1. Error detection and correction
Techniques to Increase the Reliability
Copyright © Curt Hill

2. The Story Starts with Parity
The original error detection scheme
Originally used on Teletypes in 1930s
Transmitting data over telephone/telegraph lines
Add one bit to check if rest of data was correct
Next used for tape drives in 1950s
Nine track tapes had eight data and one parity
Before that were seven and eight track tapes
Copyright © Curt Hill

3. How Does Parity Work? Parity was set to even or odd
The data bits were summed
Using even parity the parity bit was set to make the sum even
If data is: set the parity bit to 0
It is already even
If data is: set the parity bit to 1
To make it even
Copyright © Curt Hill

4. Example Consider the following data: 01101100 01110110
Even parity:
Odd parity:
Even parity:
Odd parity:
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5. Options How many bits were being transmitted in a character?
How fast were they being transmitted?
What parity was being used?
How many start and stop bits?
The answers to all these questions constitute a Protocol
The protocol did not matter as both the sender and receiver agreed on the same one
Copyright © Curt Hill

6. Parity checking It is easy to check and generate
If there is a one bit error it will detect it but give you no clue where it is
An even number of bits error is undetectable, but odd number of bits error is detectable
The number of data bits is irrelevant
A single parity bit can be put on 8 bits or 50
However, as the number of bits gets larger the protection gets smaller
Copyright © Curt Hill

7. Semiconductor Memory
Many machines also used parity to check their memory
The original IBM PC and many successors used 9 bits to store an 8 bit byte
If a parity error was detected the machine halted with a parity error
Even that generation was not as reliable as hoped
Copyright © Curt Hill

8. Is this the best we can do?
If we employ multiple error detection bits we can detect multiple bit errors and correct single bit errors
How do we correct an error?
Since each bit may only have two states 0 or 1 all we have to know is which bit is bad
Correct it by reversing it
How do we find location of the error?
Copyright © Curt Hill

9. Basic Scheme Four data bits (0-3) Three syndrome bits (a-c)
Each parity bit protects just three of the four
A protects 0-2 with parity
B protects 1-3 with parity
C protects 0,1,3 with parity
Each bit is protected by two or three of the parity bits
The number of parity bits indicate which bit was in error
Copyright © Curt Hill

10. Four data and three parityb 1 2 3
Error Parity
a, b
a, b, c
a, c
b, c
a a
b b
c c 4 c
Copyright © Curt Hill

11. More
Single error bit correction, by flipping the bit that is indicated
It takes one more bit to detect two bit errors
Otherwise it will be mistaken for a one bit error and corrected
The three bits are collectively called the syndrome
This scheme uses three error bits for four data bits and is somewhat wasteful
Copyright © Curt Hill

12. Who is responsible?
Richard Hamming developed the first error correction codes
Known as Hamming codes
Worked at Bell Labs
Was concerned about transmitting digital data at high speeds
Telephone calls were often digitized by this time
Parity is sufficient for telegraph which is relative low speed, but not telephone
Copyright © Curt Hill

13. Error Correction
In order to get error correction we need: 2K-1>M+K where M is the number of data bits and K is the number of check (error correction) bits
For four data bits we need four syndrome bits to get error correction 24-1 = > 4 + 4
Copyright © Curt Hill

14. Observations Notice the formula 2K-1>M+K
Exponentials grow rather faster than sums
Thus adding one to the number of syndrome bits doubles the number of protected bits
Large word sizes are proportionally easier to protect than small
Copyright © Curt Hill

15. Summary The number of bits needed is summarized by this table:
Data bits
SEC
% diff
SEC/DED 8 4
50% 5
62.5% 16
31.3% 6
37.5% 32
18.8% 7
18.75% 64
10.9%
12.5%
Copyright © Curt Hill

16. Layout Lets try eight data bits and four bits of error correction
The four bits can generate 16 values 0-15
We want value zero to represent no error
If the four bit value contains a single one bit that will indicate that the error is in the check bits and thus no correction is needed
If the four bit value contains more than one bit then we want this number to tell us the bit that is off
Copyright © Curt Hill

17. Layout Picture
Each check bit guards the bits that have that bit as a position
The C8 checks every bit which has a 1 in the 8s bit M8 M7 M6 M5 C8 M4 M3 M2 C4 M1 C2 C1
Copyright © Curt Hill

18. Computing Check Bits
Each check bit is the even parity of four or five bits
They are calculated as follows
The computed syndrome bits are compared with the received syndrome and should give the bit number
Copyright © Curt Hill

19. An Example Consider the data bits: 0 1 0 0 1 1 1 0
Locate c c 0 c c
Even parity for C8 bit – 1
c 0 c c
Even parity for C4 bit – 1
c c
Even parity for C2 bit – 1 c
Even parity for C1 bit – 1
Copyright © Curt Hill

20. Suppose An Error The original word now come back as:
Compute the new check bits
C8 = 0
C4 = 1
C2 = 1
C1 = 0
C8 and C1 disagree
Sum them, the error is in position 9, which is data bit M5
Copyright © Curt Hill

21. Some examples of use
IBM 30xx use an 8bit SEC-DED for each 64 bits, hence they have 12% overhead
DEC VAX uses a 7 bit SEC-DED for each 32 bits, hence they have a 22% overhead
Some versions of RAID also use this to guarantee recoverability in the case of disk errors
Copyright © Curt Hill

22. Addendum
Recently these concepts have been applied to internet/wireless data packets
When a mobile phone drops a packet it must ask for it again
This greatly reduces perceived bandwidth
A study in Boston showed that 3% of packets for mobile phones were lost
Packet loss on a fast moving train is typically 5%
Copyright © Curt Hill

23. Packets
The solution is to put error correction codes for nearby packets in adjacent packets
Known as coded TCP
In a study doing this with 2% packet loss boosted apparent bandwidth from 1 to 16 Mbit
If loss rates are low this does not help but losses in wireless networks are always present
Copyright © Curt Hill