parikshit gopalan microsoft adam r. klivans ut austin david zuckerman ut austin

Parikshit GopalanParikshit Gopalan Microsoft Microsoft Adam R. KlivansAdam R. Klivans UT AustinUT Austin

David ZuckermanDavid Zuckerman UT AustinUT Austin

0 1

0

0 1

1

1 0

List-Decoding Reed-Muller List-Decoding Reed-Muller Codes over Small FieldsCodes over Small Fields

Error Correcting CodesError Correcting Codes

Communication over a Noisy Channel:Communication over a Noisy Channel:

Adversary corrupts 10% of the bits.Adversary corrupts 10% of the bits.

Problem:Problem: Recover the (entire) message.Recover the (entire) message.

Soln:Soln: Introduce redundancyIntroduce redundancy.

Error-Correcting CodesError-Correcting Codes

Cellphones

Satellite Broadcast

Deep-space communicationInternet

Audio CDs Bar-codes

Codes from PolynomialsCodes from PolynomialsEncoding: Alice wants to send (a,b).

Let L(x) = ax +b.

Send L(1), L(2), …, L(7).

Codes from PolynomialsCodes from PolynomialsAdversary: Corrupts two values.

Decoding: Find the (unique) line that passes through 5 points.

Codes from PolynomialsCodes from Polynomials Low-degree polynomials differ in many Low-degree polynomials differ in many places.places.

Relative distanceRelative distance :: Hamming Hamming distance/lengthdistance/length

min distance:min distance: min {min {C,C’)|codewords C,C’)|codewords C,C’}C,C’}

Codes from PolynomialsCodes from Polynomials Low-degree polynomials differ in many Low-degree polynomials differ in many places.places.

Relative distanceRelative distance :: Hamming Hamming distance/lengthdistance/length

min distance:min distance: min {min {C,C’)|codewords C,C’)|codewords C,C’}C,C’}

Reed-Solomon codes: Univariate polynomials.

Reed-Muller codes: Multivariate polynomials.

Messages:Messages: Polynomials of degree Polynomials of degree rr in in mm variables over variables over {0,1}{0,1}..

– Q(XQ(X11,X,X22,X,X33) = X) = X11XX22 + X + X33

Encoding: Encoding: Truth table.Truth table.– 0101011001010110

Minimum distance:Minimum distance: = 2 = 2-r-r..

Hadamard codes:Hadamard codes: r =1r =1..

Reed-Muller Codes Reed-Muller Codes [Muller’54, [Muller’54, Reed’54]Reed’54]

0 1

0

0 1

1

1 0

Messages:Messages: Polynomials of degree Polynomials of degree rr in in mm variables over variables over {0,1}{0,1}..

– Q(XQ(X11,X,X22,X,X33) = X) = X11XX22 + X + X33

Encoding: Encoding: Truth table.Truth table.– 0101011001010110

Minimum distance:Minimum distance: = 2 = 2-r-r..

Hadamard codes:Hadamard codes: r =1r =1..

Reed-Muller Codes Reed-Muller Codes [Muller’54, [Muller’54, Reed’54]Reed’54]

1 1

0

0 0

1

0 1

Decoding Decoding ´́ Polynomial Polynomial ReconstructionReconstruction

Problem: Given data points, find a low degree polynomial that fits best.

Well studied problem, numerous applications.

1 1

0

0 0

1

0 1

0 1

0

0 1

1

0 1

The Decoding ProblemThe Decoding Problem

Received work Received work R:{0,1}R:{0,1}mm {0,1} {0,1}..

Unique Decoding:Unique Decoding: Find Find CC such that such that (R,C) < (R,C) < /2/2..

List Decoding: List Decoding: [Elias’57, Wozencraft’58] [Elias’57, Wozencraft’58] Find all Find all CC such that such that (R,C) < (R,C) < ..Few such Few such CC..

Johnson bound:Johnson bound: List is small up to List is small up to J(J() ) wherewhere

J(J() = (1-√(1-2) = (1-√(1-2))/2 = ))/2 = 22/2 + ... < /2 + ... <

CC

C’C’

The Computational ModelThe Computational Model

Global Decoding:Global Decoding:

0 1 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 1

Given R as input. Run time poly in n = 2m.

R

Local Local Decoding:Decoding:

Given an oracle for R.

Run time poly in m = log n.

Rx R(x)

Decoding Reed-Muller Decoding Reed-Muller codescodes

Unique Decoding:Unique Decoding: Majority Logic Decoder. Majority Logic Decoder. [Reed’54][Reed’54]

Local List Decoding:Local List Decoding:

Hadamard codes Hadamard codes (r = 1)(r = 1).. [Goldreich-[Goldreich-Levin’89] Levin’89]

Alternate algorithms:Alternate algorithms: [ Levin, Rackoff, [ Levin, Rackoff, Kushilevitz-Mansour, …]Kushilevitz-Mansour, …]

No algorithms known forNo algorithms known for r r ¸̧ 2 2..Good algorithms for large fields Good algorithms for large fields (r < |(r < |

F|)F|).. [Goldreich-Rubinfeld-Sudan, Arora-[Goldreich-Rubinfeld-Sudan, Arora-Sudan, Sudan-Trevisan-Vadhan]Sudan, Sudan-Trevisan-Vadhan]

Our ResultsOur ResultsMain Result: Local List-Decoding RM codes for r ¸ 2. Works up to Minimum Distance 2-r - - .

Returns list of size Returns list of size -O(r)-O(r) in time in time poly(mpoly(mrr, , -r-r))..

Improves Improves Majority Logic DecodingMajority Logic Decoding for for r r ¸̧ 2 2..

Generalizes Generalizes [Goldreich-Levin’89][Goldreich-Levin’89]. .

Beats the Beats the Johnson boundJohnson bound..–For For r =2r =2,, 0.146 0.146 versus versus 0.250.25..

List-size becomes exponential at List-size becomes exponential at 22-r-r..

Our ResultsOur Results

Global List-Decoding:Global List-Decoding: Deterministic algorithm for Deterministic algorithm for r r ¸̧ 2 2.. Works up to distance Works up to distance J(2J(2..22-r-r) - ) - . .

– Beyond the minimum distance.Beyond the minimum distance.– For For r =2r =2,, ½ - ½ - versus versus ¼ - ¼ - ..

Returns list of size Returns list of size -O(m)-O(m) in time in time poly(poly(-O(m)-O(m)))..– Brute force needs time Brute force needs time O(2O(2mmrr))..

New combinatorial bound.New combinatorial bound.

Local List-DecodingLocal List-Decoding

{0,1}{0,1}mm labeled bylabeled by received word received word RR..

Fix codeword Fix codeword QQ so that so that (Q,R) < (Q,R) < - - . .

Local List-DecodingLocal List-Decoding

{0,1}{0,1}nn labeled bylabeled by received word received word RR..

Fix codeword Fix codeword QQ so that so that (Q,R) < (Q,R) < - - . .

R(x) R(x) Q(x) Q(x)

R(x) = Q(x)R(x) = Q(x)

A Self-Corrector A Self-Corrector [Goldreich-[Goldreich-Levin]Levin]

Goal:Goal: Find Find Q(b)Q(b) whp.whp.

Pick a small Pick a small subspace subspace AA randomly.randomly.

AssumeAssume we know we know QQ on on AA. .

bb


Goal:Goal: Find Find Q(b)Q(b) whp.whp.

Pick a small Pick a small subspace subspace AA randomly.randomly.

AssumeAssume we know we know QQ on on AA..

bb


Goal:Goal: Find Find Q(b)Q(b) whp. whp.

Pick Pick AA randomly.randomly.

We know We know QQ on on AA..

Error onError on b + A < b + A < (very likely).(very likely).

bb





Error onError on b + A < b + A < (very likely)(very likely)..

Error on combined Error on combined subspacesubspace < < /2/2..





Error onError on b + A < b + A < (very likely)(very likely)..

Error on combined Error on combined subspacesubspace < < /2/2..

Unique Decode!Unique Decode!

Interpolating SetsInterpolating Sets

QQ of degreeof degree r r efficiently computable efficiently computable fromfrom Q(b), b Q(b), b B=B(r) B=B(r).. r=1:r=1: 0, e0, e11, e, e22,…, e,…, emm.. GeneralGeneral r: r: allall b b of weightof weight r. r.

Pick one random Pick one random AA. Use . Use AA to self- to self-correct all correct all bb in interpolating set in interpolating set BB..

Union bound Union bound whp correct on all of whp correct on all of BB.. Can improve via Can improve via Noisy Interpolating SetsNoisy Interpolating Sets

[Dvir,Shpilka][Dvir,Shpilka]..

Self-Corrector

Overall AlgorithmOverall Algorithm

R:{0,1}m ! {0,1}

Interpolator

advice

Generating our own AdviceGenerating our own Advice

Advice: Q Advice: Q restricted torestricted to A. A.

AA could have could have dimension dimension log mlog m..

Only Only mm choices choices for for r =1r =1..

Too many Too many choices when choices when r r ¸̧ 2 2..

dim(A) = log(1/)= 1/poly(m)

Generating our own AdviceGenerating our own Advice

Advice: Q Advice: Q restricted torestricted to A. A.

AA could have could have dimension dimension k=log k=log mm..

Error on Error on AA is is <<, , whp.whp.

Decode on Decode on AA in in time time poly(2poly(2kk))..

Global List-Decoding: Case Global List-Decoding: Case r=2r=2

Problem:Problem: GivenGiven R: {0,1}R: {0,1}kk {0,1} {0,1}, find all, find all QQ of degreeof degree 22 so thatso that (Q,R) < ¼(Q,R) < ¼. .

Run time polynomial in block-lengthRun time polynomial in block-length 2 2kk..


Problem:Problem: GivenGiven R: {0,1}R: {0,1}kk {0,1} {0,1}, find all, find all QQ of degreeof degree 22 so thatso that (Q,R) < (Q,R) < ..

ll(():): Worst case list-size.Worst case list-size.

Algorithm runs in time Algorithm runs in time poly(2poly(2kk,,ll(())))..

Works for allWorks for all ..

Does not imply bounds on list-size.Does not imply bounds on list-size.


Problem:Problem: GivenGiven R: {0,1}R: {0,1}kk {0,1} {0,1}, find all, find all QQ so thatso that (Q,R) < (Q,R) < . .

Xk = 0

Xk = 1

0

1

= ½(= ½(00 + + 11).).

LetLet 00 11. .

SoSo 00 , , 11 2 2

RecoverRecover QQ00 fromfrom XXkk = 0 = 0. . (degree(degree 22, error, error ).).

RecoverRecover L L fromfrom XXkk = 1 = 1. . (degree(degree 11, error, error 22).).

Q0

Q0 + L

Q=QQ=Q00(X(X11,…,X,…,Xk-1k-1) + X) + XkkL(XL(X11,…,X,…,Xk-k-

11))


Problem:Problem: GivenGiven R: {0,1}R: {0,1}kk {0,1} {0,1}, find all, find all QQ so thatso that (Q,R) < (Q,R) < . .

Xk = 0

Xk = 1

0

1

= ½(= ½(00 + + 11).).

LetLet 00 11. .

Don’t know whetherDon’t know whether 00 11

Try all possibilities.Try all possibilities.

Overhead is Overhead is 22k k ..

Q0

Q0 + L

Bounds on List-SizeBounds on List-Size

Problem:Problem: GivenGiven R: {0,1}R: {0,1}kk {0,1} {0,1}, bound , bound number of quadratic polys.number of quadratic polys. QQ s.t.s.t. (Q,R) < (Q,R) < 1/41/4. .

Goal: Goal: Bound ofBound of 22O(k)O(k)..

Johnson bound: Johnson bound: 22O(k)O(k) for distance for distance J(¼) = J(¼) = 0.1560.156..

Can we improve the distance ofCan we improve the distance of RM(2,k) RM(2,k) ??

Analogy: Inter-Star Analogy: Inter-Star DistanceDistance

Proxima Centauri:Proxima Centauri: 4.2 light-4.2 light-years.years.

Inter-Star Distance?Inter-Star Distance?

Within 100,000 light-years Within 100,000 light-years µµ Milky WayMilky Way..

Intergalactic DistanceIntergalactic Distance

Andromeda:Andromeda: 2.5 million light years away. 2.5 million light years away.

Inter-Star Distance?Inter-Star Distance?

Local Group of Galaxies, Local Supercluster, Local Group of Galaxies, Local Supercluster, ……

Bounds on List-SizeBounds on List-SizeProblem:Problem: GivenGiven R: {0,1}R: {0,1}kk {0,1} {0,1}, bound , bound number of quadratic polys.number of quadratic polys. QQ s.t.s.t. (Q,R) < (Q,R) < 1/41/4. .

Goal: Goal: Bound ofBound of 22O(k)O(k)..

Johnson bound: Johnson bound: 22O(k)O(k) for distance for distance J(¼) = J(¼) = 0.1560.156..

Can we improve the distance ofCan we improve the distance of RM(2,k) RM(2,k) ??

Yes, for a Yes, for a 22-O(k)-O(k)-dense subset of -dense subset of RM(2,k)RM(2,k)..Thm:Thm: Every quadratic form can be written Every quadratic form can be written asas Q = LQ = L11LL22 + …L + …L2t-12t-1LL2t2t + L + L00

wherewhere L Liis are LI ands are LI and 1 1 ·· t t ·· k/2 k/2. .

Rank of Q

Rank versus WeightRank versus WeightWeight Distribution of Quadratic Forms

0.05 0.15 0.25 0.35 0.45 0.5 0.55 0.65 0.75 0.85 0.95

Distance

Number of Codewords

Thm:Thm: List-size is List-size is 22O(k)O(k) at distance at distance ¼¼..

Rank 1 forms.

Only 22k.

Rank 2 forms.Weight 0.375.

J(0.375) = ¼.

Bounding the List-sizeBounding the List-size

R

R


R


Each remaining pair at dist.Each remaining pair at dist. 0.3750.375..

List-sizeList-size 22kk by Johnson bound.by Johnson bound.

R


Bounding the List-Size.Bounding the List-Size.

• 22kk balls by Johnson bound.balls by Johnson bound.

• Each ball contains at mostEach ball contains at most 222k2k codewords.codewords.

• Overall at most Overall at most 223k3k codewords at radius codewords at radius ¼¼..

We need We need k = O(log m)k = O(log m) for local decoding. for local decoding.

Self-Corrector

Overall Local List-DecoderOverall Local List-Decoder

R:{0,1}m ! {0,1}

Interpolator

advice

Self-Corrector

Overall Local List-DecoderOverall Local List-Decoder

R:{0,1}m ! {0,1}

Interpolator

Global List-Decoder

Extension to Higher DegreeExtension to Higher Degree

No analogue of rank.No analogue of rank.[Kasami-Tokura]:[Kasami-Tokura]: Characterizes codewords Characterizes codewords with weight with weight ·· 2 21-r1-r.. List-decoding up to radiusList-decoding up to radius 22-r-r - - in in poly(m, poly(m, -1-1))..

More Global List-Decoding More Global List-Decoding ……

Weight Distribution of Quadratic Forms

0.05 0.15 0.25 0.35 0.45 0.5 0.55 0.65 0.75 0.85 0.95

Distance

Number of Codewords

Thm:Thm: List-size is List-size is 22O(k)O(k) at distance at distance ½ - ½ - ..

Rank 1 forms.

Only 22k.

Rank c forms.Only 22ck.

List-Decoding for (Sm)all List-Decoding for (Sm)all Fields?Fields?

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

Key property of the field size 2:Key property of the field size 2:

2 = Min. Distance/Unique Decoding radius2 = Min. Distance/Unique Decoding radius

The FThe F22 Case Case

The FThe F33 Case Case

Error drops by

2/3.

Error drops by

½.

List-Decoding for (Sm)all List-Decoding for (Sm)all Fields?Fields?

Problem:Problem: What is the list-decoding radius?What is the list-decoding radius?

Lies between q/(q-1).and .

Conjecture: It approaches .

List-decoding List-decoding radius for degreeradius for degree rr

Unique-decoding Unique-decoding radius for degreeradius for degree r-1r-1≥

Over FOver F33:: for for rr even, even, ¾¾for for rr odd. odd.Over FOver F44:: 0.660.66, ¾, ¾, , Incomparable to the Johnson bound.Incomparable to the Johnson bound.

Future DirectionsFuture Directions

List-decoding radius for general fields?List-decoding radius for general fields?-resolve conjecture. -resolve conjecture. Gopalan: r=2Gopalan: r=2..

RM beyond min distance: nearest codeword?RM beyond min distance: nearest codeword?Other codes list-decodable past Johnson Other codes list-decodable past Johnson bound?bound?

- Extractor codes - Extractor codes [Ta-Shma, Z][Ta-Shma, Z]

- Folded RS codes - Folded RS codes [Guruswami-Rudra, Parvaresh-[Guruswami-Rudra, Parvaresh-Vardy]Vardy]

- Group homomorphism codes - Group homomorphism codes [Dinur, Grigorescu, [Dinur, Grigorescu, Kopparty, Sudan]Kopparty, Sudan]

- Tensor product, interleaved codes - Tensor product, interleaved codes [Gopalan, [Gopalan, Guruswami, Raghavendra]Guruswami, Raghavendra]

parikshit gopalan microsoft adam r. klivans ut austin david zuckerman ut austin

Documents