Error Correcting Codes

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Error Correcting Codes
W

• A basic description of the Hamming (7,4) Linear
  Error Correcting Code, and stacking cannon balls
  in n-dimensions.

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The Problem Noise
W
h
Message 1 1 1 1
Message 1 1 0 1
Noise 0 0 1 0
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Poor solutions
W
h
a

• Repeats
• Data 1 1 1 1
• Message
• 1 1 1 1
• 1 1 1 1
• 1 1 1 1

• Single CheckSum -
• Truth table
• General form
• Data1 1 1 1
• Message1 1 1 1 0

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Why they are poor
W h a t

• Repeat 3 times
• This divide W by 3
• It divides overall capacity by at least a factor
  of 3x.
• Single Checksum
• Allows an error to be detected but requires the
  message to be discarded and resent.
• Each error reduces the channel capacity by at
  least a factor of 2 because of the thrown away
  message.

Shannon Efficiency
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Hammings Solution
W h a t i

• Encoding
• Multiple Checksums
• Messagea b c d
• r (abd) mod 2
• s (abc) mod 2
• t (bcd) mod 2
• Coder s a t b c d

Message1 0 1 0 r(100) mod 2 1
s(101) mod 2 0 t(010) mod 2 1 Code
1 0 1 1 0 1 0
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Simulation
W h a t i s
Stochastic Simulation
100,000 iterations Add Errors to (7,4) data No
repeat randoms Measure Error Detection
Results

• Error Detection
• One Error 100
• Two Errors 100
• Three Errors 83.43
• Four Errors 79.76

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How it works 3 dots
W h a t i s t

• Only 3 possible words
• Distance Increment 1

One Excluded State (red)
Two valid code words (blue)

• It is really a checksum.
• Single Error Detection
• No error correction

This is a graphic representation of the Hamming
Distance
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Hamming Distance
W h a t i s t h

• Definition
• The number of elements that need to be changed
  (corrupted) to turn one codeword into another.
• The hamming distance from
• 0101 to 0110 is 2 bits
• 1011101 to 1001001 is 2 bits
• butter to ladder is 4 characters
• roses to toned is 3 characters

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Another Dot
W h a t i s t h e

• The code space is now 4.
• The hamming distance is still 1.

Allows Error DETECTION for Hamming Distance
1. Error CORRECTION for Hamming Distance 1
For Hamming distances greater than 1 an error
gives a false correction.
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Even More Dots
W h a t i s t h e M
Allows Error DETECTION for Hamming Distance
2. Error CORRECTION for Hamming Distance 1.

• For Hamming distances greater than 2 an error
  gives a false correction.
• For Hamming distance of 2 there is an error
  detected, but it can not be corrected.

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Multi-dimensional Codes
W h a t i s t h e M a

• Code Space
• 2-dimensional
• 5 element states
• Circle packing makes more efficient use of the
  code-space

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Cannon Balls
W h a t i s t h e M a t
Efficient Circle packing is the same as
efficient 2-d code spacing
Efficient Sphere packing is the same as
efficient 3-d code spacing
Efficient n-dimensional sphere packing is the
same as n-code spacing

• http//wikisource.org/wiki/Cannonball_stacking
• http//mathworld.wolfram.com/SpherePacking.html

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More on Codes
W h a t i s t h e M a t r

• Hamming (11,7)
• Golay Codes
• Convolutional Codes
• Reed-Solomon Error Correction
• Turbo Codes
• Digital Fountain Codes

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An Example
W h a t i s t h e M a t r i

• We will
• Encode a message
• Add noise to the transmission
• Detect the error
• Repair the error

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Encoding the message
W h a t i s t h e M a t r i x ?
But why?
To encode our message
we multiply this matrix
You can verify that
Hamming1 0 0 01 0 0 0 0 1 1 Hamming0 1 0
00 1 0 0 1 0 1 Hamming0 0 1 00 0 1 0 1 1
0 Hamming0 0 0 10 0 0 1 1 1 1
By our message
Where multiplication is the logical AND And
addition is the logical XOR
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Add noise

• If our message is
• Message 0 1 1 0
• Our Multiplying yields
• Code 0 1 1 0 0 1 1
• Lets add an error,
• so Pick a digit to mutate

Code gt 0 1 0 0 0 1 1
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Testing the message

• The matrix used to decode is
• To test if a code is valid
• Does DecoderCodeT
• 0 0 0
• Yes means its valid
• No means it has error/s

• We receive the erroneous string
• Code 0 1 0 0 0 1 1
• We test it
• DecoderCodeT
• 0 1 1
• And indeed it has an error

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Repairing the message

• To repair the code we find the collumn in the
  decoder matrix whose elements are the row results
  of the test vector
• We then change

• DecodercodeT is
• 0 1 1
• This is the third element of our code
• Our repaired code is
• 0 1 1 0 0 1 1

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Decoding the message

• We trim our received code by 3 elements and we
  have our original message.
• 0 1 1 0 0 1 1 gt 0 1 1 0

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Questions?