Reed-Solomon CodesProbability of Not Decoding a

Symbol Correctly

By: G. GrizzardNorth Carolina State University

Advising Professor: Dr. J. KomoClemson University

2002 SURE Program

Definitions• Reed Solomon Code – error

correcting code developed from qm digits of the general form (qm-1,k) where qm-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 (qm-1,k) code dmin=(qm-1)–k+1

Definitions (Cont)• Weight- total number of non-zero

digits in a word• Aj – the total number of weight-j words• Bj – 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

Definitions (Cont)• Extended Code- RS codes can be

singly extended RS(qm,k) or doubly extended RS (qm+1,k)

• Errors Only – RS code which only corrects errors

• Errors and Erasures – RS code which corrects both errors and erasures

Purpose• Find the probability of a symbol

not being decoded correctly Ps(E) +Ps(F)

• Show Ps(E) is a lower bound for the probability of not decoding a symbol correctly

Types of RS Codes• Errors Only(EO) – RS(qm-1,k)• Extended Errors Only (EO) –

RS(qm,k)• Errors and Erasures(E&E) – RS(qm-

1,k)• Extended Errors and Erasures

(E&E) – RS(qm,k)

Approximations• Ps(E)

– One Summation– Two Summation– Term Approximation

• Ps(E)+Ps(F)– j inside– dmin outside

RS Code - EO• RS(qm-1,k) is cyclic so

• Thus Ps(E) can be expressed asjmj A

1qjkB

n

dj 0k

jkjs

min

21mind

PjAn1 (E)P

Ps(E) for RS(31,21) EO

% Error – RS(31,21) EO

Extended RS Code – EO• RS(qm,k) is not cyclic however, able

to justify using

• Thus expression for Ps(E) is:jmj A

qjkB

m

min

21mind

q

dj 0k

jkjms PjA

q1(E)P

Ps(E) for RS(32,22) EO

% Error – RS(32,22) EO

RS(qm-1,k) & RS(qm,k) – E&E

• Formula for calculating Ps(E) is much more complicated (5 Summations)

• Apply for RS(qm-1,k) • Apply for RS(qm,k)

jmj A1q

jkB

jmj AqjkB

RS(31,21) – E&E

Finding Ps(F) – RS(qm-1,k) EO•Count Number of weight-j words in the decoding spheres

Ps(E)+Ps(F) - RS(31,21) - EO

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

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

Questions??