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 GCP, DoI, AUEB 11.1

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 GCP, DoI, AUEB 11.2

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)?

D

001

C

011

A

000

B

010

H

101

G

111

E

100

F

110 Computer System Organization GCP, DoI, AUEB 11.3

Hamming Distance of 2

001

B

011

A

000 010

D

101 100 • • What happens with 1 bit in error?

What happens with 2 bits in error?

Computer System Organization GCP, DoI, AUEB

C

110 111 11.4

Hamming Distance of 3

001

A

000 101 010 011

B

111 100 • • What happens with 1 bit in error?

2 bits in error? 3 bits in error?

Computer System Organization GCP, DoI, AUEB 110 11.5

Properties of Distance of Codes

 Code words have m + r bits (m data, r check)    Detecting single bit errors  Code must have distance >= 2 Detecting d single bit errors  Code must have distance >= d+1 Correcting d single bit errors  Code must have distance >= 2d+1  Correcting a single bit error: d = 1, min. distance = 3 (bits) Computer System Organization GCP, DoI, AUEB 11.6

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 GCP, DoI, AUEB 11.7

Parity Bit Concept

 Given the word: 10011011 – add “parity bit”   Even Parity: even # of 1’s: Odd Parity: odd # of 1’s: 1 10011011 0 10011011 Computer System Organization GCP, DoI, AUEB 11.8

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 powers of 2: 1 + 2 … + 2 j = n into a sum of      Bit 9 is checked by 1 and 8 ( 9 = 1 + 2 3 ) 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 P

– Parity (assume even parity)

D

– Data

3 2 1

Computer System Organization GCP, DoI, AUEB 11.9

Example: Hamming Code (11,7)

Computer System Organization GCP, DoI, AUEB 11.10

Error!

Retrieved Stored Computer System Organization GCP, DoI, AUEB 11.11

Determining Bit in Error

Computer System Organization GCP, DoI, AUEB 11.12