Parikshit Gopalan On Vacation Adam R' Klivans UT Austin David Zuckerman UT Austin

1
Parikshit Gopalan On Vacation Adam R. Klivans UT
Austin David Zuckerman UT Austin
List-Decoding Reed-Muller codes over Small Fields.
1
0
0
1
1
0
0
1
2
Error Correcting Codes
Communication over a Noisy Channel
Adversary corrupts 10 of the bits. Problem
Recover the (entire) message. Soln Introduce
redundancy.
3
Error-Correcting Codes
Cellphones
Satellite Broadcast
Audio CDs
Bar-codes
4
Codes from Polynomials
Encoding Alice wants to send (a,b). Let L(x)
ax b. Send L(1), L(2), , L(7).
5
Codes from Polynomials
Adversary Corrupts two values. Decoding Find
the (unique) line that passes through 5 points.
6
Codes from Polynomials

• Low degree ) few degrees of freedom.
• Evaluations have redundancy.
• Decoding Polynomial reconstruction.
• Reed-Solomon codes Univariate polynomials.
• Reed-Muller codes Multivariate polynomials.

7
Reed-Muller Codes Muller54, Reed54

• Messages Polynomials of degree r in n variables
  over 0,1.
• Q(X1,X2,X3) X1X2 X3
• Encoding Truth table.
• 00011110
• Relative distance ? 2-r.
• Low-degree polynomials differ in many
  places.
• Hadamard codes r 1.

1
0
0
1
1
0
0
1
8
Reed-Muller Codes Muller54, Reed54

• Messages Polynomials of degree r in n variables
  over 0,1.
• Q(X1,X2,X3) X1X2 X3
• Encoding Truth table.
• 00011110
• Relative distance ? 2-r.
• Low-degree polynomials differ in many
  places.
• Hadamard codes r 1.

1
0
0
1
1
0
0
1
9
Reed-Muller Codes Muller54, Reed54

• Messages Polynomials of degree r in n variables
  over 0,1.
• Q(X1,X2,X3) X1X2 X3
• Encoding Truth table.
• 00011110
• Relative distance ? 2-r.
• Low-degree polynomials differ in many
  places.
• Hadamard codes r 1.

0
1
0
1
0
0
1
1
10
Decoding Polynomial Reconstruction
0
1
0
1
0
1
0
1
1
0
0
0
0
1
1
1
Problem Given data points, find a low degree
polynomial that fits best. Well studied problem,
numerous applications.
11
The Decoding Problem

• Local Decoding Model
• Running time poly(n).
• Nearest Codeword Find the nearest codeword to
  R.
• Too hard?
• Unique Decoding Promise that ?(R,C) lt ?/2.
• Too easy?

R 0,1n ! 0,1
x
R(x)
12
The Decoding Problem

• Local Decoding Model
• Running time poly(n).
• List Decoding Elias57, Wozencraft58
• ?(R,C) is such that the list of Cs is small.
• Find the whole list.
• Johnson bound List is small up to J(?) where
• J(?) ?/2 ?2/2 ... lt d

R 0,1n ! 0,1
x
R(x)
13
Decoding Reed-Muller codes

• Unique Decoding
• Majority Logic Decoder. Reed54
• List Decoding
• Hadamard codes (r 1). Goldreich-Levin89
• Alternate algorithms Levin, Rackoff,
  Kushilevitz-Mansour,
• No algorithms known for r 2.
• Good algorithms for large fields (d lt F).
  Goldreich-Rubinfeld-Sudan, Arora-Sudan,
  Sudan-Trevisan-Vadhan

14
In this Work

• Main Result List-Decoding Reed-Muller codes for
  r 2. Works up to Minimum Distance 2-r.

• Improves on Majority Logic Decoding Reed54
  for r 2.
• Generalizes Goldreich-Levin89.
• Beats the Johnson bound.
• List-size becomes exponential at 2-r.

15
The Quadratic Case.

• 0,1n labeled by received word R.
• Fix codeword Q so that ?(Q,R) lt ¼.

16
The Quadratic Case.

• 0,1n labeled by received word R.
• Fix codeword Q so that ?(Q,R) lt ¼.

R(x) ? Q(x)
R(x) Q(x)
17
A Self-Corrector

• Goal Reduce error.
• Pick a small subspace A randomly.
• Assume we know Q on A.

18
A Self-Corrector

• Goal Reduce error.
• Pick a small subspace A randomly.
• Assume we know Q on A.

19
A Self-Corrector

• Goal Reduce error.
• We know Q on A.
• Pick b A randomly.
• Error on b A lt ¼ (very likely).

20
A Self-Corrector

• Goal Reduce error.
• We know Q on A.
• Pick b A randomly.
• Error on b A lt ¼ (very likely).
• Error on combined subspace lt 1/8.

21
A Self-Corrector

• Goal Reduce error.
• We know Q on A.
• Pick b A randomly.
• Error on b A lt ¼ (very likely).
• Error on combined subspace lt 1/8.
• Unique Decode!

22
A Self-Corrector

• Goal Reduce error.
• We know Q on A.
• Pick b A randomly.
• Error on b A lt ¼ (very likely).
• Error on combined subspace lt 1/8.
• Unique Decode!

23
A Self-Corrector

• Imagine Self-correct every shift.
• Works if error lt ¼.
• Might fail for some shifts.

24
A Self-Corrector

• Imagine Self-correct every shift.
• Works if error lt ¼.
• Might fail for some shifts.
• Can make fraction of bad shifts lt 1/8.
• Unique Decode!

25
Overall Algorithm
R0,1n ! 0,1
26
Generating our own Advice

• Advice Q restricted to A.
• A could have dimension log n.
• Only n choices for r 1.
• Too many choices when r 2.

dim(A) log(1/?) ? 1/poly(n)
27
Generating our own Advice

• Advice Q restricted to A.
• A has dimension log n.
• Error on A lt ¼.
• - List-Decode.
• - Guess from list.

• Find the list.
• Bound its size.

Q1 Q2 Q100
28
Global List-Decoding
Problem Given R 0,1k ? 0,1, find all Q of
degree 2 so that ?(Q,R) lt ¼. Run time
polynomial in block-length 2k.
29
Global List-Decoding

• Problem Given R 0,1k ? 0,1, find all Q of
  degree 2 so that ?(Q,R) lt ?.
• l(?) Worst case list-size.
• Algorithm runs in time poly. in 2k and l(?).
• Works for all ?.
• Does not imply bounds on list-size.

30
Global List-Decoding
Problem Given R 0,1k ? 0,1, find all Q so
that ?(Q,R) lt ?.
? ½(?0 ?1). Let ?0 lt ?1. So ?0 lt ?, ?1 lt 2?.
Q0 L
?1
Xk 1
Q0
?0
Q Q0(X1,,Xk-1) XkL(X1,,Xk-1)
Xk 0

• Recover Q0 from Xk 0. (degree 2, error ?).
• Recover L from Xk 1. (degree 1, error 2?).

31
Global List-Decoding
Problem Given R 0,1k ? 0,1, find all Q so
that ?(Q,R) lt ?.

• Guess a 2 0,1 s.t. Xk a has less error.
• Q Qa(X1,, Xk-1) (a Xk)L(X1,,Xk-1)
• Recover Qa from Xk a. (degree 2, error ?).
• Recover L from Xk 1 a. (degree 1, error 2?).

• Analysis Run time of 2O(k)l(?)r
• Lemma All lists bounded by l(?).

32
Generating our own Advice

• Advice Q restricted to A.
• A has dimension log n.
• Error on A lt ¼.
• - List-Decode.
• - Guess from list.

• Need to bound the list-size at radius ¼.

Q1 Q2 Q100
33
Bounds on List-Size
Problem Given R 0,1k ? 0,1, bound number of
quadratic polys. Q s.t. ?(Q,R) lt 1/4. Goal
Bound of 2O(k). Johnson bound 2O(k) for distance
J(¼) 0.156. Can we improve the distance of
RM(2,k) ?
34
A Natural Analogy
35
Distance of the Universe?
Proxima Centauri 4.2 light-years.
36
Distance of the Universe?
Within 100,000 light-years µ Milky Way.
37
Distance of the Universe?
Andromeda 2.5 million light years away.
38
Distance of the Universe?
Local Group of Galaxies, Local Supercluster,
39
Bounds on List-Size
Problem Given R 0,1k ? 0,1, bound number of
quadratic polys. Q s.t. ?(Q,R) lt 1/4. Goal
Bound of 2O(k). Johnson bound 2O(k) for distance
J(¼) 0.156. Can we improve the distance of
RM(2,k) ? Yes, by ignoring relatively few
codewords.
Thm Every quadratic form can be written as Q
L1L2 L2t-1L2t L0 where Lis are LI and 1 t
k/2.
Rank of Q
40
Rank versus Weight
Rank 2 forms. Weight 0.375. J(0.375) ¼.
Rank 1 forms. Only 22k.
Thm List-size is 2O(k) at distance ¼.
41
Bounding the List-size
R
42
Bounding the List-size
R
43
Bounding the List-size
R
44
Bounding the List-size
R
45
Bounding the List-size
R
46
Bounding the List-size
R
47
Bounding the List-size
R
Each remaining pair at dist. 0.375. List-size 2k
by Johnson bound.
48
Bounding the List-size
R
49
Bounding the List-Size.

• 2k balls by Johnson bound.
• Each ball contains 2O(k) codewords.
• Overall 2O(k) codewords at radius ¼.
• We need k O(log n) for local decoding.

50
Generating our own Advice

• Advice Q restricted to A.
• A has dimension log n.
• Error on A lt ¼.
• - List-Decode.
• - Guess from list.

• List-size poly(n) at radius ¼.

Q1 Q2 Q100
51
Overall Algorithm
R0,1n ! 0,1
52
Overall Algorithm
R0,1n ! 0,1
List-Decoder
53
Extension to Higher Degree

• No analogue of rank.
• Kasami-Tokura Characterizes codewords with
  weight 21-r.
• List-decoding up to radius 2-r - ? in poly(n,
  ?-1).

54
More Global List-Decoding
Rank c forms. Only 2ck.
Rank 1 forms. Only 22k.
Thm List-size is 2O(k) at distance ½ - ?.
55
List-Decoding for (Sm)all Fields?

• Algorithms that work up to the list-decoding
  radius.
• Unclear what that radius is.

• Two properties of the number 2
• 2 Min. Distance/Unique Decoding radius
• 2 Field size.

56
The F2 Case
Error drops by ½.
The F3 Case
Error drops by 2/3.
57
List-Decoding for (Sm)all Fields?
Over F3 ? for r even, ¾? for r odd. Over F4
0.66?, ¾?, ?.Incomparable to the Johnson bound.
Problem What is the list-decoding radius?
Thank You!