����� � ���� � ��� �� ���� ���� ��� ����������� � � � �������� � � � � � � �������������������������� ������������������������ ������ ���� ����� �� List-decoding Reed-Solomon codes: re-encoding techniques and Wu algorithm via simultaneous polynomial approximations Vincent Neiger § , † , ‡ ´ Claude-Pierre Jeannerod § Eric Schost † Gilles Villard § § AriC, LIP, ´ Ecole Normale Sup´ erieure de Lyon, France † ORCCA, Computer Science Department, Western University, London, ON, Canada ‡ Supported by the international mobility grant Explo’ra doc from R´ egion Rhˆ one-Alpes Journ´ ees nationales de calcul formel CIRM, Luminy, France, November 5, 2014 Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 1 / 22
Outline Decoding of Reed-Solomon codes via polynomial approximations 1 Re-encoding technique via polynomial approximations 2 Wu reduction via polynomial approximations 3 Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 2 / 22
Decoding of Reed-Solomon codes via polynomial approximations Outline Decoding of Reed-Solomon codes via polynomial approximations 1 Re-encoding technique via polynomial approximations 2 Wu reduction via polynomial approximations 3 Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 3 / 22
Decoding of Reed-Solomon codes via polynomial approximations Reed-Solomon codes At most e = n − t errors during transmission of a code word encoding noise w = w 0 + · · · + w k X k − − − − − − → ( w ( x 1 ) , . . . , w ( x n )) − − − → y = ( y 1 , . . . , y n ) i.e. # { i | w ( x i ) � = y i } � e or # { i | w ( x i ) = y i } � t Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 4 / 22
Decoding of Reed-Solomon codes via polynomial approximations Decoding of Reed-Solomon codes Polynomial Reconstruction Input: x 1 , . . . , x n the n distinct evaluation points in K k the degree bound, e = n − t the error-correction radius ( y 1 , . . . , y n ) the received word in K n Output: All polynomials w in K [ X ] such that deg w � k and # { i | w ( x i ) = y i } � t Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 5 / 22
Decoding of Reed-Solomon codes via polynomial approximations Key equations & Unique decoding Master, Interpolation and error-locator polynomials G ( X ) = � Λ( X ) = � 1 � i � n ( X − x i ) , R ( x i ) = y i , i | error ( X − x i ) Key equations: for every i , Λ( x i ) R ( x i ) = Λ( x i ) w ( x i ) Modular key equation Λ R = Λ w mod G where deg(Λ) � e , deg(Λ w ) � e + k , Λ monic. Unique decoding: e + k < n − e ⇔ e < n − k ⇒ unique rational solution Λ w Λ = w 2 computed in O ˜( n ) using e.g. the Extended Euclidean algorithm [Modern Computer Algebra, von zur Gathen - Gerhard, 2013] Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 6 / 22
Decoding of Reed-Solomon codes via polynomial approximations List-decoding: Guruswami-Sudan algorithm √ If e < n − k 2 , unique decoding. If e < n − kn , polynomial-time decoding. Recall: deg w � k and # { i | w ( x i ) = y i } � t [Guruswami - Sudan, 1999] Interpolation step compute a polynomial Q ( X , Y ) such that: Q ( X , w ) has many roots Q ( X , w ) has small degree − → w solution ⇒ Q ( X , w ) = 0 Root-finding step find all Y -roots of Q ( X , Y ), keep those that are solutions Here we focus on the Interpolation step. Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 7 / 22
Decoding of Reed-Solomon codes via polynomial approximations The interpolation step Interpolation With Multiplicities Input: number of points n , degree weight k , weighted-degree bound b =mt points { ( x i , y i ) } 1 � i � n in K 2 ( x i ’s distinct) list-size ℓ , multiplicity m ( m � ℓ ) Output: a nonzero polynomial Q in K [ X , Y ] such that ( i ) deg Y Q � ℓ, (list-size condition) deg X Q ( X , X k Y ) < b , ( ii ) (weighted-degree condition) ( iii ) ∀ i , Q ( x i , y i ) = 0 with multiplicity m (vanishing condition) Guruswami-Sudan: t 2 > kn ⇒ solution exists for some well-chosen m , ℓ − → linear system, compute a solution in polynomial time Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 8 / 22
Decoding of Reed-Solomon codes via polynomial approximations Simultaneous polynomial approximations [Roth - Ruckenstein, 2000] [Zeh - Gentner - Augot, 2011] vanishing condition ⇔ system of modular equations: write Q ( X , Y ) = Q 0 ( X ) + Q 1 ( X ) Y + · · · + Q ℓ ( X ) Y ℓ for i ∈ { 1 , . . . , n } , Q ( x i , y i ) = 0 with multiplicity m Q 0 + · · · + Q m − 1 R m − 1 + · · · + Q ℓ R ℓ = 0 mod G m Q 1 + · · · + Q m − 1 mR m − 2 + · · · + Q ℓ ℓ R ℓ − 1 = 0 mod G m − 1 ⇐ ⇒ . . ... . . = 0 mod G ··· . . � � R ℓ − m +1 = 0 mod G ℓ Q m − 1 + · · · + Q ℓ m − 1 where G = � 1 � i � n ( X − x i ) and ∀ i , R ( x i ) = y i . Dimensions of linearized problem: N = � M = 1 2 m ( m + 1) n equations , 0 � j � ℓ ( b − jk ) unknowns Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 9 / 22
Decoding of Reed-Solomon codes via polynomial approximations Algorithms based on linearization Strategy: use degree bounds to linearize the problem � � Q (0) · · · Q ( b − 1) | Q (0) · · · Q ( b − k − 1) | · · · | Q (0) · · · Q ( b − ℓ k − 1) 0 0 1 1 ℓ ℓ vanishing condition ⇔ solution to an under-determined linear system [Guruswami - Sudan, 1999] Structure “not used”, cost O (( m 2 n ) ω ) ( ω = exponent of mat. mult.) [Roth - Ruckenstein, 2000] [Zeh - Gentner - Augot, 2011] Mosaic-Hankel system, cost O ( ℓ m 4 n 2 ) using [Feng - Tzeng, 1991] [Chowdhury - Jeannerod - Neiger - Schost - Villard, 2014] Mosaic-Hankel system, cost O ˜( ℓ ω − 1 m 2 n ) using [Bostan - Jeannerod - Schost, 2007] Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 10 / 22
Decoding of Reed-Solomon codes via polynomial approximations Algorithms based on reduced lattice bases Based on polynomial lattice reduction [Alekhnovich, 2002] [Reinhard, 2003] [Beelen - Brander, 2010] [Bernstein, 2011] [Cohn - Heninger, 2011] Compute a known basis of approximants Use lattice reduction to find a small-degree approximant Cost O ˜( ℓ ω mn ) using [Giorgi - Jeannerod - Villard, 2003] (probabilistic) or [Gupta - Sarkar - Storjohann - Valeriote, 2012] Based on order basis computation Mirror all polynomials − → simultaneous Hermite-Pad´ e equations Compute an order basis of the resulting matrix of power series Cost O ˜( ℓ ω − 1 m 2 n ) using [Zhou - Labahn, 2012] Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 11 / 22
Re-encoding technique via polynomial approximations Outline Decoding of Reed-Solomon codes via polynomial approximations 1 Re-encoding technique via polynomial approximations 2 Wu reduction via polynomial approximations 3 Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 12 / 22
Re-encoding technique via polynomial approximations When some y i ’s are zero (case m = 1) Recall Q ( x i , y i ) = Q 0 ( x i ) + Q 1 ( x i ) y i + · · · + Q ℓ y ℓ i Assume y 1 = y 2 = · · · = y i 0 = 0, then for i � i 0 , Q ( x i , y i ) = 0 ⇔ Q 0 ( x i ) = 0 Thus Q 0 = G 0 � (for every i � i 0 , Q ( x i , y i ) = 0) ⇔ Q 0 Q 0 of degree < b − i 0 , where G 0 = � for some � 1 � i � i 0 ( X − x i ) − → Equations for points i = 1 , . . . , i 0 are pre-solved Then remains an easier approximation problem � Q 0 + Q 1 R / G 0 + · · · + Q ℓ R ℓ / G 0 = 0 mod ( G / G 0 ) Smaller dimensions: M − i 0 equations, N − i 0 unknowns Vincent Neiger ( ENS de Lyon ) Re-encoding and Wu algorithm via polynomial approximation Luminy, JNCF 2014 13 / 22
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