1. Efficient Soft-Decision Decoding of Reed- Solomon Codes Clemson University Center for Wireless Communications SURE 2006 Presented By: Sierra Williams Claflin University

2. Outline Background Methods Results Future Work

3. Introduction Applications of Reed-Solomon codes Storage Devices Wireless or Mobile Communications Digital Television High Speed Modems Reason for Research Minimize the number of errors

4. Introduction Block Error Control Codes Block Encoder k- symbol block n- symbol block Uncoded Data Stream Coded Data Stream

5. Introduction An (n,k,d) q Reed-Solomon code n is # of symbols in block k is the message symbols d is the minimum distance q is # of elements in Galois field Corrects t = (n-k)/2 errors or s= n-k erasures

6. Introduction Example An (8,4,5) 8 Reed-Solomon code GF(8)= {0, 1, α, α 2, α 3, α 4,α 5,α 6 } t = 2 (Correct double errors) s =4 (Correct 4 erasures)

7. Introduction Coherent Multiple Frequency Shift Keying (MFSK) Transmission Map elements of GF(8) to 8 different frequencies Therefore, r(t) = s(t) +n(t), where n(t) is AWGN (Additive White Gaussian noise) cos (ω 0 t), s 0 (t)= cos (ω 0 t), s i+1 (t)=, i = 0,1,…,6

8. Introduction Correlation receiver for coherent MFSK Yields 8 soft-decision outputs for each transmitted frequency e.g. If s 0 t transmitted the correlation outputs would be r 0 = + n 0 and r i = n i, i = 1,2,…,7 where n i is a Gaussian random number

9. Methods The C++ Program Generates 8 sets of 8 random numbers Value of signal added to first element as noise Sort each array Hard-decision error Finding beta and receiver array elements Determine codeword

10. Methods

11. Results Using the list decoding approaches maximum likelihood with fewer operations

12. Future Work Using not only the least likely to list decode but 2 nd least likely and so on.

13. Acknowledgments Rahul Amin Dr. John Komo Clemson University SURE Program

14. Questions?