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Error Detection and Correction
Fixing 0101X011
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Computer Errors RAM isn't perfect
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Computer Errors Networks aren't either
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Computer Errors How the heck do you read 1s and 0's off this?
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A message 4 bit message: 1
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A message 4 bit message: An errror: 1 1
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Trick 1 : Repetition To avoid misunderstanding, repeat yourself…
Copy 1 Copy 2 Copy 3 1
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Trick 1 : Repetition An error: Copy 1 Copy 2 Copy 3 1
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Trick 1 : Repetition Most common message wins: Copy 1 Copy 2 Copy 3 1
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Trick 1 : Repetition What if every message is wrong: Copy 1 Copy 2
1
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Trick 1 : Repetition Most common bit wins: Copy 1 Copy 2 Copy 3 1
1 Corrected 1
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Trick 1 : Repetition More errors: Copy 1 Copy 2 Copy 3 1 "Corrected" 1
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Trick 1 : Repetition Best 3 out of 5? Copy 1 Copy 2 Copy 3 Copy 4
1 Corrected 1
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Overhead 1 Message size : 4 bits Including repetition : 12 bits
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Overhead Message size : 4 bits Including repetition : 12 bits
Message size : 4 bits Including repetition : 12 bits 200% overhead 10Mb download is now 30Mb!
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Trick 2 : Redundancy Redundancy : more information than strictly required Common linguistic trick: He took his seat She took her seat They took their seats
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Trick 2 : Redundancy Redundancy : more information than strictly required Common linguistic trick: He took his seat She took her seat They took their seats
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Trick 2 : Redundancy Repetion is redundancy
Can we be redundant more efficently?
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Hamming Distance Hamming Distance : number of different bits 1010
1 2 3 4 1010 0010, 1110, 1000, 1011 0110, 0000, 0011, 1100, 1111, 1001 1101, 0001, 0111, 0100 0101
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Normal Binary 4 bits : 16 possible values: 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
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Normal Binary Using all patterns – any error looks like a different message 1 bit errors for 1010 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
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Normal Binary Only use half the patterns All "good" patterns have a distance of 2 from each other 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
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Normal Binary "Good" patterns have distance of 2: 1 bit error is obviously an error 1 bit errors for 1010 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
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Hamming Code Message Coded Message 0000 0001 0010 0011 0000000
Use extra bits to "space out" messages 4 bit message with 3 error correction bits: Message Coded Message 0000 0001 0010 0011
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Hamming Code 7 bits could be 27 = 128 codes Only use 16 of them
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Hamming Code Every good message has distance of 3+ from other good messages:
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Our Message We get: 0110110 Which message was it meant to be? 0000000
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Errors Assuming Then Started with valid code word Only one error
1 bit from one valid word 2+ bits from another valid code word Valid Code A Valid Code B Valid Code C Error
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Our Message We get: 0110110 Find the code with distance of 1 0000000
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Errors Assuming 1 error bit, we can identify correct message: Received codes After decoding , , , , ,
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Hamming Code Overhead Message size : 4 bits Code word: 7 bits
512bit message can be encoded with 522bits: 2% overhead!
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Trick 3 : Checksums Parity 1 extra bit used to make odd num of 1's
Odd or even number of 1's 1 extra bit used to make odd num of 1's data checkbit
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Trick 3 : Checksums Message: 00001 All 1 bit errors:
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Trick 3 : Checksums Checksum for decimal number: Message: 46756
Add digits, mod by 10: Message: 46756 = 28 28 mod (clock size) 10 = 8 Coded message:
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Trick 3 : Checksums Coded message: 467568 Error message: 461568
= 22 22 mod (clock size) 10 = 2!!! we have a problem
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Two Errors Coded message: 467568 Error message: 421568 Check: 421568
= 18 18 mod (clock size) 10 = 8!!! we missed it
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Staircse Code Multiply each digit by its place: 12345 Message: 46756
4 x x x x x 5 = 87 87 mod (clock size) 10 = 7 Coded message:
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Two Errors w Stair Case Coded message: 467567 Error message: 421567
Check: 4 x x x x x 5 = 61 61 mod (clock size) 10 = 1!!! we caught it
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Real Life Stair Case ISBN – books:
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Trick 4: Pinpoint How did I do it?
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Trick 4: Pinpoint How did I do it?
Every Row & Col should have odd # of black squares
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Trick 4: Pinpoint How did I do it?
Every Row & Col should have odd # of black squares
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Trick 4: Pinpoint Message / Checksum
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Trick 4 : Pinpoint With decimal values: 4 7 1 3 9 2 6
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Trick 4 : Pinpoint With decimal values: 4 7 1 5 9 2 6
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Trick 4 : Pinpoint With decimal values: 4 7 1 5 9 2 6 4 9
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4 9 4 7 1 3 9 2 6 Trick 4 : Pinpoint With decimal values: off by 2
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Hamming Code Hamming Codes as pinpoint parity checks:
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Real Life Checksum Last digit of credit card number calculated to
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