Error Detecting and Error Correcting Codes

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Error Detecting and Error Correcting Codes

• By
• R. W. Hamming

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What is this about?

• How to transit bits over a possibly noisy
  communication channel
• Noisy communication channel may introduce a
  variety of errors
• In transmitting information from one place to
  another digital machines use codes which are
  simply sets of symbols to which meaning or values
  are attached

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Transmitting Bits
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Applications

• RAM in pc
• I/O bus on high performance servers
• Serial data transmission

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Motivation

• Large scale computing machines in which a large
  number of operations must be performed without a
  single error in the end result.

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Systematic Codes

• Transmitting equipment handles information in the
  binary form of sequences of 0s 1s
• As codes in which each code symbol has exactly n
  binary digits,
• m digits are associated with the information
• k digits are used for error detection and
  correction
• kn-m

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Redundancy

• As the ratio of the number of binary digits used
  to the minimum number necessary to convey the
  same information
• R n/m
• Measure the efficiency of the code for
  transmission of information

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Single Error Detecting Codes

• Constructing n binary digits code
• In the first n-1 positions, put n-1 digits of
  information
• In the n-th position, place either 0 or 1 so that
  the entire n positions have an even number of 1s

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Example

• n-1 information digits
• 100
• n-th position
• 1001
• Received code
• 1000
• Error detected

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Parity Check

• Determine whether or not the symbol has any
  single error
• Even parity check uses the even number of 1s to
  determine the setting of the check position
• Odd parity check uses odd number of 1s
• May check over selected positions only

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Single Error Correcting Codes

• Constructing the code
• Assign m as information positions
• Assign k (n-m) as check positions
• Determine the values in the k positions
• Even parity checks over selected information
  positions

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• Received coded symbol
• Apply k-parity checks on a coded symbol
• Parity check assigns the value observed in its
  check position, 0 is written
• Checking Number The sequence of k 0s and 1s
  written from right to left in a line

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Check Number

• Requirements of checking number
• Give the position of the any single error, with
  zero value meaning no error in the symbol
• Check must describe mk1 different things
• 2k gt mk1
• 2m lt 2n/(n1)

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Parity check positions

• Checking number is obtained digit by digit from
  right to left
• Check for even parity inorder
• The checking number gives the position of any
  error in a code symbol

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• Any position which has a 1 on the right of its
  binary representation must cause the first check
  to fail
• 1 1
• 3 11
• 5 101
• 7 111

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Parity Check Positions

• First parity check position must use positions
  that have a 1 on the extreme right
• 1,3,5,7,9
• Second which have 1s for the second digit from
  the right of their binary rep
• 2,3,6,7,10,11,..
• Third 4,5,6,7,12,.

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• Choice of the check positions 1,2,4,8..
• Advantage of making the setting of the check
  positions independent of each other

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Single Error Correcting Double Error Detecting
Codes

• Start with a single error correcting code
• Add one more position for checking all the
  previous positions, using an even parity check

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Cases

• No error All parity checks are satisfied
• Single error The last parity check fails in all
  situations. The original checking number gives
  the position of the error
• Two errors Last parity check is satisfied, and
  the checking number indicates some kind of error

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Geometrical Model

• Symbols of the code with vertices of a unit
  n-dimensional cube
• Distance A single error in a code point changes
  one coordinate, two errors, two coordinates, and
  in general d errors result in a difference in d
  coordinates

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• Distance D(x,y), distance between two points x
  and y as the number of coordinates for which x
  and y are different
• Least number of edges which must be traversed in
  going from x and y

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Single Error Detecting Codes

• Packing the maximum number of points in a unit
  n-dimensional cube such that no two points are
  closer than 2 units from each other
• If the packing is optimal, then each of the two
  sub-cubes has half the points

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Single Error Correcting Codes

• In Codes we considered
• A distinction was made between information and
  check positions
• In Geometrical model
• No real distinction between the various
  coordinates

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Systematic Codes

• Symbol lengths are all equal
• The positions checked are independent of the
  information contained in the symbol
• The checks are independent of each other
• Use parity checks

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Single Error Correcting Double Error Detecting
Codes

• To prove the codes constructed are of minimum
  redundancy

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Conclusions

• Construction of systematic codes
• Single error detection
• Single error correction
• Single error correction and double error
  detection