1. Soft decoding, dual BCH codes, & better -biased list decodable codes
Venkat Guruswami (U. of Washington)
Atri Rudra (U. at Buffalo)
Start with a slide, with work in blah blah. Worked in coding theory. Specific contribs blah and blah. State in course of work solved and closed a major theory problem– state this upfront

2. Error-correcting codes
C(x) x
Mapping C : kn
Dimension k, block length n
n≥ k
Rate = k/n  1
Efficient means polynomial in n
Decoding Complexity
 : fraction of errors y
Try to justify why n is increasing. Just say efficient is polynomial in n. Avoid saying family of codes. Mention what R is later. Write on the slide instead of just saying it. x
Give up

3. Codes for Complexity Binary codes
Correct =½ -  fraction of worst case errors
  0 ( can depend on n)
Unique decoding
Cannot correct more than ¼ fraction of errors

4. The list decoding problem
Given a code and an error parameter 
For any received word w
Output all codewords c such that c and w disagree in at most  fraction of places
(, L)-list decodable code
(½-, poly(n))-list decodable code exist!
Maybe remove capacity, mention spell checker

5. Applications of list decoding
Hardcore predicates from one way functions [Goldreich-Levin 89; Impagliazzo 97; Ta-Shama-Zuckerman 01]
Worst-case vs. average-case hardness [Cai-Pavan-Sivakumar 99; Goldreich-Ron-Sudan 99; Sudan-Trevisan- Vadhan 99; Impagliazzo-Jaiswal-Kabanets 06]
Pseudorandomness [Trevisan 99; Shaltiel-Umans 01; Ta-Shama-Zuckerman 01; Guruswami-Umans-Vadhan 07]
Membership-Comparable Sets [Sivakumar 99]
Approximating NP-Witnesses [Gal-Halevi-Lipton-Petrank 99, Kumar-Sivakumar99] 5 5

6. Approximating NP-witness
L is NP-complete
Polytime computable relation R
It is NP-hard to compute y given x
Approximating the certificate?
Compute w such that y and w differ in  (½-)|y| positions
[Kumar-Sivakumar 99] NP-hard to approximate
x  L   y such that R(x,y) holds

7. Kumar-Sivakumar reduction
NP-hard to do approximation for relation R’
C is list decodable from ½- fraction of errors
Main idea: Replace y with C(y)
R’ is also polytime computable
Decode C from zero error in poly time
Keep track of required properties of C
x  L   z such that R’(x,z) holds
R’(x,z) holds if  y s.t. R(x,y) holds and z=C(y)
x  L   y such that R(x,y) holds

8. Kumar-Sivakumar reduction
Assume can compute w s.t. w and z differ in  (½-)|z| positions in poly time
Run list decoder for C on w
Returns a list y1,…., ym
Check if for any i, R(x, yi) holds
Can check if x in L in poly time!
Assume x in L
C is
(½-,poly(|z|)
list decodable
C can be list decoded in polytime
R’(x,z) holds if  y s.t. R(x,y) holds and z=C(y)

9. Some possible concerns…
R’ is the not the same as R!
[Gal-Halevi-Lipton-Petrank 99] considers hardness of approximation for the same R
Crucially uses properties of R and L

10. Required properties of C
C : {0,1}k  {0,1}n
n = poly(k,1/)
C can be list decoded from (½-)n errors in poly(n) time
C can be constructed in poly(n) time
Not necessary in the NP-witness application
Given these constraints, how small can n be?

11. Binary list decodable code
Say n is O(kb/ea)
a and b are constants
For NP-witness application, smallest e  1/n1/a
(½-, poly(n))-list decodable code with n in O(k/e2) exist
a cannot be < 2
Existence proved via random coding argument
What is the smallest a, given b is some constant?

12. Efficient list decodable binary codes
a = 4, b = 2; i.e. n in O(k2/e4)
[Guruswami, Sudan 00]
The code is explicit
Hadamard “concatenated” with Reed-Solomon code
e-biased code
All non-zero codewords have [½-, ½+] frac. of 1s
Important pseduorandom object
a=3, b=1; i.e. n in O(k/e3)
[Guruswami, R ]
Construction and decoding time > n1/e
Only useful for constant e

13. Our main result a = 3+g, b = 3; i.e. n in O(k3/e3+g) g>0
The code is explicit
Dual BCH “concatenated” with “Folded” Reed-Solomon
e-biased code
All algorithms run in time poly(k,1/)
More or less recovers the a=3 result
[Guruswami, R ]

14. (Folded) Reed-Solomon codes
View mesg. as f(X) = m0+m1X+…+mk-1Xk-1
F* ={1,g1,g2,…,gN-1} (q=N+1=|F|)
Reed-Solomon (RS) codeword
s-order Folded RS codeword (N=ms)
Optimal list decodable properties for large alphabet
f(1)
f(g2)
f(g3)
f(g4)
f(gN-1)
f(1)
f(gs)
f(g(m-1)s )
f(gs-1)
f(g2s-1)
f(ms-1)
f(g)
f(gs+1)
f(g(m-1)s+1 )

15. How do we get binary codes ?
Concatenation of codes [Forney 66]
C1: (GF(2k))K  (GF(2k))N (“Outer” code)
C2: GF(2)k  (GF(2))n (“Inner” code)
C1 C2: (GF(2))kK (GF(2))nN
Typically k=O(log N)
Brute force decoding for inner code m m1 m2 mK w1 w2 wN
C1(m)
C2(w1)
C2(w2)
C2(wN)
C1  C2(m) 15

16. List Decoding concatenated code
One natural decoding algorithm
Divide up the received word into blocks of length n
Find closest C2 codeword for each block
Run list decoding algorithm for C1
Loses Information! 16

17. How do we “list decode” from lists ?
List Decoding C2
GF(2)n y1 y2 yN T1 T2 TN
GF(2)k
How do we “list decode” from lists ? 17

18. The list recovery problem
Given a code and an error parameter 
For any set of lists T1,…,TN such that
|Ti|  t, for every i
Output all codewords c such that ci 2 Ti for at least 1- fraction of i’s
List decoding is special case with t=1 18

19. List Recovering Algorithm for C1
List Decoding C1C2 y1 y2 yN
List decode C2 T1 T2 TN
List Recovering Algorithm for C1 19

20. [Guruswami, R. 06] Result Pick C1 to be folded RS of rate e
Has optimal list recoverability
Pick C2 to be “suitably” chosen binary code of rate e2
C1C2 List decodable from ½- frac of errors

21. A Closer look… List Recovering Algorithm for C1y1 y2
Dist(C2(a1),y1)  ½-
Dist(C2(a2),y1)  ½- yN
at a1
List decode C2 from ½- frac of errors
What if Dist(C2(a1),y1) << Dist (C2(a2),y1)? T1 T2 TN
List Recovering Algorithm for C1 21

22. Weighted List Recovery
{a1,…,aq} y1 y2 yN
Output codewords w/ small weighted distance a1 aq
wN,1
wN,q
w1,1
w1,q
w2,1
w2,q T1 T2 TN
List Recovering Algorithm for C1
Soft Decoding 22

23. List Recovery as a special case
wi,j
½- 1
List decode C2 y1 y2 yN T1 T2 TN
List Recovering Algorithm for C1
Dist(C2(ai),yj) 23

24. A natural weight function
Give more weight to “closer” symbols
List decode from ½- frac of errors
C1 is RS and C2 is Hadamard
Hadamard codewords are
evaluations of linear
functions
wi,j 1
Dist(C2(ai),yj) 24

25. Uses properties of Hadamard code
RS concat Hadamard
Need to show the following for every j:
i wi,j2  O(1)
RS code of rate e2
Follows from Parseval’s identity
N  O(K/ e2)
n= q
Hadamard code
Final length = nN = N2
 O(K2/ e4) [GS00]
Uses properties of Hadamard code
wi,j 1
Dist(C2(ai),yj) 25

26. Using order s folded RS as outer code
Need to show the following for every j:
i wi,js+1  O(1) ….. (*)
s ≥ 1
Folded RS of rate e1+1/s
Hadamard as inner code
(*) is true
n = qs = Ns
As Folded RS alphabet
size is qs
wi,j 1
Too large 
Dist(C2(ai),yj) 26

27. Using dual BCH as inner code
Need to show the following for every j:
i wi,js+1  O(1) ….. (*)
s ≥ 1
Folded RS of rate e1+1/s
Dual BCH as inner code
(*) from [Kaufman-Litsyn 05]
n  N2 suffices
Final length = N3
 O(k3/e3+3/s)
wi,j 1
Dist(C2(ai),yj) 27

28. Some comments Final codes are e-biased
Dual BCH have non-zero weights close to ½
Weil-Carlitz-Uchiyama bound
Dual BCH is a generalization of Hadamard
Our result needed to generalize both
Outer code (RS  folded RS)
Inner code (Hadamard  dual BCH)
Can replace folded RS with Parvaresh-Vardy codes

29. Open questions Improve on the cubic dependence on e
Needs new algorithmic ideas
Use information about outer codes while decoding inner codes
Rate e2 using concatenated codes possible
[Guruswami, R. 08]
Any application that also uses e-biasedness?