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85-546: A cyclic redundancy check ( CRC ) is an error-detecting code commonly used in digital networks and storage devices to detect accidental changes to digital data. Blocks of data entering these systems get a short check value attached, based on the remainder of a polynomial division of their contents. On retrieval, the calculation is repeated and, in the event the check values do not match, corrective action can be taken against data corruption. CRCs can be used for error correction (see bitfilters ). CRCs are so called because

170-440: A clustered file system , using file systems that employ block level checksums such as ZFS , storage arrays that compute parity calculations such as exclusive or or use a cryptographic hash function and even having a watchdog timer on critical subsystems. Physical integrity often makes extensive use of error detecting algorithms known as error-correcting codes . Human-induced data integrity errors are often detected through

255-451: A codeword . When a codeword is received or read, the device either compares its check value with one freshly calculated from the data block, or equivalently, performs a CRC on the whole codeword and compares the resulting check value with an expected residue constant. If the CRC values do not match, then the block contains a data error. The device may take corrective action, such as rereading

340-552: 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 ECC computer memory on systems that require special provisions for reliability. The maximum proportion of errors or missing bits that can be corrected

425-411: A 3-bit CRC: The algorithm acts on the bits directly above the divisor in each step. The result for that iteration is the bitwise XOR of the polynomial divisor with the bits above it. The bits not above the divisor are simply copied directly below for that step. The divisor is then shifted right to align with the highest remaining 1 bit in the input, and the process is repeated until the divisor reaches

510-516: A computer file system may be configured on a fault-tolerant RAID array, but might not provide block-level checksums to detect and prevent silent data corruption . As another example, a database management system might be compliant with the ACID properties, but the RAID controller or hard disk drive's internal write cache might not be. This type of integrity is concerned with the correctness or rationality of

595-413: A data value is derived based on algorithm, contributors and conditions. It also specifies the conditions on how the data value could be re-derived. Data integrity is normally enforced in a database system by a series of integrity constraints or rules. Three types of integrity constraints are an inherent part of the relational data model : entity integrity, referential integrity and domain integrity. If

680-412: A database correctly rejecting mutually exclusive possibilities). Moreover, upon later retrieval , ensure the data is the same as when it was originally recorded. In short, data integrity aims to prevent unintentional changes to information. Data integrity is not to be confused with data security , the discipline of protecting data from unauthorized parties. Any unintended changes to data as the result of

765-587: A database supports these features, it is the responsibility of the database to ensure data integrity as well as the consistency model for the data storage and retrieval. If a database does not support these features, it is the responsibility of the applications to ensure data integrity while the database supports the consistency model for the data storage and retrieval. Having a single, well-controlled, and well-defined data-integrity system increases: Modern databases support these features (see Comparison of relational database management systems ), and it has become

850-405: A desired error detection power. The BCH codes are a powerful class of such polynomials. They subsume the two examples above. Regardless of the reducibility properties of a generator polynomial of degree  r , if it includes the "+1" term, the code will be able to detect error patterns that are confined to a window of r contiguous bits. These patterns are called "error bursts". The concept of

935-420: A divisor that guarantees good error-detection properties. In this analysis, the digits of the bit strings are taken as the coefficients of a polynomial in some variable x —coefficients that are elements of the finite field GF(2) (the integers modulo 2, i.e. either a zero or a one), instead of more familiar numbers. The set of binary polynomials is a mathematical ring . The selection of the generator polynomial

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1020-512: 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. The two main categories of ECC codes are block codes and convolutional codes . There are many types of block codes; Reed–Solomon coding

1105-908: 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,

1190-404: 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. In one extreme, a strong code (with low code-rate) can induce an important increase in the receiver SNR (signal-to-noise-ratio) decreasing

1275-443: A more uniform distribution of errors. 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. Systems using LDPC codes therefore typically employ additional interleaving across the symbols within a code word. For turbo codes, an interleaver is an integral component and its proper design

1360-411: A name of the form CRC- n -XXX as in the table below. The simplest error-detection system, the parity bit , is in fact a 1-bit CRC: it uses the generator polynomial  x + 1 (two terms), and has the name CRC-1. A CRC-enabled device calculates a short, fixed-length binary sequence, known as the check value or CRC , for each block of data to be sent or stored and appends it to the data, forming

1445-542: A particular protocol can impose pre-inversion, post-inversion and reversed bit ordering as described above. For example, the CRC32 used in Gzip and Bzip2 use the same polynomial, but Gzip employs reversed bit ordering, while Bzip2 does not. Note that even parity polynomials in GF(2) with degree greater than 1 are never primitive. Even parity polynomial marked as primitive in this table represent

1530-722: A piece of data, given a particular context. This includes topics such as referential integrity and entity integrity in a relational database or correctly ignoring impossible sensor data in robotic systems. These concerns involve ensuring that the data "makes sense" given its environment. Challenges include software bugs , design flaws, and human errors. Common methods of ensuring logical integrity include things such as check constraints , foreign key constraints , program assertions , and other run-time sanity checks. Physical and logical integrity often share many challenges such as human errors and design flaws, and both must appropriately deal with concurrent requests to record and retrieve data,

1615-464: A polynomial according to the application requirements and the expected distribution of message lengths. The number of distinct CRCs in use has confused developers, a situation which authors have sought to address. There are three polynomials reported for CRC-12, twenty-two conflicting definitions of CRC-16, and seven of CRC-32. The polynomials commonly applied are not the most efficient ones possible. Since 1993, Koopman, Castagnoli and others have surveyed

1700-447: A primitive polynomial multiplied by ( x + 1 ) {\displaystyle \left(x+1\right)} . The most significant bit of a polynomial is always 1, and is not shown in the hex representations. Error correcting code 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 ). The redundancy allows

1785-525: A received message can easily be verified by performing the above calculation again, this time with the check value added instead of zeroes. The remainder should equal zero if there are no detectable errors. The following Python code outlines a function which will return the initial CRC remainder for a chosen input and polynomial, with either 1 or 0 as the initial padding. Note that this code works with string inputs rather than raw numbers: Mathematical analysis of this division-like process reveals how to select

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1870-457: 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) neither locally correctable codes (LCCs), q-query LCCs are bounded exponentially while LDCs can have subexponential lengths. Interleaving is frequently used in digital communication and storage systems to improve

1955-402: A simpler decoder combined with an interleaver . An example of such an algorithm is based on neural network structures. Data integrity Data integrity is the maintenance of, and the assurance of, data accuracy and consistency over its entire life-cycle . It is a critical aspect to the design, implementation, and usage of any system that stores, processes, or retrieves data. The term

2040-407: A storage, retrieval or processing operation, including malicious intent, unexpected hardware failure, and human error , is failure of data integrity. If the changes are the result of unauthorized access, it may also be a failure of data security. Depending on the data involved this could manifest itself as benign as a single pixel in an image appearing a different color than was originally recorded, to

2125-462: Is n bits long. For a given n , multiple CRCs are possible, each with a different polynomial. Such a polynomial has highest degree n , which means it has n + 1 terms. In other words, the polynomial has a length of n + 1 ; its encoding requires n + 1 bits. Note that most polynomial specifications either drop the MSb or LSb , since they are always 1. The CRC and associated polynomial typically have

2210-409: 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). 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

2295-404: 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

2380-575: Is an easily reversible function, which makes it unsuitable for use in digital signatures. Thirdly, CRC satisfies a relation similar to that of a linear function (or more accurately, an affine function ): where c {\displaystyle c} depends on the length of x {\displaystyle x} and y {\displaystyle y} . This can be also stated as follows, where x {\displaystyle x} , y {\displaystyle y} and z {\displaystyle z} have

2465-627: 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

2550-436: Is broad in scope and may have widely different meanings depending on the specific context even under the same general umbrella of computing . It is at times used as a proxy term for data quality , while data validation is a prerequisite for data integrity. Data integrity is the opposite of data corruption . The overall intent of any data integrity technique is the same: ensure data is recorded exactly as intended (such as

2635-442: Is called an n -bit CRC when its check value is n -bits. For a given n , multiple CRCs are possible, each with a different polynomial. Such a polynomial has highest degree n , and hence n + 1 terms (the polynomial has a length of n + 1 ). The remainder has length n . The CRC has a name of the form CRC- n -XXX. The design of the CRC polynomial depends on the maximum total length of the block to be protected (data + CRC bits),

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2720-430: Is completely lost and the missing letters can be recovered with minimal guesswork. Use of interleaving techniques increases total delay. This is because the entire interleaved block must be received before the packets can be decoded. 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

2805-501: Is crucial for good performance. 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: 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. Transmission without interleaving : Here, each group of

2890-413: 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

2975-548: Is no authentication, an attacker can edit a message and recompute the CRC without the substitution being detected. When stored alongside the data, CRCs and cryptographic hash functions by themselves do not protect against intentional modification of data. Any application that requires protection against such attacks must use cryptographic authentication mechanisms, such as message authentication codes or digital signatures (which are commonly based on cryptographic hash functions). Secondly, unlike cryptographic hash functions, CRC

3060-709: Is noteworthy for its widespread use in compact discs , DVDs , and hard disk drives . Other examples of classical block codes include Golay , BCH , Multidimensional parity , and Hamming codes . Hamming ECC is commonly used to correct NAND flash memory errors. 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. NOR Flash typically does not use any error correction. Classical block codes are usually decoded using hard-decision algorithms, which means that for every input and output signal

3145-645: Is omitted. So the polynomial x 4 + x + 1 {\displaystyle x^{4}+x+1} may be transcribed as: In the table below they are shown as: CRCs in proprietary protocols might be obfuscated by using a non-trivial initial value and a final XOR, but these techniques do not add cryptographic strength to the algorithm and can be reverse engineered using straightforward methods. Numerous varieties of cyclic redundancy checks have been incorporated into technical standards . By no means does one algorithm, or one of each degree, suit every purpose; Koopman and Chakravarty recommend selecting

3230-425: 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

3315-519: Is the most important part of implementing the CRC algorithm. The polynomial must be chosen to maximize the error-detecting capabilities while minimizing overall collision probabilities. The most important attribute of the polynomial is its length (largest degree(exponent) +1 of any one term in the polynomial), because of its direct influence on the length of the computed check value. The most commonly used polynomial lengths are 9 bits (CRC-8), 17 bits (CRC-16), 33 bits (CRC-32), and 65 bits (CRC-64). A CRC

3400-471: 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. The fundamental principle of ECC is to add redundant bits in order to help the decoder to find out

3485-405: The check (data verification) value is a redundancy (it expands the message without adding information ) and the algorithm is based on cyclic codes . CRCs are popular because they are simple to implement in binary hardware , easy to analyze mathematically, and particularly good at detecting common errors caused by noise in transmission channels. Because the check value has a fixed length,

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3570-411: The function that generates it is occasionally used as a hash function . CRCs are based on the theory of cyclic error-correcting codes . The use of systematic cyclic codes, which encode messages by adding a fixed-length check value, for the purpose of error detection in communication networks, was first proposed by W. Wesley Peterson in 1961. Cyclic codes are not only simple to implement but have

3655-733: The 32-bit polynomial were in their 1975 publications: Technical Report 2956 by Brayer for Mitre, published in January and released for public dissemination through DTIC in August, and Hammond, Brown and Liu's report for the Rome Laboratory, published in May. Both reports contained contributions from the other team. During December 1975, Brayer and Hammond presented their work in a paper at the IEEE National Telecommunications Conference:

3740-476: The CRC as an error-detecting code gets complicated when an implementer or standards committee uses it to design a practical system. Here are some of the complications: These complications mean that there are three common ways to express a polynomial as an integer: the first two, which are mirror images in binary, are the constants found in code; the third is the number found in Koopman's papers. In each case, one term

3825-711: The CRC-32C (Castagnoli) polynomial. The design of the 32-bit polynomial most commonly used by standards bodies, CRC-32-IEEE, was the result of a joint effort for the Rome Laboratory and the Air Force Electronic Systems Division by Joseph Hammond, James Brown and Shyan-Shiang Liu of the Georgia Institute of Technology and Kenneth Brayer of the Mitre Corporation . The earliest known appearances of

3910-537: The IEEE CRC-32 polynomial is the generating polynomial of a Hamming code and was selected for its error detection performance. Even so, the Castagnoli CRC-32C polynomial used in iSCSI or SCTP matches its performance on messages from 58 bits to 131 kbits, and outperforms it in several size ranges including the two most common sizes of Internet packet. The ITU-T G.hn standard also uses CRC-32C to detect errors in

3995-454: 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

4080-425: The associated code is able to detect any single-bit or double-bit errors. We can improve this situation. If we use the generator polynomial g ( x ) = p ( x ) ( 1 + x ) {\displaystyle g(x)=p(x)(1+x)} , where p {\displaystyle p} is a primitive polynomial of degree r − 1 {\displaystyle r-1} , then

4165-428: The benefit of being particularly well suited for the detection of burst errors : contiguous sequences of erroneous data symbols in messages. This is important because burst errors are common transmission errors in many communication channels , including magnetic and optical storage devices. Typically an n -bit CRC applied to a data block of arbitrary length will detect any single error burst not longer than n bits, and

4250-426: 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

4335-424: The bits representing the input in a row, and position the ( n + 1 )-bit pattern representing the CRC's divisor (called a " polynomial ") underneath the left end of the row. In this example, we shall encode 14 bits of message with a 3-bit CRC, with a polynomial x + x + 1 . The polynomial is written in binary as the coefficients; a 3rd-degree polynomial has 4 coefficients ( 1 x + 0 x + 1 x + 1 ). In this case,

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4420-521: The block or requesting that it be sent again. Otherwise, the data is assumed to be error-free (though, with some small probability, it may contain undetected errors; this is inherent in the nature of error-checking). CRCs are specifically designed to protect against common types of errors on communication channels, where they can provide quick and reasonable assurance of the integrity of messages delivered. However, they are not suitable for protecting against intentional alteration of data. Firstly, as there

4505-443: 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

4590-418: The coefficients are 1, 0, 1 and 1. The result of the calculation is 3 bits long, which is why it is called a 3-bit CRC. However, you need 4 bits to explicitly state the polynomial. Start with the message to be encoded: This is first padded with zeros corresponding to the bit length n of the CRC. This is done so that the resulting code word is in systematic form. Here is the first calculation for computing

4675-809: 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 ), 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 3GPP MBMS (see fountain codes ). Turbo coding

4760-462: 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

4845-749: The database itself, which automatically ensures the accuracy and integrity of the data so that no child record can exist without a parent (also called being orphaned) and that no parent loses their child records. It also ensures that no parent record can be deleted while the parent record owns any child records. All of this is handled at the database level and does not require coding integrity checks into each application. Various research results show that neither widespread filesystems (including UFS , Ext , XFS , JFS and NTFS ) nor hardware RAID solutions provide sufficient protection against data integrity problems. Some filesystems (including Btrfs and ZFS ) provide internal data and metadata checksumming that

4930-405: The de facto responsibility of the database to ensure data integrity. Companies, and indeed many database systems, offer products and services to migrate legacy systems to modern databases. An example of a data-integrity mechanism is the parent-and-child relationship of related records. If a parent record owns one or more related child records all of the referential integrity processes are handled by

5015-444: 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

5100-399: The desired error protection features, and the type of resources for implementing the CRC, as well as the desired performance. A common misconception is that the "best" CRC polynomials are derived from either irreducible polynomials or irreducible polynomials times the factor  1 + x , which adds to the code the ability to detect all errors affecting an odd number of bits. In reality, all

5185-407: The factors described above should enter into the selection of the polynomial and may lead to a reducible polynomial. However, choosing a reducible polynomial will result in a certain proportion of missed errors, due to the quotient ring having zero divisors . The advantage of choosing a primitive polynomial as the generator for a CRC code is that the resulting code has maximal total block length in

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5270-400: The fraction of all longer error bursts that it will detect is approximately (1 − 2) . Specification of a CRC code requires definition of a so-called generator polynomial . This polynomial becomes the divisor in a polynomial long division , which takes the message as the dividend and in which the quotient is discarded and the remainder becomes the result. The important caveat is that

5355-501: 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. 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

5440-514: The latter of which is entirely a subject on its own. If a data sector only has a logical error, it can be reused by overwriting it with new data. In case of a physical error, the affected data sector is permanently unusable. Data integrity contains guidelines for data retention , specifying or guaranteeing the length of time data can be retained in a particular database (typically a relational database ). To achieve data integrity, these rules are consistently and routinely applied to all data entering

5525-728: The loss of vacation pictures or a business-critical database, to even catastrophic loss of human life in a life-critical system . Physical integrity deals with challenges which are associated with correctly storing and fetching the data itself. Challenges with physical integrity may include electromechanical faults, design flaws, material fatigue , corrosion , power outages , natural disasters, and other special environmental hazards such as ionizing radiation , extreme temperatures, pressures and g-forces . Ensuring physical integrity includes methods such as redundant hardware, an uninterruptible power supply , certain types of RAID arrays, radiation hardened chips, error-correcting memory , use of

5610-404: The maximal total block length is 2 r − 1 − 1 {\displaystyle 2^{r-1}-1} , and the code is able to detect single, double, triple and any odd number of errors. A polynomial g ( x ) {\displaystyle g(x)} that admits other factorizations may be chosen then so as to balance the maximal total blocklength with

5695-419: 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 come very close to

5780-435: The original user data to be extracted from the other, uncorrupted received symbols that also depend on the same user 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

5865-510: 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. 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: Though simple to implement and widely used, this triple modular redundancy

5950-482: The payload (although it uses CRC-16-CCITT for PHY headers ). CRC-32C computation is implemented in hardware as an operation ( CRC32 ) of SSE4.2 instruction set, first introduced in Intel processors' Nehalem microarchitecture. ARM AArch64 architecture also provides hardware acceleration for both CRC-32 and CRC-32C operations. The table below lists only the polynomials of the various algorithms in use. Variations of

6035-408: 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

6120-427: The polynomial coefficients are calculated according to the arithmetic of a finite field , so the addition operation can always be performed bitwise-parallel (there is no carry between digits). In practice, all commonly used CRCs employ the finite field of two elements, GF(2) . The two elements are usually called 0 and 1, comfortably matching computer architecture. A CRC is called an n -bit CRC when its check value

6205-645: 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 . 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

6290-455: The relations a piece of data can have to other pieces of data, such as a Customer record being allowed to link to purchased Products , but not to unrelated data such as Corporate Assets . Data integrity often includes checks and correction for invalid data, based on a fixed schema or a predefined set of rules. An example being textual data entered where a date-time value is required. Rules for data derivation are also applicable, specifying how

6375-448: The right-hand end of the input row. Here is the entire calculation: Since the leftmost divisor bit zeroed every input bit it touched, when this process ends the only bits in the input row that can be nonzero are the n bits at the right-hand end of the row. These n bits are the remainder of the division step, and will also be the value of the CRC function (unless the chosen CRC specification calls for some postprocessing). The validity of

6460-525: The same length as a result, even if the CRC is encrypted with a stream cipher that uses XOR as its combining operation (or mode of block cipher which effectively turns it into a stream cipher, such as OFB or CFB), both the message and the associated CRC can be manipulated without knowledge of the encryption key; this was one of the well-known design flaws of the Wired Equivalent Privacy (WEP) protocol. To compute an n -bit binary CRC, line

6545-584: 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 : 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 : The term "AnExample" ends up mostly unintelligible and difficult to correct. With interleaving : No word

6630-470: The sense that all 1-bit errors within that block length have different remainders (also called syndromes ) and therefore, since the remainder is a linear function of the block, the code can detect all 2-bit errors within that block length. If r {\displaystyle r} is the degree of the primitive generator polynomial, then the maximal total block length is 2 r − 1 {\displaystyle 2^{r}-1} , and

6715-401: The space of polynomials between 3 and 64 bits in size, finding examples that have much better performance (in terms of Hamming distance for a given message size) than the polynomials of earlier protocols, and publishing the best of these with the aim of improving the error detection capacity of future standards. In particular, iSCSI and SCTP have adopted one of the findings of this research,

6800-446: The system, and any relaxation of enforcement could cause errors in the data. Implementing checks on the data as close as possible to the source of input (such as human data entry), causes less erroneous data to enter the system. Strict enforcement of data integrity rules results in lower error rates, and time saved troubleshooting and tracing erroneous data and the errors it causes to algorithms. Data integrity also includes rules defining

6885-411: The theoretical maximum given by the Shannon channel capacity under the hypothesis of an infinite length frame. 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

6970-413: 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

7055-508: The use of simpler checks and algorithms, such as the Damm algorithm or Luhn algorithm . These are used to maintain data integrity after manual transcription from one computer system to another by a human intermediary (e.g. credit card or bank routing numbers). Computer-induced transcription errors can be detected through hash functions . In production systems, these techniques are used together to ensure various degrees of data integrity. For example,

7140-463: 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

7225-443: 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: 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

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