1. 1 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 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. 2 Prior work Hadamard code: n = 2 k, Goldreich-Levin local list decoder  -biased codes:  3 constructions with n  k 2 /  4 (but without efficient list- decoding) [Alon,Hastad,Goldreich,Peralta’02] [G.,Sudan’00]: n  k 2 /  4 with efficient list decoding  Reed-Solomon concatenated with Hadamard (one of the AGHP constructions)  crucial use of soft decoding of Reed-Solomon codes Our result: n  k 3 /  3+ , construction is folded Reed-Solomon concatenated with dual BCH

3. 3 RS+Hadamard soft decoding f0f0 f1f1 aNaN a1a1 a2a2 f k-1 f(X) =  f j X j RS(f) N = 2 m H(a 1 )H(a 2 ) H(a N ) y1y1 y2y2 yNyN a i = f(  i )  GF(2 m ) len(H(a i ))= N Received word y i decoded to a  GF(2 m ) with weight w i,a =1/2 -  (y i,H(a)) Parseval: For each i,  a 2 GF(2 m ) w 2 i,a  O(1)  RS soft decoder succeeds when RS rate k/N  O(  2 ). Final block length n = N 2  k 2 /  4

4. 4 Folded RS soft decoding Order s folded RS code provides similar soft decoding guarantee only assuming (s+1)-moment  a w i,a  O(1)  Further, rate k/N is better:   1+1/s  But alphabet size is Q=(2 m ) s, so Hadamard encoding has length N s & final block length = N s+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 2 2m (= N 2 ) bits  Final block length = N 3  ( k/  1+1/s ) 3 S+1

5. 5 Dual BCH code dBCH : (GF(2 b )) t  {0,1} (  1,  2,…,  t )  [Tr(  1 x+  2 x 3 +  3 x 5 +…+  t x 2t-1 )] x  GF(2 )  Hadamard encoding of a  GF(2 b ) is {Tr(ax) for x  GF(2 b )} 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: E c [ f(wt[c]) ] = 2 -n  i C n i 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 2b2b b

6. 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. 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