information theory Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, ...

and

coding theory 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 computer data storage, data sto ...

with applications in

computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...

and

telecommunications Telecommunication, often used in its plural form or abbreviated as telecom, is the transmission of information over a distance using electronic means, typically through cables, radio waves, or other communication technologies. These means of ...

, error detection and correction (EDAC) or error control are techniques that enable reliable delivery of

digital data Digital data, in information theory and information systems, is information represented as a string of Discrete mathematics, discrete symbols, each of which can take on one of only a finite number of values from some alphabet (formal languages ...

over unreliable

communication channel A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used for infor ...

s. 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.

Definitions

''Error detection'' is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver. ''Error correction'' is the detection of errors and reconstruction of the original, error-free data.

History

In classical antiquity,

copyist A copyist is a person who makes duplications of the same thing. The modern use of the term is mainly confined to music copyists, who are employed by the music industry to produce neat copies from a composer or arranger's manuscript. However, the ...

s of the

Hebrew Bible The Hebrew Bible or Tanakh (;"Tanach"
. '' stichs (lines of verse). As the prose books of the Bible were hardly ever written in stichs, the copyists, in order to estimate the amount of work, had to count the letters. This also helped ensure accuracy in the transmission of the text with the production of subsequent copies. Between the 7th and 10th centuries CE a group of Jewish scribes formalized and expanded this to create the Numerical Masorah to ensure accurate reproduction of the sacred text. It included counts of the number of words in a line, section, book and groups of books, noting the middle stich of a book, word use statistics, and commentary. Standards became such that a deviation in even a single letter in a Torah scroll was considered unacceptable. The effectiveness of their error correction method was verified by the accuracy of copying through the centuries demonstrated by discovery of the

Dead Sea Scrolls The Dead Sea Scrolls, also called the Qumran Caves Scrolls, are a set of List of Hebrew Bible manuscripts, ancient Jewish manuscripts from the Second Temple period (516 BCE – 70 CE). They were discovered over a period of ten years, between ...

in 1947–1956, dating from . The modern development of error correction codes is credited to

Richard Hamming Richard Wesley Hamming (February 11, 1915 – January 7, 1998) was an American mathematician whose work had many implications for computer engineering and telecommunications. His contributions include the Hamming code (which makes use of a Ha ...

in 1947. A description of

Hamming's code In computer science and telecommunications, 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 ...

appeared in

Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, computer scientist, cryptographer and inventor known as the "father of information theory" and the man who laid the foundations of th ...

's ''A Mathematical Theory of Communication'' and was quickly generalized by Marcel J. E. Golay.

Principles

All error-detection and correction schemes add some redundancy (i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message and to recover data that has been determined to be corrupted. Error detection and correction schemes can be either systematic or non-systematic. In a systematic scheme, the transmitter sends the original (error-free) data and attaches a fixed number of ''check bits'' (or ''parity data''), which are derived from the data bits by some encoding algorithm. If error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some point during the transmission. If error correction is required, a receiver can apply the decoding algorithm to the received data bits and the received check bits to recover the original error-free data. In a system that uses a non-systematic code, the original message is transformed into an encoded message carrying the same information and that has at least as many bits as the original message. Good error control performance requires the scheme to be selected based on the characteristics of the communication channel. Common

channel model A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used for infor ...

s include

memoryless In probability and statistics, memorylessness is a property of probability distributions. It describes situations where previous failures or elapsed time does not affect future trials or further wait time. Only the geometric and exponential d ...

models where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in bursts. Consequently, error-detecting and -correcting codes can be generally distinguished between ''random-error-detecting/correcting'' and ''burst-error-detecting/correcting''. Some codes can also be suitable for a mixture of random errors and burst errors. If the channel characteristics cannot be determined, or are highly variable, an error-detection scheme may be combined with a system for retransmissions of erroneous data. This is known as

automatic repeat request Automatic repeat request (ARQ), also known as automatic repeat query, is an error-control method for data transmission that uses acknowledgements (messages sent by the receiver indicating that it has correctly received a message) and timeout ...

(ARQ), and is most notably used in the Internet. An alternate approach for error control is hybrid automatic repeat request (HARQ), which is a combination of ARQ and error-correction coding.

Types of error correction

There are three major types of error correction:

Automatic repeat request

Automatic repeat request Automatic repeat request (ARQ), also known as automatic repeat query, is an error-control method for data transmission that uses acknowledgements (messages sent by the receiver indicating that it has correctly received a message) and timeout ...

(ARQ) is an error control method for data transmission that makes use of error-detection codes, acknowledgment and/or negative acknowledgment messages, and timeouts to achieve reliable data transmission. An ''acknowledgment'' is a message sent by the receiver 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. Three types of ARQ protocols are Stop-and-wait ARQ, Go-Back-N ARQ, and Selective Repeat ARQ. ARQ is appropriate if the communication channel has varying or unknown capacity, such as is the case on the Internet. However, ARQ requires the availability of a back channel, results in possibly increased latency due to retransmissions, and requires the maintenance of buffers and timers for retransmissions, which in the case of network congestion can put a strain on the server and overall network capacity.A. J. McAuley, ''Reliable Broadband Communication Using a Burst Erasure Correcting Code'', ACM SIGCOMM, 1990. For example, ARQ is used on shortwave radio data links in the form of ARQ-E, or combined with multiplexing as ARQ-M.

Forward error correction

Forward error correction (FEC) is a process of adding redundant data such as an error-correcting code (ECC) to a message so that it can be recovered by a receiver even when a number of errors (up to the capability of the code being used) are introduced, either during the process of transmission or on storage. Since the receiver does not have to ask the sender for retransmission of the data, a

backchannel Backchannel is the use of networked computers to maintain a real-time online conversation alongside the primary group activity or live spoken remarks. The term was coined from Backchannel (linguistics), the linguistics term to describe listeners' ...

is not required in forward error correction. Error-correcting codes are used in lower-layer communication such as

cellular network A cellular network or mobile network is a telecommunications network where the link to and from end nodes is wireless network, wireless and the network is distributed over land areas called ''cells'', each served by at least one fixed-locatio ...

, high-speed

fiber-optic communication Fiber-optic communication is a form of optical communication for transmitting information from one place to another by sending pulses of infrared or visible light through an optical fiber. The light is a form of carrier wave that is modul ...

and

Wi-Fi Wi-Fi () is a family of wireless network protocols based on the IEEE 802.11 family of standards, which are commonly used for Wireless LAN, local area networking of devices and Internet access, allowing nearby digital devices to exchange data by ...

, as well as for reliable storage in media such as

flash memory Flash memory is an Integrated circuit, electronic Non-volatile memory, non-volatile computer memory storage medium that can be electrically erased and reprogrammed. The two main types of flash memory, NOR flash and NAND flash, are named for t ...

hard disk A hard disk drive (HDD), hard disk, hard drive, or fixed disk is an electro-mechanical data storage device that stores and retrieves digital data using magnetic storage with one or more rigid rapidly rotating hard disk drive platter, pla ...

and RAM. Error-correcting codes are usually distinguished between convolutional codes and

block code In coding theory, block codes are a large and important family of Channel coding, 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. Th ...

s: * ''Convolutional codes'' are processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the Viterbi decoder allows optimal decoding. * ''Block codes'' are processed on a block-by-block basis. Early examples of block codes are repetition codes,

Hamming code In computer science and telecommunications, 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 ...

s and multidimensional parity-check codes. They were followed by a number of efficient codes, Reed–Solomon codes being the most notable due to their current widespread use. Turbo codes and

low-density parity-check code Low-density parity-check (LDPC) codes are a class of error correction codes which (together with the closely-related turbo codes) have gained prominence in coding theory and information theory since the late 1990s. The codes today are widely us ...

s (LDPC) are relatively new constructions that can provide almost optimal efficiency. Shannon's theorem is an important theorem in forward error correction, and describes the maximum information rate at which reliable communication is possible over a channel that has a certain error probability or

signal-to-noise ratio Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in deci ...

(SNR). This strict upper limit is expressed in terms of the channel capacity. More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a discrete memoryless channel can be made arbitrarily small, provided that the

code rate In telecommunication and information theory, the code rate (or information rateHuffman, W. Cary, and Pless, Vera, ''Fundamentals of Error-Correcting Codes'', Cambridge, 2003.) of a forward error correction code is the proportion of the data-stre ...

is smaller than the channel capacity. The code rate is defined as the fraction ''k/n'' of ''k'' source symbols and ''n'' encoded symbols. The actual maximum code rate allowed depends on the error-correcting code used, and may be lower. This is because Shannon's proof was only of existential nature, and did not show how to construct codes that are both optimal and have efficient encoding and decoding algorithms.

Hybrid schemes

Hybrid ARQ is a combination of ARQ and forward error correction. There are two basic approaches: * Messages are always transmitted with FEC parity data (and error-detection redundancy). A receiver decodes a message using the parity information and requests retransmission using ARQ only if the parity data was not sufficient for successful decoding (identified through a failed integrity check). * Messages are transmitted without parity data (only with error-detection information). If a receiver detects an error, it requests FEC information from the transmitter using ARQ and uses it to reconstruct the original message. The latter approach is particularly attractive on an erasure channel when using a rateless erasure code.

Types of error detection

Error detection is most commonly realized using a suitable

hash function A hash function is any Function (mathematics), function that can be used to map data (computing), data of arbitrary size to fixed-size values, though there are some hash functions that support variable-length output. The values returned by a ...

(or specifically, a checksum, cyclic redundancy check or other algorithm). A hash function adds a fixed-length ''tag'' to a message, which enables receivers to verify the delivered message by recomputing the tag and comparing it with the one provided. There exists a vast variety of different hash function designs. However, some are of particularly widespread use because of either their simplicity or their suitability for detecting certain kinds of errors (e.g., the cyclic redundancy check's performance in detecting

burst error In telecommunications, a burst error or error burst is a contiguous sequence of symbols, received over a communication channel, such that the first and last symbols are in error and there exists no contiguous subsequence of ''m'' correctly receiv ...

s).

Minimum distance coding

A random-error-correcting code based on minimum distance coding can provide a strict guarantee on the number of detectable errors, but it may not protect against a preimage attack.

Repetition codes

A repetition code is a coding scheme that repeats the bits across a channel to achieve error-free communication. Given a stream of data to be transmitted, the data are divided into blocks of bits. Each block is transmitted some predetermined number of times. For example, to send the bit pattern , the four-bit block can be repeated three times, thus producing . If this twelve-bit pattern was received as – where the first block is unlike the other two – an error has occurred. A repetition code is very inefficient and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g., in the previous example would be detected as correct). The advantage of repetition codes is that they are extremely simple, and are in fact used in some transmissions of numbers stations.

Parity bit

A ''parity bit'' is a bit that is added to a group of source bits to ensure that the number of set bits (i.e., bits with value 1) in the outcome is even or odd. It is a very simple scheme that can be used to detect single or any other odd number (i.e., three, five, etc.) of errors in the output. An even number of flipped bits will make the parity bit appear correct even though the data is erroneous. Parity bits added to each ''word'' sent are called transverse redundancy checks, while those added at the end of a stream of ''words'' are called longitudinal redundancy checks. For example, if each of a series of m-bit ''words'' has a parity bit added, showing whether there were an odd or even number of ones in that word, any word with a single error in it will be detected. It will not be known where in the word the error is, however. If, in addition, after each stream of n words a parity sum is sent, each bit of which shows whether there were an odd or even number of ones at that bit-position sent in the most recent group, the exact position of the error can be determined and the error corrected. This method is only guaranteed to be effective, however, if there are no more than 1 error in every group of n words. With more error correction bits, more errors can be detected and in some cases corrected. There are also other bit-grouping techniques.

Checksum

A ''checksum'' of a message is a

modular arithmetic In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...

sum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a ones'-complement operation prior to transmission to detect unintentional all-zero messages. Checksum schemes include parity bits,

check digit A check digit is a form of redundancy check used for Error detection and correction, error detection on identification numbers, such as bank account numbers, which are used in an application where they will at least sometimes be input manually. It ...

s, and longitudinal redundancy checks. Some checksum schemes, such as the Damm algorithm, the Luhn algorithm, and the Verhoeff algorithm, are specifically designed to detect errors commonly introduced by humans in writing down or remembering identification numbers.

Cyclic redundancy check

A ''cyclic redundancy check'' (CRC) is a non-secure

designed to detect accidental changes to digital data in computer networks. It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a ''generator polynomial'', which is used as the

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

in a polynomial long division over a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

, taking the input data as the

dividend A dividend is a distribution of profits by a corporation to its shareholders, after which the stock exchange decreases the price of the stock by the dividend to remove volatility. The market has no control over the stock price on open on the ex ...

. The

remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In a ...

becomes the result. A CRC has properties that make it well suited for detecting

s. CRCs are particularly easy to implement in hardware and are therefore commonly used in

computer network A computer network is a collection of communicating computers and other devices, such as printers and smart phones. In order to communicate, the computers and devices must be connected by wired media like copper cables, optical fibers, or b ...

s and storage devices such as hard disk drives. The parity bit can be seen as a special-case 1-bit CRC.

Cryptographic hash function

The output of a ''cryptographic hash function'', also known as a ''message digest'', can provide strong assurances about

data integrity Data integrity is the maintenance of, and the assurance of, data accuracy and consistency over its entire Information Lifecycle Management, life-cycle. It is a critical aspect to the design, implementation, and usage of any system that stores, proc ...

, whether changes of the data are accidental (e.g., due to transmission errors) or maliciously introduced. Any modification to the data will likely be detected through a mismatching hash value. Furthermore, given some hash value, it is typically infeasible to find some input data (other than the one given) that will yield the same hash value. If an attacker can change not only the message but also the hash value, then a ''keyed hash'' or message authentication code (MAC) can be used for additional security. Without knowing the key, it is not possible for the attacker to easily or conveniently calculate the correct keyed hash value for a modified message.

Digital signature

Digital signatures can provide strong assurances about data integrity, whether the changes of the data are accidental or maliciously introduced. Digital signatures are perhaps most notable for being part of the HTTPS protocol for securely browsing the web.

Error correction code

Any error-correcting code can be used for error detection. A code with ''minimum

Hamming distance In information theory, the Hamming distance between two String (computer science), strings or vectors of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number ...

'', ''d'', can detect up to ''d'' − 1 errors in a code word. Using minimum-distance-based error-correcting codes for error detection can be suitable if a strict limit on the minimum number of errors to be detected is desired. Codes with minimum Hamming distance ''d'' = 2 are degenerate cases of error-correcting codes and can be used to detect single errors. The parity bit is an example of a single-error-detecting code.

Applications

Applications that require low latency (such as telephone conversations) cannot use

(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 usable. 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. Applications that use ARQ must have a return channel; applications having no return channel cannot use ARQ. Applications that require extremely low error rates (such as digital money transfers) must use ARQ due to the possibility of uncorrectable errors with FEC. Reliability and inspection engineering also make use of the theory of error-correcting codes, as well as natural language.

Internet

In a typical

TCP/IP The Internet protocol suite, commonly known as TCP/IP, is a framework for organizing the communication protocols used in the Internet and similar computer networks according to functional criteria. The foundational protocols in the suite are ...

stack, error control is performed at multiple levels: * Each Ethernet frame uses CRC-32 error detection. Frames with detected errors are discarded by the receiver hardware. * The

IPv4 Internet Protocol version 4 (IPv4) is the first version of the Internet Protocol (IP) as a standalone specification. It is one of the core protocols of standards-based internetworking methods in the Internet and other packet-switched networks. ...

header contains a checksum protecting the contents of the header. Packets with incorrect checksums are dropped within the network or at the receiver. * The checksum was omitted from the

IPv6 Internet Protocol version 6 (IPv6) is the most recent version of the Internet Protocol (IP), the communication protocol, communications protocol that provides an identification and location system for computers on networks and routes traffic ...

header in order to minimize processing costs in network routing and because current link layer technology is assumed to provide sufficient error detection (see also RFC 3819). * UDP has an optional checksum covering the payload and addressing information in the UDP and IP headers. Packets with incorrect checksums are discarded by the network stack. The checksum is optional under IPv4, and required under IPv6. When omitted, it is assumed the data-link layer provides the desired level of error protection. * TCP provides a checksum for protecting the payload and addressing information in the TCP and IP headers. Packets with incorrect checksums are discarded by the network stack and eventually get retransmitted using ARQ, either explicitly (such as through three-way handshake) or implicitly due to a timeout.

Deep-space telecommunications

The development of error-correction codes was tightly coupled with the history of deep-space missions due to the extreme dilution of signal power over interplanetary distances, and the limited power availability aboard space probes. Whereas early missions sent their data uncoded, starting in 1968, digital error correction was implemented in the form of (sub-optimally decoded) convolutional codes and Reed–Muller codes.K. Andrews et al., ''The Development of Turbo and LDPC Codes for Deep-Space Applications'', Proceedings of the IEEE, Vol. 95, No. 11, Nov. 2007. The Reed–Muller code was well suited to the noise the spacecraft was subject to (approximately matching a bell curve), and was implemented for the Mariner spacecraft and used on missions between 1969 and 1977. The

Voyager 1 ''Voyager 1'' is a space probe launched by NASA on September 5, 1977, as part of the Voyager program to study the outer Solar System and the interstellar medium, interstellar space beyond the Sun's heliosphere. It was launched 16 days afte ...

and Voyager 2 missions, which started in 1977, were designed to deliver color imaging and scientific information from

Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...

and

Saturn Saturn is the sixth planet from the Sun and the second largest in the Solar System, after Jupiter. It is a gas giant, with an average radius of about 9 times that of Earth. It has an eighth the average density of Earth, but is over 95 tim ...

. This resulted in increased coding requirements, and thus, the spacecraft were supported by (optimally Viterbi-decoded) convolutional codes that could be concatenated with an outer Golay (24,12,8) code. The Voyager 2 craft additionally supported an implementation of a Reed–Solomon code. The concatenated Reed–Solomon–Viterbi (RSV) code allowed for very powerful error correction, and enabled the spacecraft's extended journey to

Uranus Uranus is the seventh planet from the Sun. It is a gaseous cyan-coloured ice giant. Most of the planet is made of water, ammonia, and methane in a Supercritical fluid, supercritical phase of matter, which astronomy calls "ice" or Volatile ( ...

and

Neptune Neptune is the eighth and farthest known planet from the Sun. It is the List of Solar System objects by size, fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 t ...

. After ECC system upgrades in 1989, both crafts used V2 RSV coding. The Consultative Committee for Space Data Systems currently recommends usage of error correction codes with performance similar to the Voyager 2 RSV code as a minimum. Concatenated codes are increasingly falling out of favor with space missions, and are replaced by more powerful codes such as Turbo codes or LDPC codes. 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 missions close to Earth, the nature of the

noise Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...

in the

is different from that which a spacecraft on an interplanetary mission experiences. Additionally, as a spacecraft increases its distance from Earth, the problem of correcting for noise becomes more difficult.

Satellite broadcasting

The demand for satellite transponder bandwidth continues to grow, fueled by the desire to deliver television (including new channels and

high-definition television High-definition television (HDTV) describes a television or video system which provides a substantially higher image resolution than the previous generation of technologies. The term has been used since at least 1933; in more recent times, it ref ...

) and IP data. Transponder availability and bandwidth constraints have limited this growth. Transponder capacity is determined by the selected

modulation Signal modulation is the process of varying one or more properties of a periodic waveform in electronics and telecommunication for the purpose of transmitting information. The process encodes information in form of the modulation or message ...

scheme and the proportion of capacity consumed by FEC.

Data storage

Error detection and correction codes are often used to improve the reliability of data storage media. A parity track capable of detecting single-bit errors was present on the first magnetic tape data storage in 1951. The optimal rectangular code used in group coded recording tapes not only detects but also corrects single-bit errors. Some

file format A file format is a Computer standard, standard way that information is encoded for storage in a computer file. It specifies how bits are used to encode information in a digital storage medium. File formats may be either proprietary format, pr ...

s, particularly archive formats, include a checksum (most often CRC32) to detect corruption and truncation and can employ redundancy or parity files to recover portions of corrupted data. Reed-Solomon codes are used in

compact disc The compact disc (CD) is a Digital media, digital optical disc data storage format co-developed by Philips and Sony to store and play digital audio recordings. It employs the Compact Disc Digital Audio (CD-DA) standard and was capable of hol ...

s to correct errors caused by scratches. Modern hard drives use Reed–Solomon codes to detect and correct minor errors in sector reads, and to recover corrupted data from failing sectors and store that data in the spare sectors.

RAID RAID (; redundant array of inexpensive disks or redundant array of independent disks) is a data storage virtualization technology that combines multiple physical Computer data storage, data storage components into one or more logical units for th ...

systems use a variety of error correction techniques to recover data when a hard drive completely fails. Filesystems such as ZFS or Btrfs, as well as some

implementations, support data scrubbing and resilvering, which allows bad blocks to be detected and (hopefully) recovered before they are used. The recovered data may be re-written to exactly the same physical location, to spare blocks elsewhere on the same piece of hardware, or the data may be rewritten onto replacement hardware.

Error-correcting memory

Dynamic random-access memory Dynamics (from Greek language, Greek δυναμικός ''dynamikos'' "powerful", from δύναμις ''dynamis'' "power (disambiguation), power") or dynamic may refer to: Physics and engineering * Dynamics (mechanics), the study of forces and t ...

(DRAM) may provide stronger protection against soft errors by relying on error-correcting codes. Such error-correcting memory, known as ''ECC'' or ''EDAC-protected'' memory, is particularly desirable for mission-critical applications, such as scientific computing, financial, medical, etc. as well as extraterrestrial applications due to the increased

radiation In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or a material medium. This includes: * ''electromagnetic radiation'' consisting of photons, such as radio waves, microwaves, infr ...

in space. Error-correcting memory controllers traditionally use

s, although some use triple modular redundancy. Interleaving allows distributing the effect of a single cosmic ray potentially upsetting multiple physically neighboring bits across multiple words by associating neighboring bits to different words. As long as a single-event upset (SEU) does not exceed the error threshold (e.g., a single error) in any particular word between accesses, it can be corrected (e.g., by a single-bit error-correcting code), and the illusion of an error-free memory system may be maintained. In addition to hardware providing features required for ECC memory to operate,

operating system An operating system (OS) is system software that manages computer hardware and software resources, and provides common daemon (computing), services for computer programs. Time-sharing operating systems scheduler (computing), schedule tasks for ...

s usually contain related reporting facilities that are used to provide notifications when soft errors are transparently recovered. One example is the

Linux kernel The Linux kernel is a Free and open-source software, free and open source Unix-like kernel (operating system), kernel that is used in many computer systems worldwide. The kernel was created by Linus Torvalds in 1991 and was soon adopted as the k ...

's ''EDAC'' subsystem (previously known as ''Bluesmoke''), which collects the data from error-checking-enabled components inside a computer system; besides collecting and reporting back the events related to ECC memory, it also supports other checksumming errors, including those detected on the PCI bus. A few systems also support memory scrubbing to catch and correct errors early before they become unrecoverable.

References

External links

The on-line textbook: Information Theory, Inference, and Learning Algorithms
by David J.C. MacKay, contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including

s, turbo codes, and fountain codes.
ECC Page
– implementations of popular ECC encoding and decoding routines Detection and correction