1. Reed-Solomon Codes Probability of Not Decoding a Symbol Correctly By: G. Grizzard North Carolina State University Advising Professor: Dr. J. Komo Clemson University 2002 SURE Program

2. Definitions Reed Solomon Code – error correcting code developed from q m digits of the general form (q m -1,k) where q m -1 is the length of the code and k is the number of message digits Maximum Distance Separable (MDS) – requires the maximum possible minimum distance between code words for any (q m -1,k) code d min =(q m -1)–k+1

3. Definitions (Cont) Weight- total number of non-zero digits in a word A j – the total number of weight-j words B j – the total weight of the message blocks associated with all words of weight-j [1] [1] Wicker Stephen, Error Control Systems for Digital Communications, Printice Hall, New Jersey, 1995, pp242

4. Definitions (Cont) Extended Code- RS codes can be singly extended RS(q m,k) or doubly extended RS (q m +1,k) Errors Only – RS code which only corrects errors Errors and Erasures – RS code which corrects both errors and erasures

5. Purpose Find the probability of a symbol not being decoded correctly P s (E) +P s (F) Show P s (E) is a lower bound for the probability of not decoding a symbol correctly

6. Types of RS Codes Errors Only(EO) – RS(q m -1,k) Extended Errors Only (EO) – RS(q m,k) Errors and Erasures(E&E) – RS(q m -1,k) Extended Errors and Erasures (E&E) – RS(q m,k)

7. Approximations Ps(E) –One Summation –Two Summation –Term Approximation Ps(E)+Ps(F) –j inside –d min outside

8. RS Code - EO RS(q m -1,k) is cyclic so Thus P s (E) can be expressed as

9. P s (E) for RS(31,21) EO

10. % Error – RS(31,21) EO

11. Extended RS Code – EO RS(q m,k) is not cyclic however, able to justify using Thus expression for P s (E) is:

12. P s (E) for RS(32,22) EO

13. % Error – RS(32,22) EO

14. RS(q m -1,k) & RS(q m,k) – E&E Formula for calculating P s (E) is much more complicated (5 Summations) Apply for RS(q m -1,k) Apply for RS(q m,k)

15. RS(31,21) – E&E

16. Finding P s (F) – RS( q m -1,k) EO Count Number of weight-j words in the decoding spheres

17. P s (E)+P s (F) - RS(31,21) - EO

18. Conclusion The exact probability of not decoding a symbol correctly can be found by calculating the probability of a symbol failure and adding that to the probability symbol error of error The probability of symbol error provides a lower bound for the probability of not decoding correctly

19. Future Work Develop an exact expression for the probability of symbol failure for a code that considers both errors and erasures Ideally work will be done to investigate methods to force a decision even when the decoder fails so the probability of not decoding a symbol correctly will approach the probability of symbol error

20. Questions??