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High noise regime

High noise regime. Desire code C : {0,1} k  {0,1} n such that (1/2-) fraction of errors can be corrected (think  = o(1) ) Want small n Efficient construction, list decoding ( poly(k/) time) Non-constructive optimal bound: n  k/ 2

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High noise regime

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  1. High noise regime • Desire code C : {0,1}k  {0,1}nsuch that (1/2-) fraction of errors can be corrected (think  = o(1) ) • Want small n • Efficient construction, list decoding (poly(k/) time) • Non-constructive optimal bound: n  k/2 • Zyablov bound achieves n  k/3, but construction time is exponential in 1/ • Such codes have many complexity theory applications • Hardcore predicates, extractors & pseudorandom generators, worst-case to average-case reduction, approximating NP witnesses, hardness of approximation.

  2. Prior work • Hadamard code: n = 2k, Goldreich-Levin local list decoder • -biased codes: • 3 constructions with n  k2/4 (but without efficient list-decoding) [Alon,Hastad,Goldreich,Peralta’02] • [G.,Sudan’00]: n  k2/4 withefficient list decoding • Reed-Solomon concatenated with Hadamard(one of the AGHP constructions) • crucial use of soft decoding of Reed-Solomon codes Our result: n  k3/3+ , construction is folded Reed-Solomon concatenated with dual BCH

  3. H(aN) H(a1) H(a2) RS+Hadamard soft decoding fk-1 f(X) =  fj Xj f0 f1 RS(f) N = 2m ai = f(i)  GF(2m) a1 a2 aN len(H(ai))=N Received word yN y1 y2 yi decoded to a  GF(2m) with weight wi,a=1/2 - (yi,H(a)) Parseval: For each i, a 2 GF(2m) w2i,a O(1)  RS soft decoder succeeds when RS rate k/N O(2). Final block length n = N2  k2/4

  4. Folded RS soft decoding • Order s folded RS code provides similar soft decoding guarantee only assuming (s+1)-moment a wi,a  O(1) • Further, rate k/N is better:  1+1/s • But alphabet size is Q=(2m)s, so Hadamard encoding has length Ns & final block length = Ns+1(too big) • Can use less redundant inner code • Dual of BCH code with distance 2t+1 has small 2t-moment • Hadamard is t=1 case • Maps log Q (= ms = 2mt) bits to 22m (= N2) bits • Final block length = N3  ( k/1+1/s )3 S+1

  5. Dual BCH code 2b • dBCH : (GF(2b))t {0,1} • (1,2,…,t) [Tr(1x+2x3+3x5 +…+tx2t-1)]x  GF(2 ) • Hadamard encoding of a  GF(2b)is {Tr(ax) for x  GF(2b)} Proving the moment bound: • Dual of dBCH has distance at least 2t+1 • [Kaufman,Litsyn’05] If dual of C has distance > d, weight distribution of C looks binomial to degree d polynomials: • Ec [ f(wt[c]) ] = 2-n i Cni f(i) if deg(f)  d • Use this with f(i) = (1/2- i/n)2t to bound, for any y, the sum c (1/2-dist(y,c)/n)2t b

  6. Comments • Needed to generalized both • outer code (folded RS in place of RS), and • inner code (dual BCH in place of Hadamard) • Soft decoding guarantee of folded RS code meshes perfectly with appropriate moment bound of dual BCH “coset weight distribution” • Works also with Parvaresh-Vardy codes, the precursor of folded RS code (unlike results 1 and 2)

  7. Summary • Can make good progress on binary list decoding using powerful list recovery & soft list decoding algorithms for folded RS codes • The algorithmic results 1 & 3 (Blokh-Zyablov & list-decodable -biased codes) seem best achievable via techniques which decode inner blocks independently • Need global way to reason about decoding the inner codes, taking into account outer code structure

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