразрядной схемы проверки на четность.
В данной схеме d разрядов пакета данных
разделяются на г строк и j столбцов,
образуя прямоугольную матрицу. Значение
четности вычисляется для каждой строки
и каждого столбца. Получающиеся в
результате i +j +1 битов четности образуют
разряды обнаружения ошибок кадра
канального уровня.
Предположим теперь, что в исходном
блоке данных из d разрядов происходит
однократная ошибка. В такой двухмерной
схеме контроля четности об ошибке будут
одновременно сигнализировать контрольные
разряды строки и столбца. Таким образом,
получатель сможет не только обнаружить
сам факт ошибки, но и по номерам строки
и столбца найти поврежденный бит данных
и исправить его! На рисунке показан
пример, в котором поврежден бит в позиции
(2, 2) — он изменил свое значение с 1 на 0.
Такую одиночную ошибку получатель может
не только обнаружить, но и исправить.
Хотя нас, в первую очередь, интересует
обнаружение и исправление ошибок в
исходных d разрядах, данная схема
позволяет также обнаруживать и исправлять
одиночные ошибки в самих битах четности.
Кроме того, данная двухмерная схема
контроля четности позволяет обнаруживать
(но не исправлять!) любые комбинации из
двух одиночных ошибок (то есть двойные
ошибки) в пакете.
Способность приемника обнаруживать
и исправлять ошибки иногда называют
прямым исправлением ошибок (Forward Error
Correction, FEC). Подобные приемы широко
применяются в устройствах хранения и
воспроизведения звука, например на
лазерных компакт-дисках. В сетях методы
обнаружения и исправления ошибок могут
использоваться сами по себе, а также в
сочетании с автоматическими запросами
на повторную передачу. Методы обнаружения
и исправления ошибок очень полезны так
как позволяют снизить необходимое
количество повторных передач. Кроме
того (что, возможно, даже важнее), эти
методы позволяют получателю немедленно
исправлять ошибки. Таким образом,
получатель данных может не ждать, пока
отправитель получит его сообщение об
ошибке и вышлет пакет еще раз, что может
быть существенным преимуществом в
сетевых приложениях реального времени.
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