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### CS231Lecture8

Course: CS 231, Fall 2008
School: Vanderbilt
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Word Count: 661

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Organization Computer CS 231-01 Error Detection and Correction Part 3 Dr. William H. Robinson September 10, 2004 http://eecs.vanderbilt.edu/courses/cs231/ 1 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 2 Review: Error Correction...

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Organization Computer CS 231-01 Error Detection and Correction Part 3 Dr. William H. Robinson September 10, 2004 http://eecs.vanderbilt.edu/courses/cs231/ 1 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 2 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 ] 3 4 1 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 5 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. 6 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 7 8 2 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 9 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 10 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 12 11 3 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 13 correct incorrect incorrect correct 14 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 15 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 16 Bit 10 4 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...

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