1. Linear Block Code
指導教授：黃文傑 博士
學生：吳濟廷

2. OUTLINE Introduction Encoding Decoding Conclusion Generator matrix
Parity check matrix
Syndrome
Error correction
Conclusion

4. Source v.s. Channel Coding
Why source coding ?
Eliminate redundancy in the data
Send same information in fewer bits
Why channel coding ?
Combat channel effects
Coding gain

5. Code taxonomy
Today

6. Linear Block Code A (n,k) block code , where
Block : encoder accepts a block of message symbols and generates a block of codeword
Linear : addition of any two valid codeword results in another valid codeword

7. Code Rate
Code rate
Code rate increases , error correcting capability decreases
more bandwidth efficiency
Code rate decreases , error correcting capability increases
waste of bandwidth

8. Vector Space
In a binary field
addition
multiplication

9. Vector Space and Subspace
Vector space : set of all binary n-tuples ,
Vector subspace : subset S of vector space
All-zeros vector is in S
Sum of any two vectors in S is also in S
= 16 4-tupels
Subset of

10. Vector Space structure
Packing the space with as many codewords as possible
Codewords to be as far apart from one another as possible
Linear block-code structure

11. A (6,3) Linear Block Code
But ….. How to generate all these codewords ?
message vector
6-tuples

12. Generator Matrix We could use look-up table for small k
But for larger k, we use the generator matrix G
For (6,3) code
The same method for the other messages….

13. Generator Matrix where P is the parity arrayk
(n-k) k
where P is the parity array
and is the k*k identity matrix

14. Codeword using Generator Matrix
For a (n,k) code
where

15. Parity-Check Matrix Parity-check matrix H enable us to decode
For (k*n) generator matrix G, there exist an (n-k)*n matrix H
(n-k)
(n-k) k

16. Parity-Check Matrix
It’s easy to verify that for each codeword U, generated by G and the
to check if each basis still has orthogonality

17. Syndrome Testing Let r be a received vector, where is an error pattern
Syndrome of r is defined

18. Syndrome Testing For example, if transmited and received The syndrome
And the syndrome of error pattern
error

19. Error Correction
Stand array : represent possible received vectors contains all the correctable error
Coset leader =0
coset
(n,k) standard array

20. Standard Array Coset : a set of numbers having a common syndrome
Coset leader : correctable error patterns
Received vector doesn’t mean the Tx message is for sure except the error pattern is

21. Error Correction Decoding Procedure
Calculate syndrome of r :
Locate the coset leader :
Corrected codeword :

22. Example for (6,3) code
Standard array for (6,3) code

23. Example for (6,3) code Assume transmitted received Syndrome
Use look-up table
The same as transmitted codeword !!

24. Conclusion Linear block code is easy to implement
Will be extended to space-time block code
Still some other kinds of code to introduce …