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91-452: While most RAID levels can provide good protection against and recovery from hardware defects or defective sectors/read errors ( hard errors ), they do not provide any protection against data loss due to catastrophic failures (fire, water) or soft errors such as user error, software malfunction, or malware infection. For valuable data, RAID is only one building block of a larger data loss prevention and recovery scheme – it cannot replace

182-406: A Data Loss Event was noticed. However, in most situations, there is an inverse correlation between the value of a unit of data and the length of time it takes to notice the loss of that data. Taking this into consideration, many backup strategies decrease the granularity of restorability as the time increases since the potential Data Loss Event . By this logic, recovery from recent Data Loss Events

273-434: A backup plan. RAID 0 (also known as a stripe set or striped volume ) splits (" stripes ") data evenly across two or more disks, without parity information, redundancy, or fault tolerance . Since RAID 0 provides no fault tolerance or redundancy, the failure of one drive will cause the entire array to fail, due to data being striped across all disks. This configuration is typically implemented having speed as

364-476: A data loss event is directly related to the value of the data and the length of time that it is unavailable yet needed. For an enterprise in particular, the definition of cost extends beyond the financial and can also include time. Consider: The frequency of data loss and the impact can be greatly mitigated by taking proper precautions, those of which necessary can vary depending on the type of data loss. For example, multiple power circuits with battery backup and

455-468: A lockstep ) added design considerations that provided no significant advantages over other RAID levels. Both RAID 3 and RAID 4 were quickly replaced by RAID 5. RAID 3 was usually implemented in hardware, and the performance issues were addressed by using large disk caches. RAID 4 consists of block -level striping with a dedicated parity disk. As a result of its layout, RAID 4 provides good performance of random reads, while

546-403: A 320 GB disk, the size of the array will be 120 GB × 2 = 240 GB. However, some RAID implementations would allow the remaining 200 GB to be used for other purposes. The diagram in this section shows how the data is distributed into stripes on two disks, with A1:A2 as the first stripe, A3:A4 as the second one, etc. Once the stripe size is defined during the creation of

637-407: A Hamming code is constructed by listing all columns of length m that are pair-wise independent. Thus H is a matrix whose left side is all of the nonzero n -tuples where order of the n -tuples in the columns of matrix does not matter. The right hand side is just the ( n  −  k )- identity matrix . So G can be obtained from H by taking the transpose of the left hand side of H with

728-528: A Hamming distance of four, which allows the decoder to distinguish between when at most one one-bit error occurs and when any two-bit errors occur. In this sense, extended Hamming codes are single-error correcting and double-error detecting, abbreviated as SECDED . Richard Hamming , the inventor of Hamming codes, worked at Bell Labs in the late 1940s on the Bell Model V computer, an electromechanical relay-based machine with cycle times in seconds. Input

819-403: A RAID 5 disk drive array depending upon the sequence of writing across the disks, that is: The figure shows 1) data blocks written left to right, 2) the parity block at the end of the stripe and 3) the first block of the next stripe not on the same disk as the parity block of the previous stripe. It can be designated as a Left Asynchronous RAID 5 layout and this is the only layout identified in

910-399: A RAID array's virtual disks in the presence of any two concurrent disk failures. Several methods, including dual check data computations (parity and Reed–Solomon ), orthogonal dual parity check data and diagonal parity, have been used to implement RAID Level 6." The second block is usually labeled Q, with the first block labeled P. Typically the P block is calculated as the parity (XORing) of

1001-418: A RAID 0 array, it needs to be maintained at all times. Since the stripes are accessed in parallel, an n -drive RAID 0 array appears as a single large disk with a data rate n times higher than the single-disk rate. A RAID 0 array of n drives provides data read and write transfer rates up to n times as high as the individual drive rates, but with no data redundancy. As a result, RAID 0

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1092-419: A RAID 1 array, overall write performance is equal to the speed of the slowest disk. Synthetic benchmarks show varying levels of performance improvements when multiple HDDs or SSDs are used in a RAID 1 setup, compared with single-drive performance. However, some synthetic benchmarks also show a drop in performance for the same comparison. RAID 2 , which is rarely used in practice, stripes data at

1183-455: A Reed Solomon code is used, the second parity calculation is unnecessary. Reed Solomon has the advantage of allowing all redundancy information to be contained within a given stripe. It is possible to support a far greater number of drives by choosing the parity function more carefully. The issue we face is to ensure that a system of equations over the finite field Z 2 {\displaystyle \mathbb {Z} _{2}} has

1274-560: A bunch of disks"), SPAN/BIG , and MAID ("massive array of idle disks"). Data loss Data loss is an error condition in information systems in which information is destroyed by failures (like failed spindle motors or head crashes on hard drives) or neglect (like mishandling, careless handling or storage under unsuitable conditions) in storage , transmission , or processing . Information systems implement backup and disaster recovery equipment and processes to prevent data loss or restore lost data. Data loss can also occur if

1365-440: A distance of 3, as three bits need to be flipped in the same triple to obtain another code word with no visible errors. It can correct one-bit errors or it can detect - but not correct - two-bit errors. A (4,1) repetition (each bit is repeated four times) has a distance of 4, so flipping three bits can be detected, but not corrected. When three bits flip in the same group there can be situations where attempting to correct will produce

1456-425: A field is an element of the field such that g i {\displaystyle g^{i}} is different for each non-negative i < m − 1 {\displaystyle i<m-1} . This means each element of the field, except the value 0 {\displaystyle 0} , can be written as a power of g . {\displaystyle g.} A finite field

1547-409: A generator only protect against power failures, though using an Uninterruptible Power Supply can protect drive against sudden power spikes. Similarly, using a journaling file system and RAID storage only protect against certain types of software and hardware failure. For hard disk drives, which are a physical storage medium, ensuring minimal vibration and movement will help protect against damaging

1638-505: A noisy transmission medium, a successful transmission could take a long time or may never occur. However, while the quality of parity checking is poor, since it uses only a single bit, this method results in the least overhead. A two-out-of-five code is an encoding scheme which uses five bits consisting of exactly three 0s and two 1s. This provides ( 5 3 ) = 10 {\displaystyle {\binom {5}{3}}=10} possible combinations, enough to represent

1729-424: A non-RAID setup), but in most situations it will yield a significant improvement in performance". Synthetic benchmarks show different levels of performance improvements when multiple HDDs or SSDs are used in a RAID 0 setup, compared with single-drive performance. However, some synthetic benchmarks also show a drop in performance for the same comparison. RAID 1 consists of an exact copy (or mirror ) of

1820-445: A polynomial. The effect of g i {\displaystyle g^{i}} can be thought of as the action of a carefully chosen linear feedback shift register on the data chunk. Unlike the bit shift in the simplified example, which could only be applied k {\displaystyle k} times before the encoding began to repeat, applying the operator g {\displaystyle g} multiple times

1911-540: A read request for B2 could be serviced concurrently by disk 1. RAID 5 consists of block-level striping with distributed parity. Unlike in RAID ;4, parity information is distributed among the drives. It requires that all drives but one be present to operate. Upon failure of a single drive, subsequent reads can be calculated from the distributed parity such that no data is lost. RAID 5 requires at least three disks. There are many layouts of data and parity in

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2002-407: A scrub. The redundant information is used to reconstruct the missing data, rather than to identify the faulted drive. Drives are considered to have faulted if they experience an unrecoverable read error , which occurs after a drive has retried many times to read data and failed. Enterprise drives may also report failure in far fewer tries than consumer drives as part of TLER to ensure a read request

2093-416: A set of data on two or more disks; a classic RAID 1 mirrored pair contains two disks. This configuration offers no parity, striping, or spanning of disk space across multiple disks, since the data is mirrored on all disks belonging to the array, and the array can only be as big as the smallest member disk. This layout is useful when read performance or reliability is more important than write performance or

2184-673: A suitable irreducible polynomial p ( x ) {\displaystyle p(x)} of degree k {\displaystyle k} over Z 2 {\displaystyle \mathbb {Z} _{2}} . We will represent the data elements D {\displaystyle D} as polynomials D = d k − 1 x k − 1 + d k − 2 x k − 2 + . . . + d 1 x + d 0 {\displaystyle \mathbf {D} =d_{k-1}x^{k-1}+d_{k-2}x^{k-2}+...+d_{1}x+d_{0}} in

2275-572: A system to the precise state it was in prior to the Data Loss Event is extremely difficult. Some level of compromise between granularity of recoverability and cost is necessary. Furthermore, a Data Loss Event may not be immediately apparent. An effective backup strategy must also consider the cost of maintaining the ability to recover lost data for long periods of time. A highly effective backup system would have duplicate copies of every file and program that were immediately accessible whenever

2366-532: A taped interview, Hamming said, "And so I said, 'Damn it, if the machine can detect an error, why can't it locate the position of the error and correct it?'". Over the next few years, he worked on the problem of error-correction, developing an increasingly powerful array of algorithms. In 1950, he published what is now known as Hamming code, which remains in use today in applications such as ECC memory . A number of simple error-detecting codes were used before Hamming codes, but none were as effective as Hamming codes in

2457-508: A unique solution. To do this, we can use the theory of polynomial equations over finite fields. Consider the Galois field G F ( m ) {\displaystyle GF(m)} with m = 2 k {\displaystyle m=2^{k}} . This field is isomorphic to a polynomial field F 2 [ x ] / ( p ( x ) ) {\displaystyle F_{2}[x]/(p(x))} for

2548-401: Is 0, which is incorrect. If we increase the size of the bit string to four, we can detect all two-bit errors but cannot correct them (the quantity of parity bits is even); at five bits, we can both detect and correct all two-bit errors, but not all three-bit errors. Moreover, increasing the size of the parity bit string is inefficient, reducing throughput by three times in our original case, and

2639-814: Is called a (canonical) generator matrix of a linear ( n , k ) code, and H := ( A I n − k ) {\displaystyle \mathbf {H} :={\begin{pmatrix}{\begin{array}{c|c}A&I_{n-k}\\\end{array}}\end{pmatrix}}} is called a parity-check matrix . This is the construction of G and H in standard (or systematic) form. Regardless of form, G and H for linear block codes must satisfy H G T = 0 {\displaystyle \mathbf {H} \,\mathbf {G} ^{\text{T}}=\mathbf {0} } , an all-zeros matrix. Since [7, 4, 3] = [ n ,  k ,  d ] = [2  − 1, 2  − 1 −  m , 3]. The parity-check matrix H of

2730-472: Is easier and more complete than recovery from Data Loss Events that happened further in the past. Recovery is also related to the type of Data Loss Event. Recovering a single lost file is substantially different from recovering an entire system that was destroyed in a disaster. An effective backup regimen has some proportionality between the magnitude of Data Loss and the magnitude of effort required to recover. For example, it should be far easier to restore

2821-969: Is fulfilled in a timely manner. In measurement of the I/O performance of five filesystems with five storage configurations—single SSD, RAID 0, RAID 1, RAID 10, and RAID 5 it was shown that F2FS on RAID 0 and RAID 5 with eight SSDs outperforms EXT4 by 5 times and 50 times, respectively. The measurements also suggest that the RAID controller can be a significant bottleneck in building a RAID system with high speed SSDs. Combinations of two or more standard RAID levels. They are also known as RAID 0+1 or RAID 01, RAID 0+3 or RAID 03, RAID 1+0 or RAID 10, RAID 5+0 or RAID 50, RAID 6+0 or RAID 60, and RAID 10+0 or RAID 100. In addition to standard and nested RAID levels, alternatives include non-standard RAID levels , and non-RAID drive architectures . Non-RAID drive architectures are referred to by similar terms and acronyms, notably JBOD ("just

Standard RAID levels - Misplaced Pages Continue

2912-427: Is given as an expression in terms of the number of drives, n ; this expression designates a fractional value between zero and one, representing the fraction of the sum of the drives' capacities that is available for use. For example, if three drives are arranged in RAID 3, this gives an array space efficiency of 1 − 1/ n = 1 − 1/3 = 2/3 ≈ 67% ; thus, if each drive in this example has a capacity of 250 GB, then

3003-465: Is guaranteed to have at least one generator. Pick one such generator g {\displaystyle g} , and define P {\displaystyle \mathbf {P} } and Q {\displaystyle \mathbf {Q} } as follows: As before, the first checksum P {\displaystyle \mathbf {P} } is just the XOR of each stripe, though interpreted now as

3094-400: Is guaranteed to produce m = 2 k − 1 {\displaystyle m=2^{k}-1} unique invertible functions, which will allow a chunk length of k {\displaystyle k} to support up to 2 k − 1 {\displaystyle 2^{k}-1} data pieces. If one data chunk is lost, the situation is similar to

3185-455: Is implemented in the manufacturer's storage architecture—in software, firmware, or by using firmware and specialized ASICs for intensive parity calculations. RAID 6 can read up to the same speed as RAID 5 with the same number of physical drives. When either diagonal or orthogonal dual parity is used, a second parity calculation is necessary for write operations. This doubles CPU overhead for RAID-6 writes, versus single-parity RAID levels. When

3276-425: Is in error. If only one parity bit indicates an error, the parity bit itself is in error. With m parity bits, bits from 1 up to 2 m − 1 {\displaystyle 2^{m}-1} can be covered. After discounting the parity bits, 2 m − m − 1 {\displaystyle 2^{m}-m-1} bits remain for use as data. As m varies, we get all

3367-532: Is possible to increase the minimum distance of the Hamming code to 4, which allows the decoder to distinguish between single bit errors and two-bit errors. Thus the decoder can detect and correct a single error and at the same time detect (but not correct) a double error. If the decoder does not attempt to correct errors, it can reliably detect triple bit errors. If the decoder does correct errors, some triple errors will be mistaken for single errors and "corrected" to

3458-411: Is possible to set the error-correcting bits so that the index-XOR of the whole message is 0. If the receiver receives a string with index-XOR 0, they can conclude there were no corruptions, and otherwise, the index-XOR indicates the index of the corrupted bit. An algorithm can be deduced from the following description: If a byte of data to be encoded is 10011010, then the data word (using _ to represent

3549-439: Is primarily used in applications that require high performance and are able to tolerate lower reliability, such as in scientific computing or computer gaming . Some benchmarks of desktop applications show RAID 0 performance to be marginally better than a single drive. Another article examined these claims and concluded that "striping does not always increase performance (in certain situations it will actually be slower than

3640-540: Is relatively CPU intensive, as it involves polynomial multiplication in F 2 [ x ] / ( p ( x ) ) {\displaystyle F_{2}[x]/(p(x))} . This can be mitigated with a hardware implementation or by using an FPGA . The above Vandermonde matrix solution can be extended to triple parity, but for beyond a Cauchy matrix construction is required. The following table provides an overview of some considerations for standard RAID levels. In each case, array space efficiency

3731-434: Is that a parity value of one indicates that there is an odd number of ones in the data, and a parity value of zero indicates that there is an even number of ones. If the number of bits changed is even, the check bit will be valid and the error will not be detected. Moreover, parity does not indicate which bit contained the error, even when it can detect it. The data must be discarded entirely and re-transmitted from scratch. On

Standard RAID levels - Misplaced Pages Continue

3822-441: Is that any given bit is included in a unique set of parity bits. To check for errors, check all of the parity bits. The pattern of errors, called the error syndrome , identifies the bit in error. If all parity bits are correct, there is no error. Otherwise, the sum of the positions of the erroneous parity bits identifies the erroneous bit. For example, if the parity bits in positions 1, 2 and 8 indicate an error, then bit 1+2+8=11

3913-418: Is the only original level of RAID that is not currently used. RAID 3 , which is rarely used in practice, consists of byte -level striping with a dedicated parity disk. One of the characteristics of RAID 3 is that it generally cannot service multiple requests simultaneously, which happens because any single block of data will, by definition, be spread across all members of the set and will reside in

4004-486: The Hamming(7,4) code which adds three parity bits to four bits of data. In mathematical terms, Hamming codes are a class of binary linear code. For each integer r ≥ 2 there is a code-word with block length n = 2 − 1 and message length k = 2 − r − 1 . Hence the rate of Hamming codes is R = k / n = 1 − r / (2 − 1) , which is the highest possible for codes with minimum distance of three (i.e.,

4095-469: The bit (rather than block) level, and uses a Hamming code for error correction . The disks are synchronized by the controller to spin at the same angular orientation (they reach index at the same time), so it generally cannot service multiple requests simultaneously. However, depending with a high rate Hamming code , many spindles would operate in parallel to simultaneously transfer data so that "very high data transfer rates" are possible as for example in

4186-427: The (7,4) encoded word (see Hamming(7,4) ). This can be summed up with the revised matrices: and Note that H is not in standard form. To obtain G, elementary row operations can be used to obtain an equivalent matrix to H in systematic form: For example, the first row in this matrix is the sum of the second and third rows of H in non-systematic form. Using the systematic construction for Hamming codes from above,

4277-492: The 16 possible data vectors a → {\displaystyle {\vec {a}}} is given by the standard matrix product x → = a → G {\displaystyle {\vec {x}}={\vec {a}}G} where the summing operation is done modulo-2. For example, let a → = [ 1 , 0 , 1 , 1 ] {\displaystyle {\vec {a}}=[1,0,1,1]} . Using

4368-527: The Galois field. Let D 0 , . . . , D n − 1 ∈ G F ( m ) {\displaystyle \mathbf {D} _{0},...,\mathbf {D} _{n-1}\in GF(m)} correspond to the stripes of data across hard drives encoded as field elements in this manner. We will use ⊕ {\displaystyle \oplus } to denote addition in

4459-618: The SECDED level of protection, no longer use Hamming's method, relying instead on the designs with longer codewords (128 to 256 bits of data) and modified balanced parity-check trees. The (72,64) Hamming code is still popular in some hardware designs, including Xilinx FPGA families. In 1950, Hamming introduced the [7,4] Hamming code. It encodes four data bits into seven bits by adding three parity bits. As explained earlier, it can either detect and correct single-bit errors or it can detect (but not correct) both single and double-bit errors. With

4550-495: The Thinking Machines' DataVault where 32 data bits were transmitted simultaneously. The IBM 353 also observed a similar usage of Hamming code and was capable of transmitting 64 data bits simultaneously, along with 8 ECC bits. With all hard disk drives implementing internal error correction, the complexity of an external Hamming code offered little advantage over parity so RAID 2 has been rarely implemented; it

4641-438: The addition of an overall parity bit, it becomes the [8,4] extended Hamming code and can both detect and correct single-bit errors and detect (but not correct) double-bit errors. The matrix G := ( I k − A T ) {\displaystyle \mathbf {G} :={\begin{pmatrix}{\begin{array}{c|c}I_{k}&-A^{\text{T}}\\\end{array}}\end{pmatrix}}}

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4732-422: The array has a total capacity of 750 GB but the capacity that is usable for data storage is only 500 GB. Different RAID configurations can also detect failure during so called data scrubbing . Historically disks were subject to lower reliability and RAID levels were also used to detect which disk in the array had failed in addition to that a disk had failed. Though as noted by Patterson et al. even at

4823-643: The components internally, as can maintaining a suitable drive temperature. Regular data backups are an important asset to have when trying to recover after a data loss event, but they do not prevent user errors or system failures. As such, a data backup plan needs to be established and run in unison with a disaster recovery plan in order to lower risk. Data recovery is often performed by specialized commercial services that have developed often proprietary methods to recover data from physically damaged media. Service costs at data recovery labs are usually dependent on type of damage and type of storage medium, as well as

4914-461: The data, the same as RAID 5. Different implementations of RAID 6 use different erasure codes to calculate the Q block, often one of Reed Solomon, EVENODD, Row Diagonal Parity (RDP), Mojette, or Liberation codes. RAID 6 does not have a performance penalty for read operations, but it does have a performance penalty on write operations because of the overhead associated with parity calculations. Performance varies greatly depending on how RAID 6

5005-439: The digits 0–9. This scheme can detect all single bit-errors, all odd numbered bit-errors and some even numbered bit-errors (for example the flipping of both 1-bits). However it still cannot correct any of these errors. Another code in use at the time repeated every data bit multiple times in order to ensure that it was sent correctly. For instance, if the data bit to be sent is a 1, an n = 3 repetition code will send 111. If

5096-584: The direction the data blocks are written, the location of the parity blocks with respect to the data blocks and whether or not the first data block of a subsequent stripe is written to the same drive as the last parity block of the prior stripe. The figure to the right is just one of many such layouts. According to the Storage Networking Industry Association (SNIA), the definition of RAID 6 is: "Any form of RAID that can continue to execute read and write requests to all of

5187-433: The drive in order to establish a secondary copy of the data. This can then be tested on, with recovery attempted, abolishing the risk of harming the source data. 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,

5278-512: The efficiency drops drastically as we increase the number of times each bit is duplicated in order to detect and correct more errors. If more error-correcting bits are included with a message, and if those bits can be arranged such that different incorrect bits produce different error results, then bad bits could be identified. In a seven-bit message, there are seven possible single bit errors, so three error control bits could potentially specify not only that an error occurred but also which bit caused

5369-503: The error. Hamming studied the existing coding schemes, including two-of-five, and generalized their concepts. To start with, he developed a nomenclature to describe the system, including the number of data bits and error-correction bits in a block. For instance, parity includes a single bit for any data word, so assuming ASCII words with seven bits, Hamming described this as an (8,7) code, with eight bits in total, of which seven are data. The repetition example would be (3,1) , following

5460-417: The field, and concatenation to denote multiplication. The reuse of ⊕ {\displaystyle \oplus } is intentional: this is because addition in the finite field Z 2 {\displaystyle \mathbb {Z} _{2}} represents to the XOR operator, so computing the sum of two elements is equivalent to computing XOR on the polynomial coefficients. A generator of

5551-602: The following operations: From the above matrix we have 2 = 2 = 16 codewords. Let a → {\displaystyle {\vec {a}}} be a row vector of binary data bits, a → = [ a 1 , a 2 , a 3 , a 4 ] , a i ∈ { 0 , 1 } {\displaystyle {\vec {a}}=[a_{1},a_{2},a_{3},a_{4}],\quad a_{i}\in \{0,1\}} . The codeword x → {\displaystyle {\vec {x}}} for any of

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5642-1281: The generator matrix G {\displaystyle G} from above, we have (after applying modulo 2, to the sum), x → = a → G = ( 1 0 1 1 ) ( 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 ) = ( 1 0 1 1 2 3 2 ) = ( 1 0 1 1 0 1 0 ) {\displaystyle {\vec {x}}={\vec {a}}G={\begin{pmatrix}1&0&1&1\end{pmatrix}}{\begin{pmatrix}1&0&0&0&1&1&0\\0&1&0&0&1&0&1\\0&0&1&0&0&1&1\\0&0&0&1&1&1&1\\\end{pmatrix}}={\begin{pmatrix}1&0&1&1&2&3&2\end{pmatrix}}={\begin{pmatrix}1&0&1&1&0&1&0\end{pmatrix}}} The [7,4] Hamming code can easily be extended to an [8,4] code by adding an extra parity bit on top of

5733-1391: The identity k - identity matrix on the left hand side of  G . The code generator matrix G {\displaystyle \mathbf {G} } and the parity-check matrix H {\displaystyle \mathbf {H} } are: G := ( 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 ) 4 , 7 {\displaystyle \mathbf {G} :={\begin{pmatrix}1&0&0&0&1&1&0\\0&1&0&0&1&0&1\\0&0&1&0&0&1&1\\0&0&0&1&1&1&1\end{pmatrix}}_{4,7}} and H := ( 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 1 0 0 1 ) 3 , 7 . {\displaystyle \mathbf {H} :={\begin{pmatrix}1&1&0&1&1&0&0\\1&0&1&1&0&1&0\\0&1&1&1&0&0&1\end{pmatrix}}_{3,7}.} Finally, these matrices can be mutated into equivalent non-systematic codes by

5824-456: The inception of RAID many (though not all) disks were already capable of finding internal errors using error correcting codes. In particular it is/was sufficient to have a mirrored set of disks to detect a failure, but two disks were not sufficient to detect which had failed in a disk array without error correcting features. Modern RAID arrays depend for the most part on a disk's ability to identify itself as faulty which can be detected as part of

5915-408: The intended goal. RAID 0 is normally used to increase performance, although it can also be used as a way to create a large logical volume out of two or more physical disks. A RAID 0 setup can be created with disks of differing sizes, but the storage space added to the array by each disk is limited to the size of the smallest disk. For example, if a 120 GB disk is striped together with

6006-443: The last edition of The Raid Book published by the defunct Raid Advisory Board. In a Synchronous layout the data first block of the next stripe is written on the same drive as the parity block of the previous stripe. In comparison to RAID 4, RAID 5's distributed parity evens out the stress of a dedicated parity disk among all RAID members. Additionally, write performance is increased since all RAID members participate in

6097-432: The limited redundancy that Hamming codes add to the data, they can only detect and correct errors when the error rate is low. This is the case in computer memory (usually RAM), where bit errors are extremely rare and Hamming codes are widely used, and a RAM with this correction system is an ECC RAM ( ECC memory ). In this context, an extended Hamming code having one extra parity bit is often used. Extended Hamming codes achieve

6188-467: The minimal number of bit changes needed to go from any code word to any other code word is three) and block length 2 − 1 . The parity-check matrix of a Hamming code is constructed by listing all columns of length r that are non-zero, which means that the dual code of the Hamming code is the shortened Hadamard code , also known as a Simplex code. The parity-check matrix has the property that any two columns are pairwise linearly independent . Due to

6279-652: The one before. In the case of two lost data chunks, we can compute the recovery formulas algebraically. Suppose that D i {\displaystyle \mathbf {D} _{i}} and D j {\displaystyle \mathbf {D} _{j}} are the lost values with i ≠ j {\displaystyle i\neq j} , then, using the other values of D {\displaystyle D} , we find constants A {\displaystyle A} and B {\displaystyle B} : We can solve for D i {\displaystyle D_{i}} in

6370-434: The original data is removed from the parity, the new data calculated into the parity and both the new data sector and the new parity sector are written. RAID 6 extends RAID 5 by adding a second parity block; thus, it uses block -level striping with two parity blocks distributed across all member disks. RAID 6 requires at least four disks. As in RAID 5, there are many layouts of RAID 6 disk arrays depending upon

6461-412: The original message in the presence of errors is known as an error-correcting code. This triple repetition code is a Hamming code with m = 2, since there are two parity bits, and 2 − 2 − 1 = 1 data bit. Such codes cannot correctly repair all errors, however. In our example, if the channel flips two bits and the receiver gets 001, the system will detect the error, but conclude that the original bit

6552-405: The parity bits) would be __1_001_1010, and the code word is 011100101010. The choice of the parity, even or odd, is irrelevant but the same choice must be used for both encoding and decoding. This general rule can be shown visually: Shown are only 20 encoded bits (5 parity, 15 data) but the pattern continues indefinitely. The key thing about Hamming Codes that can be seen from visual inspection

6643-467: The performance of random writes is low due to the need to write all parity data to a single disk, unless the filesystem is RAID-4-aware and compensates for that. An advantage of RAID 4 is that it can be quickly extended online, without parity recomputation, as long as the newly added disks are completely filled with 0-bytes. In diagram 1, a read request for block A1 would be serviced by disk 0. A simultaneous read request for block B1 would have to wait, but

6734-504: The physical medium containing the data is lost or stolen. Data loss is distinguished from data unavailability, which may arise from a network outage . Although the two have substantially similar consequences for users, data unavailability is temporary, while data loss may be permanent. Data loss is also distinct from data breach , an incident where data falls into the wrong hands, although the term data loss has been used in those incidents. Studies show hardware failure and human error are

6825-505: The possible Hamming codes: Hamming codes have a minimum distance of 3, which means that the decoder can detect and correct a single error, but it cannot distinguish a double bit error of some codeword from a single bit error of a different codeword. Thus, some double-bit errors will be incorrectly decoded as if they were single bit errors and therefore go undetected, unless no correction is attempted. To remedy this shortcoming, Hamming codes can be extended by an extra parity bit. This way, it

6916-611: The required security or cleanroom procedures. File system corruption can frequently be repaired by the user or the system administrator. For example, a deleted file is typically not immediately overwritten on disk, but more often simply has its entry deleted from the file system index. In such a case, the deletion can be easily reversed. Successful recovery from data loss generally requires implementation of an effective backup strategy. Without an implemented backup strategy, recovery requires reinstallation of programs and regeneration of data. Even with an effective backup strategy, restoring

7007-464: The resulting data storage capacity. The array will continue to operate so long as at least one member drive is operational. Any read request can be serviced and handled by any drive in the array; thus, depending on the nature of I/O load, random read performance of a RAID 1 array may equal up to the sum of each member's performance, while the write performance remains at the level of a single disk. However, if disks with different speeds are used in

7098-481: The same effect — potentially overwriting lost files with the temporary HTML and image files created when viewing a web page. File operations such as copying, editing, or deleting should also be avoided. Upon realizing data loss has occurred, it is often best to shut down the computer and remove the drive in question from the unit. Re-attach this drive to a secondary computer with a write blocker device and then attempt to recover lost data. If possible, create an image of

7189-466: The same logic. The code rate is the second number divided by the first, for our repetition example, 1/3. Hamming also noticed the problems with flipping two or more bits, and described this as the "distance" (it is now called the Hamming distance , after him). Parity has a distance of 2, so one bit flip can be detected but not corrected, and any two bit flips will be invisible. The (3,1) repetition has

7280-399: The same overhead of space. Parity adds a single bit that indicates whether the number of ones (bit-positions with values of one) in the preceding data was even or odd . If an odd number of bits is changed in transmission, the message will change parity and the error can be detected at this point; however, the bit that changed may have been the parity bit itself. The most common convention

7371-486: The same physical location on each disk. Therefore, any I/O operation requires activity on every disk and usually requires synchronized spindles. This makes it suitable for applications that demand the highest transfer rates in long sequential reads and writes, for example uncompressed video editing. Applications that make small reads and writes from random disk locations will get the worst performance out of this level. The requirement that all disks spin synchronously (in

7462-501: The second equation and plug it into the first to find D j = ( g m − i + j ⊕ 1 ) − 1 ( g m − i B ⊕ A ) {\displaystyle D_{j}=(g^{m-i+j}\oplus 1)^{-1}(g^{m-i}B\oplus A)} , and then D i = A ⊕ D j {\displaystyle D_{i}=A\oplus D_{j}} . Unlike P , The computation of Q

7553-428: The serving of write requests. Although it will not be as efficient as a striping (RAID 0) setup, because parity must still be written, this is no longer a bottleneck. Since parity calculation is performed on the full stripe, small changes to the array experience write amplification : in the worst case when a single, logical sector is to be written, the original sector and the according parity sector need to be read,

7644-476: The simple parity code cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are perfect codes , that is, they achieve the highest possible rate for codes with their block length and minimum distance of three. Richard W. Hamming invented Hamming codes in 1950 as a way of automatically correcting errors introduced by punched card readers. In his original paper, Hamming elaborated his general idea, but specifically focused on

7735-494: The single lost file than to recover the entire system. If data loss occurs, a successful recovery must ensure that the deleted data is not over-written. For this reason write operations to the affected storage device should be avoided. This includes not starting the system to which the affected device is connected. This is because many operating systems create temporary files in order to boot, and these may overwrite areas of lost data — rendering it unrecoverable. Viewing web pages has

7826-424: The three bits received are not identical, an error occurred during transmission. If the channel is clean enough, most of the time only one bit will change in each triple. Therefore, 001, 010, and 100 each correspond to a 0 bit, while 110, 101, and 011 correspond to a 1 bit, with the greater quantity of digits that are the same ('0' or a '1') indicating what the data bit should be. A code with this ability to reconstruct

7917-485: The two most common causes of data loss, accounting for roughly three quarters of all incidents. Another cause of data loss is a natural disaster, which is a greater risk dependent on where the hardware is located. While the probability of data loss due to natural disaster is small, the only way to prepare for such an event is to store backup data in a separate physical location. As such, the best backup plans always include at least one copy being stored off-site. The cost of

8008-402: The wrong code word. In general, a code with distance k can detect but not correct k − 1 errors. Hamming was interested in two problems at once: increasing the distance as much as possible, while at the same time increasing the code rate as much as possible. During the 1940s he developed several encoding schemes that were dramatic improvements on existing codes. The key to all of his systems

8099-493: The wrong value. Error correction is therefore a trade-off between certainty (the ability to reliably detect triple bit errors) and resiliency (the ability to keep functioning in the face of single bit errors). This extended Hamming code was popular in computer memory systems, starting with IBM 7030 Stretch in 1961, where it is known as SECDED (or SEC-DED, abbreviated from single error correction, double error detection ). Server computers in 21st century, while typically keeping

8190-506: Was fed in on punched paper tape , seven-eighths of an inch wide, which had up to six holes per row. During weekdays, when errors in the relays were detected, the machine would stop and flash lights so that the operators could correct the problem. During after-hours periods and on weekends, when there were no operators, the machine simply moved on to the next job. Hamming worked on weekends, and grew increasingly frustrated with having to restart his programs from scratch due to detected errors. In

8281-484: Was to have the parity bits overlap, such that they managed to check each other as well as the data. The following general algorithm generates a single-error correcting (SEC) code for any number of bits. The main idea is to choose the error-correcting bits such that the index-XOR (the XOR of all the bit positions containing a 1) is 0. We use positions 1, 10, 100, etc. (in binary) as the error-correcting bits, which guarantees it

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