Hamming code

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Hamming code
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Hamming Codes

• Class of error-correcting codes
• Capable of correcting all single-error patterns
• For each m gt 2, there is a Hamming code of length
  n 2m 1 with n k m parity check bits

Redundancy
3
m 3 Hamming Code

• Information bits are b1, b2, b3, b4
• Equations for parity checks b5, b6, b7

b5 b1 b3 b4 b6 b1 b2
b4 b7 b2 b3 b4

• There are 24 16 codewords
• (0,0,0,0,0,0,0) is a codeword

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Hamming (7,4) code
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Parity Check Equations

• Rearrange parity check equations

0 b5 b5 b1 b3 b4 b5 0 b6
b6 b1 b2 b4 b6 0 b7 b7
b2 b3 b4 b7

• In matrix form

• All codewords must satisfy these equations
• Note each nonzero 3-tuple appears once as a
  column in check matrix H

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Error Detection with Hamming Code
0 0 1 0 0 0 0
1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1
1 0 1
Single error detected
s H e
0 1 0 0 1 0 0
1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1
1 1 1
0 1 1
1 0 0
Double error detected
s H e

1 1 1 0 0 0 0
1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1
1 1 0
0 1 1
1 0 1
Triple error not detected
s H e
0
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Minimum distance of Hamming Code

• Previous slide shows that undetectable error
  pattern must have 3 or more bits
• At least 3 bits must be changed to convert one
  codeword into another codeword

Set of n-tuples within distance 1 of b2
Set of n-tuples within distance 1 of b1
Distance 3

• Spheres of distance 1 around each codeword do not
  overlap
• If a single error occurs, the resulting n-tuple
  will be in a unique sphere around the original
  codeword

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General Hamming Codes

• For m gt 2, the Hamming code is obtained through
  the check matrix H
• Each nonzero m-tuple appears once as a column of
  H
• The resulting code corrects all single errors
• For each value of m, there is a polynomial code
  with g(x) of degree m that is equivalent to a
  Hamming code and corrects all single errors
• For m 3, g(x) x3x1

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Error-correction using Hamming Codes

• The receiver first calculates the syndrome
• s HR H (b e) Hb He He
• If s 0, then the receiver accepts R as the
  transmitted codeword
• If s is nonzero, then an error is detected
• Hamming decoder assumes a single error has
  occurred
• Each single-bit error pattern has a unique
  syndrome
• The receiver matches the syndrome to a single-bit
  error pattern and corrects the appropriate bit

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Performance of Hamming Error-Correcting Code

• Assume bit errors occur independent of each other
  and with probability p