1. Error control coding – binary linear codes Background material for linear error control codes

2. Developing linear codes C(5,2)

3. Message vectors

4. The error group for syndrome 001

5. Checking the error group property

6. Selecting the group leader (P_b=0.01) The group leader isit occurs with the largest probability

7. 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 ! Syndrome vectorMaximum likely error vector (the group leader) 00000000 00100001 01000010 01100011 10000100 10100101 11010000 11101000 For example:

8. Constructing the syndrome decoding table 3. Carry out the multiplications 4. Group the results with respect to s (collect all e vectors 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 1. List the numbers in decimal from 2. Convert this decimal numbers to n bit binary numbers (the possible error vectors )

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

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

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. Constructing the groups and assigning the group leaders

15. The syndrome decoding table Syndrome vectorGroup leader error vector 00000000 00100001 01000010 01100011 10000100 10100101 11010000 11101000

16. Another way of constructing the error groups Ifandthen 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.

17. Example Pick

18. Example (cont’) Pick

19. Example Pick

20. Example Pick

21. Example Pick

22. Example Pick

23. Example Pick

24. The syndrome decoding table Syndrome vectorGroup leader error vector 00000000 00100001 01000010 01100011 10000100 10100101 11010000 11101000

25. The coding scheme se 00000000 00100001 01000010 01100011 10000100 10100101 11010000 11101000 Trunc 01111 01 00100 01011 100 01111 01 00100

26. The standard array Syndrome vector 0123 00000000011111011011001 00100001011101011111000 01000100011011010011011 01100011011001010111010 10000100010111001011101 10100101010101001111100 11000110010011000011111 11100111010001000111110