Error-Detecting and Error-Correcting Codes

1. Error-Detecting and Error-Correcting Codes • Motivation • Computers make errors occasionally (data gets corrupted) due to • Voltage spikes • Cosmic particles • Corrupt data causes incorrect behavior • Fix • Use some bits to hold redundant information • Data + Redundancy  Code Words • Depending of amount of redundancy (and exact properties of the codes) we can • Detect errors • Correct errors (automatically) Computer System Organization

2. Hamming Distance • Hamming Distance (between 2 codewords): • the number of bits that need to be changed (reversed) to change one codeword into the other codeword • Example: change in 1 bit creates a new (valid) codeword • Hamming Distance = 1 • Equivalently: number of bits that differ • 1111 and 1010 are 2 bits apart • 1111 and 0000 have distance 4 • Hamming Distance of a code: the minimum Hamming Distance between any two codewords of the code Computer System Organization

3. D -001 C -011 A -000 B -010 H -101 G -111 F -110 E -100 Hamming Distance of 1 • Hamming Distance of 1: change in 1 bit creates a new codeword • What happens with change of 1 bit (1 bit in error)? Computer System Organization

4. 001 B -011 A -000 010 D -101 111 C -110 100 Hamming Distance of 2 • What happens with 1 bit in error? • What happens with 2 bits in error? Computer System Organization

5. 001 011 A -000 010 101 B -111 110 100 Hamming Distance of 3 • What happens with 1 bit in error? • 2 bits in error? 3 bits in error? Computer System Organization

6. Properties of Distance of Codes • Code words have m + r bits (m data, r check) • Detectingsingle bit errors • Code must have distance >= 2 • Detecting d single bit errors • Code must have distance >= d+1 • Correctingd single bit errors • Code must have distance >= 2d+1 • Correcting a single bit error: d = 1, min. distance = 3 (bits) Computer System Organization

7. Example of Code Distance Properties • Consider the code with only 2 code words • 1111 and 0000 • Distance of 4 • 1110 • Detected as single bit error • Distance 1 from 1111 • Correctable since only one code word can have single bit error and become “1110” • This is the 1111 codeword • 1100 • Detected as 2 bit errors • Distance 2 (e.g., from 1111) • Correctable? Computer System Organization

8. Parity Bit Concept • Given the word: 10011011 – add “parity bit” • Even Parity: even # of 1’s: 110011011 • Odd Parity: odd # of 1’s: 010011011 Computer System Organization

9. Hamming’s Algorithm(Illustrated in the Single Bit Correction Case) • Bits in power of two position are check bits • Bit n is checked by bits in the decomposition of n into a sum of powers of 2: 1 + 2 … + 2j = n • Bit 9 is checked by 1 and 8 ( 9 = 1 + 23) • Bit 1 checks 1, 3, 5, 7, 9, 11 in codeword • 1+2 = 3, 1+4 = 5, etc. • 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 • 8 bit data word has codeword of the form • D D D D P D D D P D P P • 12 11 10 9 8 7 6 5 4 3 21 • P – Parity (assume even parity) • D – Data Computer System Organization

10. Example: Hamming Code (11,7) Computer System Organization

11. Error! Retrieved Stored Computer System Organization

12. Determining Bit in Error Computer System Organization