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http://faculty.chemeketa.edu/ascholer/cs160/Files/ecGrid.html

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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: 1010

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A message 4 bit message: An errror: 1010 1110

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Trick 1 : Repetition To avoid misunderstanding, repeat yourself… 1010 1010 1010 Copy 1 Copy 2 Copy 3

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Trick 1 : Repetition An error: 1010 1110 1010 Copy 1 Copy 2 Copy 3

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Trick 1 : Repetition Most common message wins: 1010 1110 1010 Copy 1 Copy 2 Copy 3

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Trick 1 : Repetition What if every message is wrong: 0010 1110 1011 Copy 1 Copy 2 Copy 3

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Trick 1 : Repetition Most common bit wins: 0010 1110 1011 Copy 1 Copy 2 Copy 3 1010Corrected

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Trick 1 : Repetition More errors: 0000 1110 0011 Copy 1 Copy 2 Copy 3 0010"Corrected"

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Trick 1 : Repetition Best 3 out of 5? 0000 1110 0011 1111 1000 Copy 1 Copy 2 Copy 3 Copy 4 Copy 5 1010Corrected

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Overhead Message size : 4 bits Including repetition : 12 bits 1010 1010 1010

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Overhead Message size : 4 bits Including repetition : 12 bits 200% overhead 10Mb download is now 30Mb! 1010 1010 1010

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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 0010, 1110, 1000, 1011 0110, 0000, 0011, 1100, 1111, 1001 1101, 0001, 0111, 0100 0101 Distance 0 1 2 3 4

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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 Distance of 1 between values… any error will look like new value 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 Add extra bits to "space out" messages 4 bit message with 3 error correction bits: MessageCoded Message 0000 0001 0010 0011 0000000 0001011 0010111 0011100

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Hamming Code 7 bits could be 2 7 = 128 codes Only use 16 of them 0000000 1000110 0001011 1001101 0010111 1010001 0011100 1011010 0100101 1100011 0101110 1101000 0110010 1110100 0111001 1111111

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Hamming Code Every code word has distance of 3+ from other messages: 0000000 1000110 0001011 1001101 0010111 1010001 0011100 1011010 0100101 1100011 0101110 1101000 0110010 1110100 0111001 1111111

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Our Message We get: 0110110 Which message was it meant to be? 0000000 1000110 0001011 1001101 0010111 1010001 0011100 1011010 0100101 1100011 0101110 1101000 0110010 1110100 0111001 1111111

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Errors Assuming – Started with valid code word – Only one error Then – 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 1000110 0001011 1001101 0010111 1010001 0011100 1011010 0100101 1100011 0101110 1101000 0110010 1110100 0111001 1111111

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Errors Assuming 1 error bit, we can identify correct message: Received codesAfter decoding 0000000, 0000001, 0000010 0000100, 0001000, 0010000 0100000, 1000000 0 0 0 0 0 0 0

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Hamming Code Overhead Message size : 4 bits Code word: 7 bits 75% overhead… 512bit message can be encoded with 522bits: 2% overhead!

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Trick 3 : Checksums Parity – Odd or even number of 1's 1 extra bit used to make odd num of 1's data checkbit 10011001000000111100

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Trick 3 : Checksums Message: 00001 All 1 bit errors: 10001 01001 00101 00011 00000

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Trick 3 : Checksums Checksum for decimal number: – Add digits, mod by 10: Message: 46756 – 4 + 6 + 7 + 5 + 6 = 28 – 28 mod (clock size) 10 = 8 Coded message: 467568

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Trick 3 : Checksums Coded message: 467568 Error message: 461568 Check: 461568 – 4 + 6 + 1 + 5 + 6 = 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 – 4 + 2 + 1 + 5 + 6 = 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 1 + 6 x 2 + 7 x 3 + 5 x 4 + 6 x 5 = 87 – 87 mod (clock size) 10 = 7 Coded message: 467567

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Two Errors w Stair Case Coded message: 467567 Error message: 421567 12345 Check: 421567 – 4 x 1 + 2 x 2 + 1 x 3 + 5 x 4 + 6 x 5 = 61 – 61 mod (clock size) 10 = 1!!! we caught it

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Real Life Stair Case ISBN – books: http://www-math.ucdenver.edu/~wcherowi/jcorner/isbn.html http://www-math.ucdenver.edu/~wcherowi/jcorner/isbn.html

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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: 471 392 76

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Trick 4 : Pinpoint With decimal values: 471 592 76

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Trick 4 : Pinpoint With decimal values: 471 592 76 4 9

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Trick 4 : Pinpoint With decimal values: 471 392 76 4 9 off by 2

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Hamming Code Hamming Codes as pinpoint parity checks: http://www.systems.caltech.edu/EE/Faculty/rjm /SAMPLE_20040708.html

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Real Life Checksum Last digit of credit card number calculated to http://tywkiwdbi.blogspot.com/2012/06/checksum-number-on-credit-card.html

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