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ERROR-CORRECTING CODES

ERROR-CORRECTING CODES. John Stone 9/21/04. Overview. Quick Definitions Error Generation Real World Example Reed-Muller Code Hadamard Matrices Hamming Distance Encoding Example Decoding Example Summary References. Quick Definitions. Error-Correcting Code

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ERROR-CORRECTING CODES

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  1. ERROR-CORRECTING CODES John Stone 9/21/04

  2. Overview • Quick Definitions • Error Generation • Real World Example • Reed-Muller Code • Hadamard Matrices • Hamming Distance • Encoding Example • Decoding Example • Summary • References Error-Correcting Codes

  3. Quick Definitions • Error-Correcting Code • Code that can detect and fix its own errors • Error (Binary) • 0 changed to 1 • 1 changed to 0 • Example • 100010101011101 • 101010100011101 Error-Correcting Codes

  4. Error Generation • Noise • Other signals • Other devices • Distance • Reduced signal strength • Greater chance of interference • Examples • A space probe that collects and transmits data back to earth • Downloading from a FTP site in China Error-Correcting Codes

  5. Real World Example • Mariner series of Mars reconnaissance vehicles • 32 possible gray levels • 5 bits / pixel • (32, 5) Reed-Muller Code • 32 bits / pixel Error-Correcting Codes

  6. Reed-Muller Code • Code produced that contains extra information so that a limited amount of errors can be corrected if needed • Uses Hadamard matrices for encoding and decoding error-correcting codes. Error-Correcting Codes

  7. Hadamard Matrices • Each row is a possible code • Each matrix has a Hamming distance Error-Correcting Codes

  8. Hamming Distance • The number of places that two binary words differ from one another • Can fix  (d – 1) / 2⌡errors Error-Correcting Codes

  9. Encoding Example Hamming Distance = 4  (4 – 1) / 2⌡ = 1 (fixable error) Example: 110  01001101 Error-Correcting Codes

  10. Decoding Example Example: 0101 1101  ? Result: 0101 1101  0100 1101  110 Error-Correcting Codes

  11. Summary • Introduction • Quick Definitions • Error Generation • Real World Example • Reed-Muller Code • Hadamard Matrices • Hamming Distance • Encoding Example • Decoding Example Error-Correcting Codes

  12. References • Fundamentals of Error-Correcting Codes • Author: Cary Huffman • Publisher: Cambridge University Press • Date: June 26, 2003 • ISBN: 0521782805 • The Theory of Error-Correcting Codes • Author: Neil J. A. Sloane • Publisher: North-Holland • Date: January 1, 1983 • ISBN: 0444851933 • Error Correcting Codes: A Mathematical Introduction • Author: John Baylis • Publisher: Chapman & Hall/CRC • Date: December 4, 1997 • ISBN: 0412786907 • Websites • http://www.eccpage.com/ • http://mathworld.wolfram.com/ Error-Correcting Codes

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