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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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Presentation on theme: "Decoding Reed-Solomon Codes using the Guruswami- Sudan Algorithm PGC 2006, EECE, NCL Student: Li Chen Supervisor: Prof. R. Carrasco, Dr. E. Chester."— Presentation transcript:

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 (Kotter’s Algorithm) Factorisation (Ruth-Ruckenstein Algorithm) Simulation Performance Complexity Analysis Algebraic-Geometric Extension Conclusion

3 Funny Talk about List Decoder Decoder—Search the lost boy named “ John” Unique decoder—Police without cooperation List decoder—Police with cooperation PoliceDecoder 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 (k

6 Guruswami-Sudan Algorithm

7 GS Overview Decode RS(5, 2): Encoding elemnts x=(x 0, x 1, x 2, x 3, x 4 ) Received word y=(y 0, y 1, y 2, y 3, y 4 ) Build Q(x, y) that goes through 5 points: Q(x, y)=y 2 -x 2 y-(-x) y-p(x)?=f(x) y-x The Decoded codeword is produced by re-evaluate p(x) over x 0, x 1, x 2, x 3, x 4 !!! Q(x, y) has a zero of multiplicity m =1 over the 5 points. GS = Interpolation + Factorisation

8 How about increase the degree of Q(x, y) ? Q 2 =(y 2 -x 2 ) 2 y-(-x) y-xy-x y-p(x)?=f(x) y-(-x) y-xy-x Q 2 (x, y) has a zero of multiplicity m =2 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 t m Output list l m Examples: RS(63, 15) with r =0.24, e =24 RS(63, 31) with r =0.49, e =16

10 Interpolation---Build Q(x, y) Multiplicity definition:(2) --- q ab =0 for a+b

11 Cont… Therefore, we have to construct a Q(x, y) that satisfies: Q(x, y)=min{Q(x, y)  F q [x, y]|D uv Q(x i, y i )=0 for i=0, ∙∙∙, n-1 and u+v

12 Kotter’s Algorithm Initialisation: G 0 ={g 0, g 1, …, g j, …,} Hasse Derivative Evaluation Find the minimal polynomial in J : Bilinear Hasse Derivative modification: For ( j  J ), if j=j *, if j≠j *, If i=n, end! Else, update i, and ( u, v )

13 Factorisation---Find p(x) p(x) satisfy: y-p(x)|Q(x, y) and deg(p(x))

14 Ruth-Ruckenstein Algorithm p(x)p(x) p(x)p(x) Q 0 (x, y) Q 1 (x, y) Q 2 (x, y) Q’s sequential transformation:(4) p i are the roots of Q i (0, y)=0.

15 Simulation Results 1----RS(63, 15) AWGNRayleigh fading Coding gain: dB 1-2.8dB

16 Simulation Result 2----RS(63, 31) AWGNRayleigh fading Coding gain: dB dB

17 Complexity Analysis RS(63, 15) RS(63, 31) Reason: Iterative Interpolation

18 Little Supplements----GS’s 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 addressed in future!!!

20 I Welcome your Questions


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