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87 lines
3.6 KiB
Plaintext
87 lines
3.6 KiB
Plaintext
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Introduction to Reed Solomon Codes:
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Henry Minsky, Universal Access Inc.
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hqm@alum.mit.edu
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[For details see Cain, Clark, "Error-Correction Coding For Digital
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Communications", pp. 205.] The Reed-Solomon Code is an algebraic code
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belonging to the class of BCH (Bose-Chaudry-Hocquehen) multiple burst
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correcting cyclic codes. The Reed Solomon code operates on bytes of
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fixed length.
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Given m parity bytes, a Reed-Solomon code can correct up to m byte
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errors in known positions (erasures), or detect and correct up to m/2
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byte errors in unknown positions.
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This is an implementation of a Reed-Solomon code with 8 bit bytes, and
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a configurable number of parity bytes. The maximum sequence length
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(codeword) that can be generated is 255 bytes, including parity bytes.
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In practice, shorter sequences are used.
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ENCODING: The basic principle of encoding is to find the remainder of
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the message divided by a generator polynomial G(x). The encoder works
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by simulating a Linear Feedback Shift Register with degree equal to
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G(x), and feedback taps with the coefficents of the generating
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polynomial of the code.
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The rs.c file contains an algorithm to generate the encoder polynomial
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for any number of bytes of parity, configurable as the NPAR constant
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in the file ecc.h.
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For this RS code, G(x) = (x-a^1)(x-a^2)(x-a^3)(x-a^4)...(x-a^NPAR)
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where 'a' is a primitive element of the Galois Field GF(256) (== 2).
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DECODING
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The decoder generates four syndrome bytes, which will be all zero if
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the message has no errors. If there are errors, the location and value
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of the errors can be determined in a number of ways.
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Computing the syndromes is easily done as a sum of products, see pp.
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179 [Rhee 89].
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Fundamentally, the syndome bytes form four simultaneous equations
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which can be solved to find the error locations. Once error locations
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are known, the syndrome bytes can be used to find the value of the
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errors, and they can thus be corrected.
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A simplified solution for locating and correcting single errors is
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given in Cain and Clark, Ch. 5.
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The more general error-location algorithm is the Berlekamp-Massey
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algorithm, which will locate up to four errors, by iteratively solving
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for the error-locator polynomial. The Modified Berlekamp Massey
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algorithm takes as initial conditions any known suspicious bytes
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(erasure flags) which you may have (such as might be flagged by
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a laser demodulator, or deduced from a failure in a cross-interleaved
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block code row or column).
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Once the location of errors is known, error correction is done using
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the error-evaluator polynomial.
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APPLICATION IDEAS
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As an example application, this library could be used to implement the
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Compact Disc standard of 24 data bytes and 4 parity bytes. A RS code
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with 24 data bytes and 4 parity bytes is referred to as a (28,24) RS
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code. A (n, k) RS code is said to have efficiency k/n. This first
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(28,24) coding is called the C2 or level 2 encoding, because in a
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doubly encoded scheme, the codewords are decoded at the second
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decoding step.
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In following the approach used by Compact Disc digital audio, the 28
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byte C2 codewords are four way interleaved and then the interleaved
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data is encoded again with a (32,28) RS code. The is the C1 encoding
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stage. This produces what is known as a "product code", and has
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excellent error correction capability due to the imposition of
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two-dimensional structure on the parity checks. The interleave helps
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to insure against the case that a multibyte burst error wipes out more
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than two bytes in each codeword. The cross-correction capability of
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the product code can provide backup if in fact there are more than 2
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uncorrectable errors in a block.
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