Efficient Soft-Decision Decoding of Reed- Solomon Codes

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 Uncoded Data Stream k- symbol block Block Encoder n- symbol block 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, α2, α3, α4 ,α5 ,α6} • t = 2 (Correct double errors) • s =4 (Correct 4 erasures)

7. s0(t)= cos (ω0t) , cos (ω0t) , si+1(t)= , i = 0,1,…,6 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)

8. Introduction • Correlation receiver for coherent MFSK • Yields 8 soft-decision outputs for each transmitted frequency • e.g. If s0t transmitted the correlation outputs would be r0 = + n0 and ri = ni , i = 1,2,…,7 where ni 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 2nd least likely and so on.

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

14. Questions?