ERROR DETECTING AND CORRECTING CODES

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ERROR DETECTING AND CORRECTING CODES -BY
R.W. HAMMING
PRESENTED BY- BALAKRISHNA DHARMANA
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INTRODUCTION

• Why do we need error detection and correction?
• Unwanted Random signals interfere with accurate
  transmission of signals
• Some simple ways of error detection and
  correction
• Sending each word again
• Sending each letter again
• Within a computer errors are rare

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• Systematic codes
• Redundancy
• R n/m
• Redundancy serves to measure the efficiency of
  the code
• Lowers the effective channel capacity

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TYPES OF CODES

• Single error detecting codes
• Single error correcting codes
• Single error correcting plus double error
  detecting codes

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• Application of these codes may be expected to
  occur under conditions-
• Unattended operation over long periods of time
• Extremely large and tightly interrelated systems
  where a single failure causes the entire
  installation
• When the signaling is not possible in the
  presence of noise

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SINGLE ERROR DETECTING CODES

• Contains n-bits
• Out of n-bits, n-1 are information bits and one
  parity bit
• Redundancy n/n-1
• As n increases probability of getting errors
  increases
• Type of check used to detect any single error is
  called parity check (even or odd)

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SINGLE ERROR CORRECTING CODES

• First assign m positions in available positions
  as information positions
• Specific positions are left to a later
  determination
• Assign k remaining positions as check positions
• Apply k parity checks

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• The result of the k parity checks from right to
  left is checking number
• Checking number must describe mk1 different
  things
• so that, 2k gt m k 1
• writing n mk, we find
• 2m lt 2n / n1

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• Now we have to determine the positions over which
  the various parity checks are to be applied
• Any position which has a 1 on the right of its
  binary representation must cause the first check
  fail.
• By examining the binary form of the various
  integers
• 1 - 1
• 3 - 11
• 5 - 101
• 7 - 111 etc

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TABLE II
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SINGLE ERROR CORRECTING PLUS DOUBLE ERROR
DETECTING CODES

• Begin with single error correcting code
• Add one more position for checking all previous
  positions using even parity check
• In the operation of the code ,
• No errors all parity checks including the last
  are satisfied
• Single error- the last parity check fails
• Two errors- last parity check is satisfied and
  indicates some kind of error

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GEOMETRICAL MODEL
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• At a given minimum distance, some of the
  correctability can be exchanged for more
  detectability.
• For example, a subset with minimum distance 5 may
  be used for
• Double error correction
• Single error correction plus triple error
  detection
• Quadruple error detection

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APPLICATION OF GEOMETRICAL MODEL TO CODES

• If code points are at a distance of at least 2
  from each other then any single error will
  carry the code point over to a point that is not
  a code point. Means single error is detectable
• If distance is at least 3 units then any single
  error will leave the point nearer to the correct
  code point than to any other code point, this
  means single error will be correctable.

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CONCLUSION

• This paper helps us to discuss the minimum
  redundancy code techniques for
• Single error detection
• Single error correction
• And single error correction plus double error
  detection
• Also gives the geometrical model of above
  techniques in depth.

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REFERENCE

• M. J. E. Golay, Correspondence, notes on Digital
  coding, Proceedings of the I.R.E., Vol. 37, p.
  657, June 1949.
• http//www.math.ups.edu/bryans/current/journal_sp
  ring_2002/300_EFejta_2002.htm
• http//www.ee.unb.ca/tervo/ee4253/hamming.htm
• http//www.cs.mdx.ac.uk/staffpages/mattsmith/modul
  es/COM1021/seminar_sheets