1 / 20

Decoding Reed-Solomon Codes using the Guruswami-Sudan Algorithm

Decoding Reed-Solomon Codes using the Guruswami-Sudan Algorithm. PGC 2006, EECE, NCL Student: Li Chen Supervisor: Prof. R. Carrasco, Dr. E. Chester. Introduction. List Decoding Guruswami-Sudan Algorithm Interpolation (Kotter’s Algorithm) Factorisation (Ruth-Ruckenstein Algorithm)

Download Presentation

Decoding Reed-Solomon Codes using the Guruswami-Sudan Algorithm

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

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 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(k<n) (C0, C1, …, Cn-1)=(f(x0), f(x1), …, f(xn-1)) transmitted message f(x)=f0x0+f1x1+∙∙∙+fk-1xk-1 kdimensional monomial basis of curve y=0 • Application: Storage device Mobile communications

  6. 1999 1997 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-x2y-(-x) y-p(x)?=f(x) y-x Q(x, y) has a zero of multiplicity m=1 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)2y-(-x) y-x y-p(x)?=f(x) y-(-x) y-x Q2(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 tm Output list lm • 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) ---qab=0 for a+b<m, Q has a zero of multiplicity m at (0, 0). • Define over a certain point (xi, yi): • ---quv=0 for u+v<m, Q has a zero of multiplicity m at (xi,yi) • quv is the Q’s (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)=min{Q(x, y)Fq[x, y]|DuvQ(xi, yi)=0 for i=0, ∙∙∙, n-1 and u+v<m} Q has a zero of multiplicity m over the n points

  12. Kotter’s Algorithm • Initialisation: G0={g0, g1, …, gj, …,} Hasse Derivative Evaluation If i=n, end! Else, update i, and (u, v) Find the minimal polynomial in J: Bilinear Hasse Derivative modification: For (jJ), if j=j*, if j≠j*,

  13. Factorisation---Find p(x) • p(x) satisfy: y-p(x)|Q(x, y) and deg(p(x))<k • p(x)=p0+p1x+∙∙∙+pk-1xk-1 ---we can deduce coefficients p0, p1, …, pk-1sequentially!!!

  14. Ruth-Ruckenstein Algorithm Q0(x, y) Q1(x, y) Q2(x, y) p(x) p(x) Q’s sequential transformation: (4) pi are the roots of Qi(0, y)=0.

  15. Simulation Results 1----RS(63, 15) AWGN Rayleigh fading Coding gain: 0.4-1.3dB 1-2.8dB

  16. Simulation Result 2----RS(63, 31) AWGN Rayleigh fading 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----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

More Related