Progressive Algebraic Soft-Decision Decoding of Reed-Solomon Codes Li Chen （陈立） , PhD, MIEEE Associate Professor, School of Information Science and Technology Sun Yat-sen University, China Joint work with Prof. Xiao Ma and Mrs. Siyun Tang Hangzhou, China 5 th , Nov, 2011
Outline Introduction Progressive ASD (PASD) algorithm Validity analysis Complexity analysis Error-correction performance Conclusions
I. Introduction of list decoding Decoding philosophy evolution of an ( n , k ) RS code Unique decoding List decoding 1 n k ( 1 ) n n k unique list 2
I. Overview of Enc. & Decd. Encoding Given a message polynomial: u ( x ) = u 0 + u 1 x + ··· + u k -1 x k- 1 ( u i ∈ GF( q )) Generate the codeword of an ( n , k ) RS code ( α i ∈ GF( q )\{0}) c = ( c 0 , c 1 , …, c n -1 ) = ( u ( α 0 ), u ( α 1 ), …, u ( α n -1 )) Decoding M Π L Factorization Interpolation Reliab. Trans. Π M Q ( x , p ( x )) Q ( x , y ) Computationally expensive step!
I. Perf. of AHD and ASD Algebraic hard-decision decoding (AHD) [Guruwammi99] (-GS Alg) Algebraic soft-decision decoding (ASD) [Koetter03] (-KV Alg) Advantage on error-correction performance
I. Introduction Price to pay: decoding complexity
I. Inspirations The algebraic soft decoding is of high complexity. It is mainly due to the iterative interpolation process ; A modernized thinking – the decoding should be flexible (i.e., channel dependent). E.g., the belief propagation algorithm high The current AHD/ASD system Initialisation Complexity Horizontal step Vertical step Hard decision ˆ low c No ˆ T 0 ? c H bad good Quality of the received word ˆ Yes, c Iterative proc. Incremental comp. Continues validat.
II. A Graphical Introduction l l 1 2 3 4 5 PASD ( ) 5 ASD ( ) c 1 c 2 c 3 c 1 c 2 c 3 c 5 c 6 c 7 c 5 c 6 c 7 c 4 c 4 r r c 8 c 9 c 10 c 8 c 9 c 10 1 { } l L c 6 2 { , } l L c c { , , , , } L c c c c c 6 9 5 6 7 9 10 3 { , , } l L c c c 6 9 10 4 { , , , } l L c c c c | L | - factorization output list size 5 6 9 10 5 { , , , , } l L c c c c c 5 6 7 9 10 Enlarging the decoding radius progressively Enlarging the factorization OLS progressively
Ⅱ . Decoding architecture Progressive Progressive Interpolation reliab. trans. Validated by CRC code! v – iteration index; l v -- designed OLS at each iteration; l T -- designed maximal OLS (~the maximal complexity that the system can tolerate); l’ – step size for updating the OLS, l v +1 = l v + l’ ;
II. Progressive approach Enlarge the decoding radius Enlarge the OLS Progressive decoding Π Series of OLS: l 1 , l 2 , ……, l v -1 , l v , ……, l T Series of M mtxs.: M 1 , M 2 , ……, M v -1 , M v , ……, M T Series of Q polys.: Q (1) , Q (2) , ……, Q ( v -1) , Q ( v ) , ……, Q ( T ) Question: Can the solution of Q ( v ) be found based on the knowledge of Q ( v -1) ?
II. Incremental interpolation constraints Multiplicity m ij ~ interpolated point ( x j , α i ) 1 Given a polynomial Q ( x , y ), m ij implies constraints of ( + 1 ) m ij m ij 2 ( r + s < m ij ) Definition 1: Let Λ ( m ij ) denote a set of interpolation constraints ( r , s ) ij indicated by m ij , then Λ (M) denotes a collection of all the sets Λ ( m ij ) defined by the entry m ij of M 2 0 0 0 1 1 Λ (M) = {(0, 0) 00 , (1, 0) 00 , (0, 1) 00 , (0, 0) 11 , Example: M M 1 2 0 (0, 0) 12 , (0, 0) 20 , (0, 0) 21 , (1, 0) 21 , (0, 1) 21 , (0, 0) 32 } 0 0 1
II. Incremental Interp. Constr. v -1 and m ij Definition 2: Let m ij v denote the entries of matrix M v -1 and M v , the incremental interpolation constraints introduced between the matrices are defined as a collection of all the residual sets between Λ ( m ij v ) and Λ ( m ij v -1 ) as: 2 0 0 1 0 0 0 1 1 0 0 1 Example: M 2 M M 1 M 2 1 2 0 1 1 1 0 0 0 1 0 0 1 Λ (M 2 ) = {(0, 0) 00 , (1, 0) 00 , (0, 1) 00 , Λ (M 1 ) = {(0, 0) 00 , (0, 0) 12 , (0, 0) 11 , (0, 0) 12 , (0, 0) 20 , (0, 0) 20 , (0, 0) 21 , (0, 0) 32 } (0, 0) 21 , (1, 0) 21 , (0, 1) 21 , (0, 0) 32 } 5 constraints 10 constraints Λ ( Δ M 2 ) = {(1, 0) 00 , (0, 1) 00 , (0, 0) 11 , 5 constraints (1, 0) 21 , (0, 1) 21 }
II. Progressive Interpolation A big chunk of interpolation task Λ ( Μ T ) can be sliced into smaller pieces. Λ ( ΔΜ 1 ) Λ ( ΔΜ 2 ) Λ ( ΔΜ 3 ) Λ ( Μ T ) ... Λ ( ΔΜ v ) ... Λ ( ΔΜ T ) Λ ( ΔΜ v ) defines the interpolation task of iteration v.
II. Incremental Computations Review on the interpolation process – iterative polynomial construction Given M v , the interpolation constraints are Λ ( Μ v ) The polynomial group is: G v = { g 0 , g 1 , …, g v }, (deg y g v = l v ) The iterative process defines the outcome of the updated group. For ( r , s ) ij ∈ Λ ( Μ v ) For each g t ∈ G v Finally, Q ( v ) = min{ g t | g t ∈ G v }
II. Incremental Computations From iteration v -1 to v … The progressive interpolation can be seen as a progressive polynomial group expansion which consists of two successive stages. Let be the outcome of iteration v -1. During the generation of , a series of with ( r , s ) ij ∈ Λ ( Μ v -1 ) are identified and stored. Expansion I: expand the number of polynomials of the group Polynomials of Δ G v perform interpolation w.r.t. constraints of Λ ( Μ v -1 ); Polynomials are re-used for the update of Δ G v . Let be the updated outcome of Δ G v and
II. Incremental Computations Expansion II: expand the size of polynomials of the group Polynomials of will now perform interpolation w.r.t. the incremental constraints Λ ( ΔΜ v ), yielding . Finally, Visualize the polynomial group expansion Expansion I E.g., l v = 5 g 1 g 2 g 3 g 4 g 5 g 6 g 1 g 2 g 3 g 4 g 5 g 6 Expansion II ASD PASD
II. Progressive Interpolation The process of progressive interpolation M 1 , M 2 , M 3 , … , M v -1 , M v , …, M T -1 , M T Λ (M 1 ), Λ (M 2 ), Λ (M 3 ), …, Λ (M v -1 ), Λ (M v ), …, Λ (M T -1 ), Λ (M T ) Λ ( Δ M T ) Λ ( Δ M v ) Λ ( Δ M 2 ) Λ ( Δ M 3 ) G 1 G 2 G 3 …… G v -1 G v G T -1 G T Δ G 3 Δ G v Δ G T Δ G 2 Q (1) Q (2) Q (3) Q ( v -1) Q ( v ) Q ( T -1) Q ( T ) If Q ( v ) ( x , u ( x )) = 0, the decoding will be terminated. Multiple factorizations are carried out in order to determine whether u ( x ) has been found!
III. Validity Analysis For any two ( r 1 , s 1 ) i 1 j 1 and ( r 2 , s 2 ) i 2 j 2 of Λ ( Μ T ): ( r 1 , s 1 ) i 1 j 1 ( r 2 , s 2 ) i 2 j 2 <==> ( r 2 , s 2 ) i 2 j 2 ( r 1 , s 1 ) i 1 j 1 The algorithm imposes a progressive interpolation order Λ ( ΔΜ 1 ) Λ ( ΔΜ 2 ) Λ ( ΔΜ 3 ) Λ ( Μ T ) ... Λ ( ΔΜ v ) ... Λ ( ΔΜ T ) The ( r , s ) ij to be satisfied. The satisfied ( r , s ) ij .
III. Validity Analysis Decoding with an OLS of l v , the solution Q ( x , y ) is seen as the minimal candidate chosen from the cumulative kernel For both of the algorithms: the same set of constraints are defined for the cumulative kernel; Consequently, they will offer the same solution of Q ( x , y ).
III. Validity Analysis In the end of Expansion I , Can and be found separately? Recall the polynomial updating rules • The minimal polynomial defines the solution of one round of poly. update w.r.t. ( r , s ) ij . • If such a group expansion procedure does not change the identity of , and can indeed be found separately.
III. Validity Analysis ( < ) Expansion I: update to w.r.t. constraint of No update is required For is in memory, Since < g * (of Δ G v ) Update is required will be re-used is not in memory, it will be picked up from Δ G v. The identity of the existing is left unchanged. Consequently, the solution of remains intact. Therefore,
IV. Complexity Analysis Average decoding complexity – average number of finite field arithmetic operation for decoding one codeword frame; --- the probability of the decoder is performing a successful decoding with an OLS of l v ; --- the decoding complexity with an OLS of l v ; The average decoding complexity is: Decoding succ. Decoding fail. is now channel dependent!
IV. Complexity Analysis where consists of and since
IV. Complexity Analysis The decoding complexity increases exponentially with the OLS (dominant exponential factor of 5); The decoding complexity is quadratic in the dimension of the code k. The decoding complexity will be smaller for a low rate code.
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