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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

ERROR-CORRECTING CODES

John Stone

9/21/04

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

Error-Correcting Codes

quick definitions
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

error generation
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

real world example
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

reed muller code
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

hadamard matrices
Hadamard Matrices
  • Each row is a possible code
  • Each matrix has a Hamming distance

Error-Correcting Codes

hamming distance
Hamming Distance
  • The number of places that two binary words differ from one another
  • Can fix  (d – 1) / 2⌡errors

Error-Correcting Codes

encoding example
Encoding Example

Hamming Distance = 4

 (4 – 1) / 2⌡ = 1 (fixable error)

Example:

110  01001101

Error-Correcting Codes

decoding example
Decoding Example

Example:

0101 1101  ?

Result:

0101 1101  0100 1101  110

Error-Correcting Codes

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

Error-Correcting Codes

references
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