Decoding Reed-Solomon Codes using the Guruswami-Sudan Algorithm

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Decoding Reed-Solomon Codes using the
Guruswami-Sudan Algorithm

• PGC 2006, EECE, NCL
• Student Li Chen
• Supervisor Prof. R. Carrasco, Dr. E. Chester

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Introduction

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

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Funny Talk about List Decoder

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

Police
Decoder
from
now
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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?

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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

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Guruswami-Sudan Algorithm
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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!!!
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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!!!
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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

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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)

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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

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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,
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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!!!

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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.
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Simulation Results 1----RS(63, 15)
AWGN
Rayleigh fading
Coding gain 0.4-1.3dB 1-2.8dB
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Simulation Result 2----RS(63, 31)
AWGN
Rayleigh fading
Coding gain 0.2-0.8dB 0.5-1.4dB
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Complexity Analysis
RS(63, 15) RS(63, 31)
Reason Iterative Interpolation
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Little Supplements----GSs AG extension
RS f(x) Q(x, y) p(x) AG f(x, y) Q(x, y,
z) p(x, y)
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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
  addressed in future!!!

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I Welcome your Questions?