Efficient Soft-Decision Decoding of Reed-

Solomon Codes

Clemson University Center for Wireless Communications

SURE 2006

Presented By:Sierra Williams

Claflin University

Outline

Background Methods Results Future Work

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

Introduction

Block Error Control Codes

Block Encoder

k- symbol block

n- symbol block

Uncoded Data Stream

Coded Data Stream

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

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)

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)

T

E2 Tt 0cos (ω0t) , s0(t)=

T

E2 Tt 0cos (ω0t) , si+1(t)= , i = 0,1,…,6

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

E

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

Methods

Results

Using the list decoding approaches maximum likelihood with fewer operations

Future Work

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

Acknowledgments

Rahul Amin Dr. John Komo Clemson University SURE Program

Questions?