Organization Computer CS 231-01 Error Detection and Correction Part 3
Dr. William H. Robinson September 10, 2004 http://eecs.vanderbilt.edu/courses/cs231/
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Topics
Give me a productive error over a boring, mundane and unproductive fact any day.
Anonymous
Coding theory Hamming algorithm and limitations General announcements Assignment
Continue reading Chapter 2 in Tanenbaum
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Review: Error Correction
Motivation:
DRAM is dense Signals are easily disturbed High Capacity higher probability of failure Transmission channels are not ideal introduce noise to signal
Coding Theory
k-bit Data 2k possibilities
g 0,0 g k 1,0
n-bit Code with dmin Generator (mapping) Function
2k Valid Codewords
g 0,n 1 = [c0 g k 1,n 1
Approach: Redundancy
Add extra information so that we can recover from errors Can we do better than just create complete copies?
Block Codes: Data coded in blocks
k data bits coded into n encoded bits Measure of overhead: Rate of Code: K/N Often called an (n,k) code
Adapted from John Kubiatowiczs CS 252 lecture notes. Copyright 2003 UCB.
[m0
mk 1 ]
cn 1 ]
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Decoding Theory
Received n-bit Codeword 2n possibilities Parity (syndrome) Function (n-k-1)-bit Syndrome 2n-k Syndromes
Code Space
General Idea: Code Vector Space
C0=f(d0)
Code Distance (Hamming Distance)
[r0
rn 1 ] H T = [s0
sn k 1 ]
d0
Not every vector in the code space is valid Hamming Distance for the code (dmin):
Minimum number of bit flips to turn one code word into another
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Number of errors that we can detect: (dmin - 1) Number of errors that we can fix: (dmin - 1)
Adapted from John Kubiatowiczs CS 252 lecture notes. Copyright 2003 UCB.
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Hamming Codes
First major class of binary codes designed for error correction Originally used in error control for longdistance telephony Encodes parity for groups of bits within data
Hamming Algorithm
In a Hamming code
r parity bits added to m-bit word Forms codeword with length (m + r) bits
Bit numbering
Starts at 1 with leftmost (high-order) bit All powers of 2 are parity bits Remaining bits are for data
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Bit Numbering for Hamming Algorithm
Given an 8-bit data word to encode Data bits (remaining bits)
Parity Bit Assignment
Bit No. 1 2 3 4 5 6 7 8 9 10 11 12
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Bit No. in Binary 8s 0 0 0 0 0 0 0 1 1 1 1 1 4s 0 0 0 1 1 1 1 0 0 0 0 1 2s 0 1 1 0 0 1 1 0 0 1 1 0 1s 1 0 1 0 1 0 1 0 1 0 1 0
Encoded by
Encodes 3, 5, 7, 9, 11 3, 6, 7, 10, 11
2, 1 5, 6, 7, 12 4, 1 4, 2 4, 2, 1 9, 10, 11, 12 8, 1 8, 2 8, 2, 1 8, 4
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0 0 0 0 0 0 0 0 0 0 0 0 __ __ __ __ __ __ __ __ __ __ __ __ 1 2 3 4 5 6 7 8 9 10 11 12
Parity bits (powers of 2)
Example
Given an 8-bit data word to encode Codeword has bits 12 (8 data, 4 parity)
Bit 1 checks: 1, 3, 5, 7, 9, 11 Bit 2 checks: 2, 3, 6, 7, 10, 11 Bit 4 checks: 4, 5, 6, 7, 12 Bit 8 checks: 8, 9, 10, 11, 12
Boolean Exclusive Sum - XOR Function
Logic operation Output is true when either input is true but not both Corresponds to addition in base 2
A B F = A B Truth table for XOR function F
A 0 0 1 1
B 0 1 0 1
F 0 1 1 0
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Class Exercise
Compute the codeword for the 8-bit data: 0x71 0111 0001 0 0 0 0 1 1 1 1 0 0 0 1 __ __ __ __ __ __ __ __ __ __ __ __ 1 2 3 4 5 6 7 8 9 10 11 12
Class Exercise
Find the bit that is incorrect in the following codeword. This 12 bit codeword encodes 8 data bits. 0x19D 0001 1001 1101
Bit 1 checks: 1, 3, 5, 7, 9, 11 Bit 2 checks: 2, 3, 6, 7, 10, 11 Bit 4 checks: 4, 5, 6, 7, 12 Bit 8 checks: 8, 9, 10, 11, 12
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correct incorrect incorrect correct
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Bit 6
Two-Bit Errors: Detectable
Correct codeword: 0x1DD 0001 1101 1101 Create two-bit error: 0001 1000 1101
Bit 1 checks: 1, 3, 5, 7, 9, 11 Bit 2 checks: 2, 3, 6, 7, 10, 11 Bit 4 checks: 4, 5, 6, 7, 12 Bit 8 checks: 8, 9, 10, 11, 12 correct incorrect incorrect incorrect
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Two-Bit Errors: Undetectable
Correct codeword: 0x1DD 0001 1101 1101 Create two-bit error: 0001 1001 1100
Bit 1 checks: 1, 3, 5, 7, 9, 11 ??? Bit 2 checks: 2, 3, 6, 7, 10, 11 Bit 4 checks: 4, 5, 6, 7, 12 Bit 8 checks: 8, 9, 10, 11, 12 correct incorrect correct incorrect
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Bit 10
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Class Exercise
Compute the codeword for the 8-bit data: 0x96 1001 0110 1 0 1 1 0 0 1 0 0 1 1 0 __ __ __ __ __ __ __ __ __ __ __ __ 1 2 3 4 5 6 7 8 9 10 11 12
Class Exercise
Find the bit that is incorrect in the following code...