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## Decoding Reed-Solomon Codes using the Guruswami-Sudan Algorithm

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### Title: Decoding Reed-Solomon Codes using the Guruswami-Sudan Algorithm Author: a3702361 Last modified by: a3702361 Created Date: 1/20/2006 7:22:25 PM – PowerPoint PPT presentation

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Title: Decoding Reed-Solomon Codes using the Guruswami-Sudan Algorithm

1
Decoding Reed-Solomon Codes using the
Guruswami-Sudan Algorithm
• PGC 2006, EECE, NCL
• Student Li Chen
• Supervisor Prof. R. Carrasco, Dr. E. Chester

2
Introduction
• List Decoding
• Guruswami-Sudan Algorithm
• Interpolation (Kotters Algorithm)
• Factorisation (Ruth-Ruckenstein Algorithm)
• Simulation Performance
• Complexity Analysis
• Algebraic-Geometric Extension
• Conclusion

3
• DecoderSearch the lost boy named John
• Unique decoderPolice without cooperation
• List decoderPolice with cooperation

Police
Decoder
from
now
4
List Decoding
• Introduced by P. Elias and J. Wozencraft
independently in 1950s
• Idea
• Unique decoder can correct r1,
• but not r2?
• List decoder can correct
• r1 and r2?

5
Reed-Solomon Codes
• Encoding ?k ?n (kltn)
• (C0, C1, , Cn-1)(f(x0), f(x1), , f(xn-1))
• transmitted message
• f(x)f0x0f1x1fk-1xk-1
• k dimensional monomial basis of curve y0
• Application
• Storage device
• Mobile communications

6
Guruswami-Sudan Algorithm
7
GS Overview
• Decode RS(5, 2)
• Encoding elemnts x(x0, x1, x2, x3, x4)
• Received word y(y0, y1, y2, y3, y4)

Build Q(x, y) that goes through 5 points Q(x,
y)y2-x2 y-(-x) y-p(x)?f(x) y-x
Q(x, y) has a zero of multiplicity m1 over the 5
points.
GS Interpolation Factorisation
The Decoded codeword is produced by re-evaluate
p(x) over x0, x1, x2, x3, x4!!!
8
How about increase the degree of Q(x, y)?
• Q2(y2-x2)2 y-(-x)
• y-x
• y-p(x)?f(x)
• y-(-x)
• y-x

Q2(x, y) has a zero of multiplicity m2 over the
5 points.
The higher degree of Q(x, y) more candidate
to be chosen as f(x) diverser point can be
included in Q(x, y) better error correction
capability!!!
9
GS Decoding Property
• Error correction upper bound (1)

Multiplicity m Error correction tm Output list lm
• Examples
• RS(63, 15) with r0.24, e24 RS(63,
31) with r0.49, e16

10
Interpolation---Build Q(x, y)
• Multiplicity definition (2)
• ---qab0 for abltm, Q has a zero of multiplicity
m at (0, 0).
• Define over a certain point (xi, yi)
• ---quv0 for uvltm, Q has a zero of multiplicity
m at (xi,yi)
• quv is the Qs (u, v) Hasse derivative
evaluation on (xi, yi)
• (3)

11
Cont
• Therefore, we have to construct a Q(x, y) that
satisfies
• Q(x, y)minQ(x, y)?Fqx, yDuvQ(xi, yi)0
• for i0, , n-1 and
uvltm
• Q has a zero of multiplicity m over the n points

12
Kotters Algorithm
• Initialisation G0g0, g1, , gj, ,

Hasse Derivative Evaluation
If in, end! Else, update i, and (u, v)
Find the minimal polynomial in J
Bilinear Hasse Derivative modification For
(j?J), if jj, if j?j,
13
Factorisation---Find p(x)
• p(x) satisfy
• y-p(x)Q(x, y) and deg(p(x))ltk
• p(x)p0p1xpk-1xk-1
• ---we can deduce coefficients p0, p1, , pk-1
sequentially!!!

14
Ruth-Ruckenstein Algorithm
Q0(x, y)
Q1(x, y) Q2(x, y)
p(x)
p(x)
Qs sequential transformation (4) pi are the
roots of Qi(0, y)0.
15
Simulation Results 1----RS(63, 15)
AWGN
Coding gain 0.4-1.3dB 1-2.8dB
16
Simulation Result 2----RS(63, 31)
AWGN
Coding gain 0.2-0.8dB 0.5-1.4dB
17
Complexity Analysis
RS(63, 15) RS(63, 31)
Reason Iterative Interpolation
18
Little Supplements----GSs AG extension
RS f(x) Q(x, y) p(x) AG f(x, y) Q(x, y,
z) p(x, y)
19
Conclusion of GS algorithm
• Correct errors beyond the (d-1)/2 boundary
• Outperform the unique decoding algorithm
• Greater potential for low rate codes
• Used for decode AG codes
• Higher decoding complexity----Need to be