Solving The Reed-Solomon Error Correction Algorithm

Solving The Reed-Solomon Error Correction Algorithm

Here are a few simple steps that can help solve the problem of the Reed-Solomon error correction algorithm.


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    The Reed-Solomon algorithm is often used to correct multi-bit errors in an exclusive page, but it can fail due to large message blocks, which currently drastically increases the time to encode or decode some data, when the block size can be doubled. . A popular Reed-Solomon code is usually RS(255, 247) with 8-bit characters.


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  • Reed-Solomon codes are a suitable group of error correction codes introduced by Sirving. Reid and Solomon Gustave in 1960.[1]They have a wide range of applications, the most important of which typically include consumer technologies such as minidiscs, CDs, DVDs, Blu-ray discs, codes, high-speed data transfer technologies such as DSL and WiMAX, broadcast systems, such as satellite communications. . , DVB and ATSC, as well as storage systems such as RAID 6.

    Reed-Solomon codes operate on a block between data, considered as a set of field finite elements, called characters. Reed-Solom codesThey can detect and recommend several character errors. By adding c = n – k check characters to the data, the Reed-Solomon code can understand (but not correct) any given combination of all erroneous characters up to t or possibly positive – find and correct Št / 2⌠‹ erroneous characters in unknown places . As an erasure code, it can probably fix up to t erasures at known points and give the algorithm time, or it can find and fix combinations of errors in addition to erasures. Reed-Solomon codes are also suitable as packet error correction codes because a sequence of b+1 consecutive minor bit errors can affect at most one symbol of size b. The definition of t is usually the responsibility of the code designer and can be freely chosen if desired.

    How do you use Reed Solomon?

    wr takes the message r(x) and divides it by the power generator polynomial g(x).If r(x)/g(x)=0, this implies a numerical error.If r(x)/g(x)≠0, then the corruption polynomial is evaluated using the following expression: r(x) implies p(x) * g(x) + e(x)The error polynomial gives the position of the defects.

    There are two main types of Reed-Solomon codes: native representation and BCH representation. The BCH representation is the most common because BC read decodersH are faster and use less memory than the d Origin decoder. Codes


    How does Reed Solomon error correction work?

    Reed-Solomon codes can detect and correct some character errors. Adding t means n – k control characters, so you can detect (but not correct)i) data, Reed-Solomon code, any grouping up to t distortions, or find and correct distortions in ⌊t/2⌋ unidentified places.

    Reed – Solomon was accidentally developed in 1960 by Irving S. Reed and Gustave Solomon, then members of the MIT Lincoln Laboratory. Their seminal paper was essentially titled “Polynomial Codes over Finite Safe Fields”. (Reid and Solomon, 1960). The original encoding scheme described in Reed & Solomon’s Accurate article used a variable polynomial based on the correspondence being encoded, knowing that a set of actually fixed values ​​(scores) intended to aid in encoding, they could encode and decode. The original theoretical decoder generated potential polynomials based on subsets of k messages (uncoded length) from n values ​​(encoded implication length) of the received message, choosing the most popular polynomial as good, which was impractical for all but the simplest cases. This was initially resolved by converting the original schema to a code-like BCH schema based onbased on a particular fixed polynomial known as the encoder-decoder, but later basic decoders were developed based on the original scheme, although they are slower than the BCH schemes. . The result of this would be that there are two main styles of Reed-Solomon codes: those that use the original scheme encoding and those that use the BCH method encoding.

    Also in 1960, a practical stationary polynomial decoder for BCH codes, developed by Daniel Gorenstein and Neil Zierler, was described by Zierler in a report by the Lincoln Laboratory of the Massachusetts Institute of Technology in January 1961, and then in a printed copy in June 1961.< sup> [2 ] The Gorenstein-Zierler decoder and related work on BCH codes is described in W. Wesley Peterson’s novel Error Correction with Codes (1961).[3] Vers 1963 Possibly (or earlier) , J. J. Stone (and others known) that Reed-Solomon codes can use the BCH scheme with a fixed generator polynomial, which makes these codes a special class of BCH codes,[4 ], but Reed-Solomon codes that work according to the original coding scheme are all is not yet a class of BCH reduction codes, and in terms of points they are not just cyclic codes.

    reed-solomon error correction algorithm

    In 1969, Alvin Berlekamp and James Massey developed an improved BCH decoding scheme known since then as the Berlekamp-Massey decoding algorithm.

    Another improved BCH scheme decoder was developed by Yasuo Sugiyama based on the extended Euclid algorithm.[5]

    In 1977, the Reed-Solomon codes were put into practice in the Voyager program, modeled on error correction chain laws. In 1982, along with the CD, came the first mass-produced commercial consumer product that used multiple interleaved Reed-Solomon codes. Today, Reed-Solomon codes are widely used in online digital storage and digital communication environments, although they are being gradually replaced by Bowes-Chowdhury-Hawkvingham (BCH) codes. In phase, Reed-Solomon codes are used in a kind of DVB (Digital Video Broadcasting) standard in DVB-S in combination with internal code folding, but BCH codes are used with LDPC in its successor DVB-S2.

    In 1986 there wasand an original decoder circuit called Berlekamp-Welch, an algorithm, has been developed.

    In 1996, variants of legal decoders called list decoders or appeasement decoders were developed by Madhu Sudan and others, and work on decoders of this type is ongoing – see Guruswami – Sudan List Decode Algorithm.

    Is Reed Solomon a perfect code?

    One might wonder how MDS codes become perfect codes, but they are undeniably incomparable; There are ideal codes because they are not MDS codes, but MDS codes need not be made up. The Reed-Solomon code is an illustration of this.

    In 2002, Shuhong Gao developed another new circuit based on the extended Euclid algorithm.[6]


    Data Store

    Reed-Solomon coding is very important in used data storage systems Time Accuracybatch errors due to paper defects.

    How many errors can Reed Solomon correct?

    The Reed-Solomon decoder can correct the uptime for t errored symbols after the codeword, where 2t means n-k. The decoder can correct all 2007 character errors in the code text: d. H Errors up to the fourth or 16 bytes anywhere in the code word can in many cases be corrected automatically.

    Reed-Solomon encoding is the key to making CDs possible. It was the first powerful error correction coding in a mass loan product, using systems similar to those used by DAT and DVD. On CD, more levels of Reed-Solomon coding, by removing the 28-channel convolutional interleaver, results in an arbitrary scheme called cross-linked Reed-Solomon coding.Common Interleaved (CIRC). The first element of the CIRC decoder is a relatively weak inner Reed-Solomon code (32.28) truncated from each code (255.251) of 8-bit symbols. This can correct up to two byte errors per 32 byte block. It is important to note that it marks several uncorrectable blocks as deleted, i.e. H-blocks that contain more than 2 error bytes. The decoded 28-byte blocks with erasure indicators are then split together with a de-interleaver to help the various outer code blocks (28, 24). As a result of deinterleaving, a particular erased block of 28 bytes of clear code becomes one erased byte in each of the 28 outer stop codes. v
    reed-solomon error correction algorithm

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