1. Data Link Layer: Error Detection and Correction
: Data Communication and Computer Networks
Asst. Prof. Chaiporn Jaikaeo, Ph.D.
Computer Engineering Department
Kasetsart University, Bangkok, Thailand
Adapted from lecture slides by Behrouz A. Forouzan
© The McGraw-Hill Companies, Inc. All rights reserved

2. Outline Overview of Data Link Layer Types of errors Redundancy
Correction vs. detection
Coding

3. Data Link Layer

4. Error Control Detecting errors Correcting errors
Forward error correction
Automatic repeat request

5. Types of Errors
Single-bit errors
Burst errors

6. Redundancy
To detect or correct errors, redundant bits of data must be added

7. Coding Process of adding redundancy for error detection or correction
Two types:
Block codes
Divides the data to be sent into a set of blocks
Extra information attached to each block
Memoryless
Convolutional codes
Treats data as a series of bits, and computes a code over a continuous series
The code computed for a set of bits depends on the current and previous input

8. XOR Operation
Main operation for computing error detection/correction codes
Similar to modulo-2 addition

9. Block Coding Message is divided into k-bit blocks
Known as datawords
r redundant bits are added
Blocks become n=k+r bits
Known as codewords

10. Example: 4B/5B Block Coding
Data
Code
0000
11110
1000
10010
0001
01001
1001
10011
0010
10100
1010
10110
0011
10101
1011
10111
0100
01010
1100
11010
0101
01011
1101
11011
0110
01110
1110
11100
0111
01111
1111
11101
k = ?
r = ?
n = ?

11. Error Detection in Block Coding

12. Notes
An error-detecting code can detect only the types of errors for which it is designed
Other types of errors may remain undetected.
There is no way to detect every possible error

13. Error Correction

14. Example: Error Correction Code
k, r, n = ?
The receiver receives 01001, what is the original dataword?

15. Hamming Distance d(01, 00) = ? d(11, 00) = ? d(010, 100) = ?
Hamming Distance between two words is the number of differences between corresponding bits.
d(01, 00) = ?
d(11, 00) = ?
d(010, 100) = ?
d(0011, 1000) = ?
How many 8-bit words are n bits away from ?

16. Minimum Hamming Distance
The minimum Hamming distance is the smallest Hamming distance between all possible pairs in a set of words.
Find the minimum Hamming Distance of the following codebook
00000
01011
10101
11110

17. Detection Capability of Code
To guarantee the detection of up to s-bit errors, the minimum Hamming distance in a block code must be
dmin = s + 1

18. Correction Capability of Code
To guarantee the correction of up to t-bit errors, the minimum Hamming distance in a block code must be
dmin = 2t + 1

19. Example: Hamming Distance
A code scheme has a Hamming distance dmin = 4. What is the error detection and correction capability of this scheme?

20. Common Detection Methods
Parity check
Cyclic Redundancy Check
Checksum

21. Parity Check Most common, least complex Single bit is added to a block
Two schemes:
Even parity – Maintain even number of 1s
E.g., 1011  10111
Odd parity – Maintain odd number of 1s
E.g., 1011  10110

22. Example: Parity Check
Suppose the sender wants to send the word world. In ASCII the five characters are coded (with even parity) as
The following shows the actual bits sent

23. Example: Parity Check Receiver receives this sequence of words:
Which blocks are accepted? Which are rejected?

24. Parity-Check: Encoding/Decoding

25. Performance of Parity Check
Can 1-bit errors be detected?
Can 2-bit errors be detected? :

26. 2D Parity Check
What is its performance?

27. 2D Parity Check: Performance

28. 2D Parity Check: Performance

29. Cyclic Redundancy Check
In a cyclic code, rotating a codeword always results in another codeword
Example:

30. CRC Encoder/Decoder

31. CRC Generator

32. Checking CRC

33. Polynomial Representation
More common representation than binary form
Easy to analyze
Divisor is commonly called generator polynomial

34. Division Using Polynomial

35. Strength of CRC M(x)xn = Q(x)G(x) + R(x)
Can be analyzed using polynomial
M(x) – Original message
G(x) – Generator polynomial of degree n
R(x) – Generated CRC
Transmitted message is
M(x)xn – R(x)
which is divisible by G(x)
M(x)xn = Q(x)G(x) + R(x)

36. Strength of CRC Received message is M(x)xn – R(x) + E(x)
where E(x) represents bit errors
Receiver does not detect any error when E(x) is divisible by G(x), which means either:
E(x) 0  No error
E(x)  0  Undetectable error

37. Strength of CRC
If G(x) contains at least two terms, then all single-bit errors can be detected
If G(x) cannot divide xt + 1 (0 t < n), then all isolated double errors can be detected
If G(x) contains a factor of (x+1), all odd-numbered errors can be detected

38. Properties of Good Polynomial
It should have at least two terms
The coefficient of the term x0 should be 1
It should not divide xt + 1, for t between 2 and n − 1
It should have the factor x + 1

39. CRC's Strength Summary All burst errors with L ≤ n will be detected
All burst errors with L = n + 1 will be detected with probability 1 – (1/2)n–1
All burst errors with L > n + 1 will be detected with probability 1 – (1/2)n

40. Example: CRC Generators
Which of the following polynomials guarantees that a single-bit error can be detected
(a) x+1
(b) x3
(c) 1

41. Example: CRC Generators
Criticize the following CRC generators x3
x10 + x9 + x5
x6+1

42. Standard Polynomials

43. Error Correction Two methods Retransmission after detecting error
Forward error correction (FEC)

44. Forward Error Correction
Consider only a single-bit error in k bits of data
k possibilities for an error
One possibility for no error
#possibilities = k + 1
Add r redundant bits to distinguish these possibilities; we need
2r  k+1
But the r bits are also transmitted along with data; hence
2r  k+r+1

45. Number of Redundant Bits
Number of data bits k
Number of redundancy bits r
Total bits
k + r 1 2 3 5 6 4 7 9 10 11

46. Hamming Code Simple, powerful FEC Widely used in computer memory
Known as ECC memory
error-correcting bits

47. Redundant Bit Calculation

48. Example: Hamming Code

49. Example: Correcting Error
Receiver receives

50. Strength of Hamming Code
Minimum Hamming Distance is 3
It can correct at most 1 bit error
It can detect at most 2 bit error
But… not both!!! (Why?)
SECDED – Extended Hamming code with one extra parity bit
Achieves minimum Hamming distance of 4
Can distinguish between one bit and two bit errors

51. Burst Error Correction