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Limits to List Decoding Reed-Solomon Codes

Limits to List Decoding Reed-Solomon Codes. Venkatesan Guruswami Atri Rudra (University of Washington). Error-Correcting Codes. Linear Code C : GF(q) k ! GF(q) n Hamming Distance or  ( u , v ) for u , v 2 GF(q) n Number of positions u and v differ

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Limits to List Decoding Reed-Solomon Codes

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  1. Limits to List Decoding Reed-Solomon Codes Venkatesan Guruswami Atri Rudra (University of Washington) STOC 2005, Baltimore

  2. Error-Correcting Codes • Linear Code C : GF(q)k! GF(q)n • Hamming Distance or (u,v) for u,v 2 GF(q)n • Number of positions u and v differ • Distance of code C, d=minx,y2GF(qk)(C(x),C(y)) • C is an [n,k,d]GF(q) code • Relative distance =d/n • This talk is about Reed-Solomon (RS) Codes STOC 2005, Baltimore

  3. List Decoding • Given r2 GF(q)n and 0·e· 1 • Output all codewords c2 C such that (c,r)·en • Combinatorial Issues • How big can the list of codewords be ? • LDR(C) largest e such that list size is poly(n) • Algorithmic issues • Can one find list of codewords in poly(n) time ? • Cannot have poly time algo beyond LDR(C) errors STOC 2005, Baltimore

  4. List Recovery • Related to List Decoding Problem • Given C: GF(q)k! GF(q)n and Liµ GF(q), 1· i· n • Find all codewords c=hc1,,cnis.t. ci2 Li8 i • |Li|· s • LRB(C)largest s for which # of codewords is poly(n) 1 2 3 n L3 L2 L1 Ln STOC 2005, Baltimore

  5. Reed-Solomon Codes • RS [n,k+1]GF(q) • Message P a poly. of degree · k over GF(q) • S µ GF(q) • RS(P) = h P(a) ia2S • n= |S| • d = n – k • For this talk S = GF(q) • 9poly timealgo for list decoding of RS codes till error bound J()=1-(1-)1/2=1- (k/N)1/2 STOC 2005, Baltimore

  6. 9 poly time algo for error bound · J() Negative Result ) above algo optimal The Big Picture for RS Polynomial Reconstruction RS List Recovery RS List Decoding STOC 2005, Baltimore

  7. Talk outline • Our main result is about combinatorial limitation of List Recovery of Reed Solomon Codes • Motivation of the problem • Main Result and Implications • Proof of the main result STOC 2005, Baltimore

  8. Combinatorial Limitations- I • For any C • LDR(C ) ¸/2 • Unique decodability Half Distance Error Bound Relative Distance () STOC 2005, Baltimore

  9. Combinatorial Limitations- II Half Distance • LDR(C)¼ is the best one can hope for • e ¸ can’t detect errors • Lots of “good” codes with LDR(C)¼ • Random Linear Codes • 2x improvement over unique decoding • Difficulty: getting explicit codes Full Distance Error Bound Relative Distance () STOC 2005, Baltimore

  10. Combinatorial Limitations- III Half Distance Full Distance • Johnson Bound • For any code C • LDR(C) ¸ J()=(1-(1-)1/2) • Exists codes for which Johnson Bound is tight • Non-linear codes [GRS00] • Linear codes [G02] Johnson Bound Error Bound Relative Distance () STOC 2005, Baltimore

  11. Going beyond the Johnson Bound Half Distance Full Distance • Go beyond Johnson Bound • Choice of code matters • Random Linear codesget there • What about well studied codes like RS codes ? • Motivation of our work Johnson Bound Error Bound Relative Distance () STOC 2005, Baltimore

  12. ? ? ? ? ? ? ? ? Algorithmic Status of RS Half Distance Full Distance • Unique decoding • [Peterson60] • List Decoding • Johnson Bound • [Sud97, GS99] • Unknown beyond JB • Some belief that LDR(RS)=(1-(1-)1/2) Johnson Bound STOC 2005, Baltimore

  13. b2 b3 bi bn General setup for GS algorithm • Polynomial Reconstruction • Pairs of numbers {(ai,bi)}, i=1..N • Finds all degree k poly P at most N-(Nk)1/2 indices i, P(ai) bi • aidistinct ) List Decoding of RS • ainot necessarily distinct ) List Recovery of RS a1 a2 a3 ai an b1 Li STOC 2005, Baltimore

  14. Main Result of this talk • Version of Johnson Bound implies LRB(RS) ¸dn/ke -1 (GS algo works in poly time in this regime) We show LRB(RS) = dn/ke-1 )For Polynomial reconstruction GS algo is optimal STOC 2005, Baltimore

  15. a1 a2 a3 ai an dn/ke Implication for List Decoding RS • Polynomial Reconstruction • In List Recovering setting N=n¢dn/ke • Number of disagreements = N-n w (Nk)1/2 • With (little more than)N-(Nk)1/2disagreement have super poly RS codewords • GS algo works for disagreement · N-(Nk)1/2 • Improvement “must” use near distinctness of ais STOC 2005, Baltimore

  16. Main Result a1 a2 a3 ai an dn/ke • n=q=pm • D=(pm-1)/(p-1) =pm-1+pm-2++p+1 • Consider RS [n,k=D+1]GF(qm)* • For each i=1,,n the list Li= GF(p) • dn/ke = p Number of deg D polys over GF(pm) which take values in GF(p) is p2m STOC 2005, Baltimore

  17. Explicit Construction of Polys 2m-1 • Pb(z) = i=0bi( zai + 1)Dwherebi2 GF(p) • a is a generator of GF(pm) • D=(pm-1)/(p-1)=pm-1++p+1 • Poly overGF(pm) • Takes values inGF(p) • Norm function: for all x2 GF(pm), xD2 GF(p) Will now prove for distinct b,Pb(z) are distinct polys over GF(pm) STOC 2005, Baltimore

  18. D j Proof Idea • By Linearity,need to show Pb(z) = i=0bi( zai + 1)D  0 )b1 = b2 == b2m-1 =0 • Coefficients of all zj must be 0 ( )i=0 bi (ai)j =0 for j=0..D • D+1 eqns and 2m vars (some of them trivial) 2m-1 D=pm-1++p+1 2m-1 STOC 2005, Baltimore

  19. D j Lucas’ Lemma • p prime and integers a and b • a=a0+a1p++arpr • b=b0+b1p++brpr ( )  ( ) ( )( ) mod p • D=1+p+ pm-1, j=j0+j1p+ +jmpm-1 ( )  0iff for alli, ji 2 {0,1} )2m equations and 2m var a a0 a1 ar b b0 b1 br STOC 2005, Baltimore

  20. b0 aj0 (aj0)2 … (aj0)2m-1 1 1 . . . 1 b1 = 0 aj1 (aj1)2 … (aj1)2m-1 Wrapping up the proof • 2m equations in 2m variables • T= { j0+j1p+ jm-1pm-1 | ji2 {0,1} } • i=0 bi (ai)j =0 for j 2 T Coefficient matrix is Vandermonde 2m-1 STOC 2005, Baltimore

  21. Other Results in the Paper • Use connection with BCH codes to get an exact estimate • Show existence of explicit received word with super poly “close by” RS codewords for certain parameters • Uses ideas from [CW04] STOC 2005, Baltimore

  22. Open Questions • Is Johnson Bound the true list decoding radius of Reed Solomon codes ? • Show RS of rate 1/L cannot be list recovered using lists of size L which are not prime powers. • What RS codes on prime fields ? STOC 2005, Baltimore

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