# Reed-Solomon error correction

Reed-Solomon error correction is a coding scheme which works by first constructing a polynomial from the data symbols to be transmitted and then sending an over-sampled plot of the polynomial instead of the original symbols themselves. Because of the redundant information contained in the over-sampled data, it is possible to reconstruct the original polynomial and thus the data symbols even in the face of transmission errors, up to a certain degree of error.

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## Overview of the method

The data points are sent as an encoded block. The total number of m-bit symbols in the encoded block is n=2m−1. Thus a Reed-Solomon code operating on 8-bit symbols has n=28−1 = 255 symbols per block. (This is a very popular value because of the prevalence of 8-bit byte-oriented computer systems.) The number k, k < n, of data symbols in the block is a design parameter. A commonly used code encodes k = 223 8-bit data symbols plus 32 8-bit parity symbols in a n = 255-symbol block; this is denoted as a (n, k) = (255,223) code. This particular code is capable of correcting up to 16 symbol errors per block.

The scheme encodes the block's message as points in a polynomial plotted over a finite field. The coefficients of the polynomial are the data symbols of the block. The plot overdetermines the coefficients, which can be recovered from subsets of the plotted points. In the same sense that one can correct a curve by interpolating past a gap, a Reed-Solomon code can bridge a series of errors in a block of data to recover the coefficients of the polynomial that drew the original curve.

## Properties of Reed-Solomon codes

The error-correcting ability of any Reed-Solomon code is determined by nk, the measure of redundancy in the block. If the locations of the errored symbols are not known in advance, then a Reed-Solomon code can correct up to (nk)/2 errored symbols, i.e., it can correct half as many errors as there are redundant symbols added to the block. Sometimes error locations are known in advance (e.g. "side information" in demodulator signal-to-noise ratios)—these are called erasures. A Reed-Solomon code is twice as powerful at erasure correction than at error correction, and any combination of errors and erasures can be corrected as long as the equation 2E + S ≤ (n−k) is satisfied, where E is the number of errors and S is the number of erasures in the block.

The properties of Reed-Solomon codes make them especially well-suited to applications where errors occur in bursts. This is because it does not matter to the code how many bits in a symbol are in error—if multiple bits in a symbol are corrupted it only counts as a single error. Alternatively, if a data stream is not characterized by error bursts or drop-outs but by random single bit errors, a Reed-Solomon code is usually a poor choice. More effective codes are available for this case. A Viterbi-decoded convolutional code is an especially good (and popular) choice when soft-decision information is available from the demodulator.

Designers are not required to use the "natural" sizes of Reed-Solomon code blocks. A technique known as "shortening" can produce a smaller code of any desired size from a larger code. For example, the widely used (255,223) code can be converted to a (160,128) code by padding the unused portion of the block (usually the beginning) with 95 binary zeroes and not transmitting them. At the decoder, the same portion of the block is loaded locally with binary zeroes. The compact disc is an example of an application of shortened Reed-Solomon codes.

## Use of Reed-Solomon codes in optical and magnetic storage

Reed-Solomon coding is very widely used in mass storage systems to correct the burst errors associated with media defects.

Reed-Solomon coding is a key component of the compact disc (CD). It was the first use of strong error correction coding in a mass-produced consumer product, and DAT and DVD use similar schemes. In the CD, two layers of Reed-Solomon coding separated by a 28-way convolutional interleaver yields a scheme called Cross-Interleaved Reed Solomon Coding (CIRC). The first element of a CIRC decoder is a relatively weak inner (32,28) Reed-Solomon code, shortened from a (255,251) code with 8-bit symbols. This code can correct up to 2 byte errors per 32-byte block. More importantly, it flags as erasures any uncorrectable blocks, i.e., blocks with more than 2 byte errors. The decoded 28-byte blocks, with erasure indications, are then spread by the deinterleaver to different blocks of the (28,24) outer code. Thanks to the deinterleaving, an erased 28-byte block from the inner code becomes a single erased byte in each of 28 outer code blocks. The outer code easily corrects this, since it can handle up to 4 such erasures per block.

The result is a CIRC that can completely correct error bursts up to 4000 bits, or about 2.5 mm on the disc surface. This code is so strong that most CD playback errors are almost certainly caused by tracking errors that cause the laser to jump track, not by uncorrectable error bursts.

## Timeline of Reed-Solomon development

The code was invented in 1960 by Irving S. Reed and Gustave Solomon, who were then members of MIT Lincoln Laboratory. Their seminal article was "Polynomial Codes over Certain Finite Fields." When it was written, digital technology was not advanced enough to implement the concept. The key to application of Reed-Solomon codes was the invention of an efficient decoding algorithm by Elwyn Berlekamp, a professor of electrical engineering at the University of California, Berkeley. Today they are used in disk drives, CDs, telecommunication and digital broadcast protocols.

## Satellite technique: Reed-Solomon + Viterbi coding

One significant application of Reed-Solomon coding was to encode the digital pictures sent back by the Voyager space probe at Uranus. Voyager introduced Reed-Solomon coding in conjunction with Viterbi-decoded convolutional codes, a practice that has since become very widespread in deep space and satellite (e.g., direct digital broadcasting) communications.

Viterbi decoders tend to produce errors in short bursts. Correcting these burst errors is a job best done by short or simplified Reed-Solomon codes.

Modern versions of concatenated Reed-Solomon/Viterbi-decoded convolutional coding were and are used on the Mars Pathfinder, Galileo, Mars Exploration Rover and Cassini missions, where they perform within about 1–1.5 dB of the ultimate limit imposed by the Shannon capacity.

These concatenated codes are now being replaced by more powerful turbo codes where the transmitted data does not need to be decoded immediately.

## External links

• An application note from 4i2i on some specific implementations (http://www.4i2i.com/reed_solomon_codes.htm)
• a presentation (http://www.cs.cornell.edu/Courses/cs722/2000sp/ReedSolomon.pdf) on Reed & Solomon's paper
• Phil Karn's FEC page (includes GPL source for various Reed-Solomon codecs) [1] (http://www.ka9q.net/code/fec/)
• A collection of links to books, online articles and source code [2] (http://www.radionetworkprocessor.com/reed-solomon.html)de:Reed-Solomon-Code
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