1. Error Detection and Correction

2. 10.1 Introduction
Data can be corrupted
Single-bit error:
Burst error:

3. Redundancy
Extra information (redundant bits) can be used to “detect” and “correct” errors
Coding (to add redundant bits)
Encoding
Decoding

4. 10.2 Block Coding
Coding schemes can be divided into two broad categories: block coding and convolution coding
Background
Modular arithmetic (modulus N): Use only a limited number of integers [0, 1, 2, …, N-1]
E.g., in modular-9 arithmetic is 4, or “(7+6) % 9 = 4”
Exclusive OR (XOR) = modular-2 arithmetic

5. Block Coding Datawords: messages divided into k-bits blocks
Codewords: k-bit datawords + r-bits redundancy
We have 2n codewords, but use only 2k datawords
There are 2n – 2k invalid codewords (i.e., unused codewords)

6. Example of Error Detection
Datawords and codewords
Example: sender encodes dataword 01 as 011
If receiver receives 011, decodes it to 01
If receiver receives 111 (i.e., there is a single bit error), which is not a valid codeword, detects the error
If receiver receivers 000 (i.e., there are two bits error), decodes it to 00  undetectable errors

7. Example of Error Correction
Datawords and codewords
Example: sender encodes dataword 01 as 01011
Receiver received 01001
Finds the codeword that is closest to 01001
01011 has a single-bit difference
00000  2-bit difference, …,  3-bit difference
Decodes it to 01

8. Hamming Distance
Hamming distance between two words is the number of differences between corresponding bits
Minimum hamming distance (dmin) is the “smallest” Hamming distance between all possible pairs in a set of words
111
100
000

9. Hamming Distance for Detection
To guarantee detection of up to “s” errors, we should have dmin > s

10. Hamming Distance for Correction
To guarantee correction of up to “t” errors, we should have dmin > 2t

11. Example Minimum Hamming distance? How many bit errors can be detected?
How many bit errors can be corrected?

12. 10.3 Linear Block Codes Precise definition is out of scope
Informally, in a linear block code, XOR of any two valid codewords creates another valid codeword
Previous two sets of codewords are linear block codes
Minimum Hamming distance of linear block codes is the minimum number of 1’s in a non-zero codeword

13. Simple Parity-Check Code
A k-bit dataword is encoded into (k+1)-bit codeword
The extra bit is chosen to make the number of 1’s in the codeword “even”
Simple parity-check code is a single-bit error-detecting code, with dmin = 2
The last bit (0 or 1) is added

14. Implementation of Simple Parity-Check Code
s0 = b3 + b2 + b1 + b0 + q0 (modulo 2)
If s0 is 1, discard the received codeword

15. Two-Dimensional Parity Check
Example: four 7-bit datawords become five 8-bit codewords
Can detect up to 3 errors
Cannot detect 4 errors

16. Each bit (e.g., a0) is covered by two parity bits (e.g., r0 and r2)
Hamming Codes
We consider hamming codes with dmin = 3
Detect two errors, or correct one single error
Choose a number m ≥ 3, then
Codeword length n = 2m – 1
Dataword length k = n – m
Number of check bits r = m
Example: m = 3, n = 7 (=23–1), k = 4 (=7–3)
Dataword “a3a2a1a0”  Codeword “a3a2a1a0 + r2r1r0”
r0 = a2 + a1+ a0 (modulo-2)
r1 = a3 + a2+ a1 (modulo-2)
r2 = a1 + a0+ a3 (modulo-2)
Each bit (e.g., a0) is covered by two parity bits (e.g., r0 and r2)

17. Hamming Codes

18. Hamming Codes Calculation of syndrome
r0 = a2 + a1+ a0 (modulo-2)
r1 = a3 + a2+ a1 (modulo-2)
r2 = a1 + a0+ a3 (modulo-2)
Calculation of syndrome
s0 = b2 + b1+ b0 + q0 (modulo-2)
s1 = b3 + b2+ b1 + q1 (modulo-2)
s2 = b1 + b0+ b3 + q2 (modulo-2)
Decision to correct a single-bit error

19. Hamming Codes
Example
Received codeword and syndrome is 000, what is the dataword?
Received codeword and syndrome is 011, what is the dataword?

20. Hamming Codes
Detect a burst error

21. 10.4 Cyclic Codes Linear block codes with the following property shift
If a codeword is cyclically shifted, the result is another codeword
Our discussion is limited to a typical example, called Cyclic Redundancy Check (CRC)
shift

22. Implementation
Case of CRC with 4-bit dataword, 7-bit codeword

23. CRC Encoder
Result of XOR

24. The same divider as encoder
CRC Decoder
Non-corrupted bits
Corrupted bits
The same divider as encoder

25. It takes n-k times to get the result
Hardware Design
CRC encoder and decoder can be easily implemented using shift registers
Encoder example: divisor = 1011
It takes n-k times to get the result

26. CRC Encoder using Shift Registers
Divisor = 1011

27. Polynomials
A pattern can be considered as coefficients of a polynomial
Adding and subtracting are done between the coefficients of the same power
Ex:

28. Division using Polynomials
Encoder
= 1001
1011 =

29. Cyclic Code Analysis
Dataword d(x), codeword c(x), divisor g(x), syndrome s(x), error e(x)
At receiver, received codeword = c(x) + e(x)
After decoding, we have
An error cannot be detected if e(x) is divisible by g(x)

30. Design of Divisor Single-bit error: e(x) = xa for some a
Divisor has more than one term and the least coefficient (i.e., of x0) is 1
Two-bit error: e(x) = xa + xb for some a, b
Divisor should not divide xt+1
Odd numbers of errors
Divisor should have the factor (x+1)
Burst errors: e(x) = xa (1+ … + xb ) for some a, b
All burst errors of b-a+1 <= r will be detected
All other errors will be detected with probability 1-(1/2)r-1-(1/2)r

31. 10.5 Checksum Often used in the Internet protocols Idea
For data (7, 11, 12, 0, 6),
send (7, 11, 12, 0, 6, -36), where 36 =
Receiver checks error by adding all elements
One’s complement
For the case, when the addition results in a large number more than n bits

32. Checksum Procedure One’s complement for 4-bit binary
9 = – 6 in one’s complement representation