1. II. Linear Block Codes

2. Last Lecture What are Linear Block Codes?
Linear Systematic Block Codes
Generator Matrix and Encoding Process
Parity Check Matrix, Syndrome & Error Detection Process
Encoding Circuit & Syndrome Circuit

3. Hamming Weight and Hamming Distance
Hamming Weight w(v):
The number of nonzero components of v
Hamming Distance d(v,w):
The number of places where v and w differ
Example:
v= ( ): w(v)=5
w=( ): d(v,w)=3
Important Remark: In GF(2)
d(v,w)=w(v+w)
In the example above v+w=( )

4. Minimum Distance of a Block Code
Minimum Distance of a block code (dmin):
The minimum Hamming distance between any two code words in the code book of the block code
For Linear Block Codes

5. Theorem
Let C be an (n,k) linear block code with parity check matrix H. For each code word of Hamming weight l, there exists l columns of H such that the vector sum of these l columns is equal to the zero vector.
PROOF:

6. Theorem (Cont’d)
Conversely, if there exists l columns of H whose vector sum is the zero vector, there exists a codeword of Hamming weight l in C.
PROOF:

7. Corollaries
Given C is a linear block code with parity check matrix H. If no d-1 or fewer columns of H add to 0, the code has minimum weight of at least d.
Given C is a linear block code with parity check matrix H. The minimum distance of C is equal to the smallest number of columns of H that sum to 0.

8. Example (7,4) Linear Block Code
No all 0 column. No two columns are identical:
dmin>2
Columns 0, 1, 3 sum to zero vector
dmin=3

9. Error Detection Error Detection Capability means:
HOW MANY ERRORS ARE GUARANTEED TO BE DETECTED AT THE RECEIVER SIDE
TWO Possibilities:
Number of errors (no. of 1s in e) is smaller than dmin
Number of errors (no. of 1s in e) is greater than dmin v r
Channel e
You could be absolutely sure that r is not a valid codeword.
There is a chance that r is a valid codeword (if e is a valid code word)
v: transmitted codeword
e: error pattern caused by channel
r: received pattern
If error pattern includes l errors: d(v,r)=l
THE ERROR IS GUARANTEED TO BE DETECTED
THE ERROR PATTERN MIGHT NOT BE DETECTED

10. Error Detection Capability
The “Error Detection Capability” of a code defines the number of errors that are GUARANTEED to be detected.
For a code with minimum Hamming distance dmin:
The error detection capability is dmin-1

11. Number of Detectable Error Patterns
An error pattern is defined as any nonzero n-tuple that could affect a transmitted code word.
For an (n,k) code:
There are 2n-1 error patterns. Note that ( ) is not an error pattern
There are 2k-1 undetectable error patterns. Note that
( ) is selected to be a valid codeword
An (n,k) code has the ability to detect 2n-2k error patterns
Example: A (7,4) code is able to detect:
128-16=112 error patterns

12. Error Correction Capability
The “error correction capability” defines the number of errors that are GUARANTEED to be corrected.
For a code with minimum Hamming distance dmin:
The error correction capability is (dmin-1)/2
x means floor(x)

13. Error Correction Capability: Proof
Assume a Block Code with dmin
Define a Positive Integer t such that: v r
Channel e
v: transmitted codeword
e: error pattern caused by channel
r: received pattern
Assume w is also a valid codeword in C
Triangle Inequality:
Assume an error pattern with t’ errors
Given that v and w are codewords
i.e., if an error pattern of t or fewer errors occur, the received vector r MUST BE closer to v than to any other codeword in C
If t’≤t

14. Error Correction & Error Detection Capability
Some codes are designed such that they can GUARANTEE correction of λ or fewer errors AND detection of up to l>λ
For a code with minimum Hamming distance dmin:
If the code could correct λ and detect up to l then
dmin≥ λ+l+1
Example:
If it is required for a code to correct 3 errors AND detect up to 6 errors then dmin must satisfy: dmin ≥10
Notes: In the example above,
If the number of errors are 3 or less: you can provide a GUARNTEE to correct all of them.
If the number of errors are from 4 to 6: you can provide a GUARNTEE to detect that the number of errors are between 4 to 6 without being able to correct ANY of them.
If the number of errors are greater than 6: there is NO GUARANTEE that you would be able to detect or correct the errors

15. Quantization and Analogy to Error Correction and Detection Operation
Correct Reception:
The value received is identical to what has been transmitted
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3
v*0=0
v*1=1
v*2=2
v*3=3
Channel

16. Quantization and Analogy to Error Correction and Detection Operation
Error Detection:
The value received IS NOT EQUAL to any of the valid representation levels
ARQ: Request retransmission
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3
v*0=0
v*1=1
v*2=2
v*3=3
Channel

17. Quantization and Analogy to Error Correction and Detection Operation
Error Detection:
The value received IS NOT EQUAL to any of the valid representation levels
ARQ: Request retransmission
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3
v*0=0
v*1=1
v*2=2
v*3=3
Channel

18. Quantization and Analogy to Error Correction and Detection Operation
Un-Detected Errors:
The value received IS a valid representation level. However, it is NOT what has been transmitted
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3
v*0=0
v*1=1
v*2=2
v*3=3
Channel

19. Quantization and Analogy to Error Correction and Detection Operation
Approximate the received value to the closest valid representation level.
FEC: The receiver defines decision zones and maps ANY received value to a valid representation level. ARQ cannot be applied
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3 D0 D1 D2 D3
v*0=0
v*1=1
v*2=2
v*3=3
Channel

20. Quantization and Analogy to Error Correction and Detection Operation
Approximate the received value to the closest valid representation level.
FEC: The receiver defines decision zones and maps ANY received value to a valid representation level. ARQ cannot be applied
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3
v*0=0
v*1=1
v*2=2
v*3=3 D0 D1 D2 D3
Channel

21. Quantization and Analogy to Error Correction and Detection Operation
False Correction:
The error pushes the received value outside the decision zone of the representation level that has been transmitted.
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3
v*0=0
v*1=1
v*2=2
v*3=3 D0 D1 D2 D3
Channel

22. Quantization and Analogy to Error Correction and Detection Operation
False Correction:
The error pushes the received value outside the decision zone of the representation level that has been transmitted.
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3
v*0=0
v*1=1
v*2=2
v*3=3 D0 D1 D2 D3
Channel

23. Quantization and Analogy to Error Correction and Detection Operation
Error Correction and Error Detection:
Defines two types of decision zone such that both FEC and ARQ could be used dependent on the received value
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3 D0 D1 D2 D3
v*0=0
v*1=1
v*2=2
v*3=3
Channel

24. Quantization and Analogy to Error Correction and Detection Operation
Error Correction and Error Detection:
Defines two types of decision zone such that both FEC and ARQ could be used dependent on the received value
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3
v*0=0
v*1=1
v*2=2
v*3=3 D0 D1 D2 D3
Channel
Error Correction

25. Quantization and Analogy to Error Correction and Detection Operation
Error Correction and Error Detection:
Defines two types of decision zone such that both FEC and ARQ could be used dependent on the received value
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3
v*0=0
v*1=1
v*2=2
v*3=3 D0 D1 D2 D3
Channel

26. Quantization and Analogy to Error Correction and Detection Operation
Error Correction and Error Detection:
Defines two types of decision zone such that both FEC and ARQ could be used dependent on the received value
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3
v*0=0
v*1=1
v*2=2
v*3=3 D0 D1 D2 D3
Channel
False Error Correction

27. Quantization and Analogy to Error Correction and Detection Operation
Error Correction and Error Detection:
Defines two types of decision zone such that both FEC and ARQ could be used dependent on the received value
Transmitter Side
Receiver Side
v0=0
v1=1
v2=2
v3=3
v*0=0
v*1=1
v*2=2
v*3=3 D0 D1 D2 D3
Channel
Error Detection and ARQ could be used