Error control coding for wireless communication technologies

1. Errorcontrolcodingforwireless communicationtechnologies Background material for linear error control codes Janos Levendovszky EU-USA Atlantis Programme FIT & Budapest University of Technology and Economics

2. Challenge Howtoachievereliable communication over an unreliablechannel ? Byerrorcontrolcoding.

3. Developing linear codes C(5,2)

4. Message vectors

5. The error group for syndrome 001

6. Checking the error group property

7. Selecting the group leader The group leader is it occurs with the largest probability

8. The syndrome detection table Based on the syndrome vector we identify the correpsonding most likely error vector and we store these pairs in an LUT ! For example:

9. Constructing the syndrome decoding table 1. List the numbers in decimal from 2. Convertthisdecimalnumberston bit binarynumbers (thepossibleerrorvectors) 3. Carry out the multiplications 4. Group the results with respect to s (collect all evectors into the same group if they belong to the same s) 5. Determine the minimum weight e in each group 6. Construct an LUT by entering the “s and the corresponding minimum weight e” pairs

10. E.g.: constructing the syndrome decoding table of a C(5,2) code List of possible error vectors

11. E.g.: constructing the syndrome decoding table of a C(5,2) code Multiplication with the generator matrix

12. E.g.: constructing the syndrome decoding table of a C(5,2) code Multiplication with the generator matrix

13. E.g.: constructing the syndrome decoding table of a C(5,2) code Multiplication with the generator matrix

14. E.g.: constructing the syndrome decoding table of a C(5,2) code Multiplication with the generator matrix

15. Constructing the groups and assigning the group leaders

16. The syndrome decoding table

17. Another way of constructing the error groups If and then 1. Pick an error vector e 2. Calculate the corresponding syndrome vector 3. Construct the error group as follows 4. Pick another error vector e” for which and go back to Step 1.

18. Example Pick

19. Example (cont’) Pick

20. Example Pick

21. Example Pick

22. Example Pick

23. Example Pick

24. Example Pick

25. The syndrome decoding table

26. The coding scheme 00100 100 01011 01111 01111 01 00100 01 Trunc

27. The standard array

28. Suggested readings D. Costello: Error control codes, Wiley, 2005, Chapter 3

29. Expected Quiz question Given a generator matrix of a linear binary systematic code, determine the error group belonging to a given error vector

30. Thank you for your attention !