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
GCP, DoI, AUEB

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
GCP, DoI, AUEB

3. 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)?
A -000
D -001
F -110
C -011
H -101
G -111
B -010
E -100
Computer System Organization
GCP, DoI, AUEB

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

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

6. 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

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
GCP, DoI, AUEB

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

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: … + 2j = n
Bit 9 is checked by 1 and 8 ( 9 = )
Bit 1 checks 1, 3, 5, 7, 9, 11 in codeword
1+2 = 3, = 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
P – Parity (assume even parity)
D – Data
Computer System Organization
GCP, DoI, AUEB

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

11. Error!
Retrieved
Stored
Computer System Organization
GCP, DoI, AUEB

12. Determining Bit in Error
Computer System Organization
GCP, DoI, AUEB