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

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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

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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 (k<n) (C 0, C 1, …, C n-1 )=(f(x 0 ), f(x 1 ), …, f(x n-1 )) transmitted message f(x)=f 0 x 0 +f 1 x 1 +∙∙∙+f k-1 x k-1 k dimensional monomial basis of curve y =0 Application: Storage device Mobile communications

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Guruswami-Sudan Algorithm 1999 1997

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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

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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!!!

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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

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Interpolation---Build Q(x, y) Multiplicity definition:(2) --- q ab =0 for a+b<m, Q has a zero of multiplicity m at (0, 0). Define over a certain point ( x i, y i ): --- q uv =0 for u+v<m, Q has a zero of multiplicity m at ( x i,y i ) q uv is the Q ’s ( u, v ) Hasse derivative evaluation on ( x i, y i ) (3)

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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<m} Q has a zero of multiplicity m over the n points

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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 )

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Factorisation---Find p(x) p(x) satisfy: y-p(x)|Q(x, y) and deg(p(x))<k p(x)=p 0 +p 1 x+∙∙∙+p k-1 x k-1 ---we can deduce coefficients p 0, p 1, …, p k-1 sequentially!!!

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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.

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

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Simulation Result 2----RS(63, 31) AWGNRayleigh 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----GS’s 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

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