lovett_RM.pptx

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Structure and pseudo-randomness in coding theory: List decoding Reed-Muller codes up to minimal distance

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What this talk is about?

• Technically: new understanding of a basic and important

family of codes

• Conceptually: structure and pseudo-randomness play

important roles in many computational domains.

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Overview

• Coding theory 101

• Regularity in coding theory

• Structural properties of polynomials

• Pseudo-randomness for polynomials

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Overview

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Decoding from errors

• The basic problem of coding theory: recovering from errors

• Goal: recover correct codeword from a noisy received word

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Unique decoding

Codeword

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Unique decoding

• Unique decoding: find the closest codeword

• Basic limitation: minimal distance of the code

• If a received word is “in between” two codewords,

we cannot distinguish which is the correct codeword

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Unique decoding

Codeword

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List decoding

• List decoding: find few closest codewords

[Elias ‘57]

• Circumvents the ½ minimal distance problem

• In general, can recover from errors up to Johnson bound

½ minimal distance < Johnson bound < minimal distance

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List decoding

Codeword

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Polynomial codes

• Most codes are based on polynomials

• In this talk, focus on the most basic families

• Reed-Solomon: univariate polynomials

• Reed-Muller: multivariate polynomials

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Why polynomial codes?

• Polynomial codes are “special”

• Do they behave better than “worst-case” analysis?

• Concretely: are they list decodable beyond the Johnson bound?

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Reed-Muller codes

• Reed-Muller codes = multivariate polynomials over finite fields

• _RM_F_(n,d)

• F – finite field

• n – number of variables • d – degree

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Minimal distance of Reed-Muller codes

• Distance: for , fraction of points where they differ

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Minimal distance of Reed-Muller codes

• Distance: for , fraction of points where they differ

• Minimal distance of a code C:

• For RM_F(n,d):

• If d<|F| then

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Hadamard codes

• Hadamard codes correspond to d=1 (linear functions)

• Minimal distance=1-1/|F|

• List decodable up to distance 1-1/|F|

[Goldreich-Levin’89, Goldreich-Rubinfeld-Sudan’00]

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Going beyond linear functions

• Our understanding of Reed-Muller codes for degrees d>1

depends on the field size

• Here, will discuss two extreme cases:

• Large fields: |F| >> d

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Large fields

• Large fields |F|>>d: minimal distance 1-d/|F|

• ½ minimal distance: (useless) • Johnson bound:

• List decodable up to , algorithmically [Sudan’97,…,Sudan-Trevisan-Vadhan’01]

• Open problem: can they handle more errors? Maybe up to the

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Small fields

• _{Breakthrough in 2008: Over F}₂_{, RM codes are list decodable}

up to minimal distance (combinatorially & algorithmically) [Gopalan-Klivans-Zuckerman’08]

• Proof doesn’t extend to larger fields: uses special

properties of the Johnson bound over F₂

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GKZ conjecture

• GKZ conjecture: RM codes are list decodable up to minimal distance over all fields

• True for d=1 [Goldreich-Levin’89, Goldreich-Rubinfeld-Sudan’00] • True for d=2 [Gopalan ‘10]

• True if p-1|d [Gopalan-Klivans-Zuckerman’08]

• This work: true for any constant , constant prime fields

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Main result (this work)

• _{Theorem: for any constant , constant prime field F, RM}_F_(n,d)

is list decodable up to its minimal distance

• That is: for any function , any

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Extension to large fields (in progress)

• _{Theorem: for any fixed , any prime field F, RM}_F_{(n,d) is} list decodable up to minimal distance

• That is: for any function , any

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Proof idea

• _{Theorem: for any constant , constant prime field F, RM}_F_(n,d)

is list decodable up to its minimal distance

• Proof introduces some new tools to coding theory

1. Regularity for codes

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Overview

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The list decoding problem, revisited

• Code: family of functions

• Received word: function

• Ball around received word:

• Goal: upper bound for all g, for

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Regularity for list decoding

•

Lemma: for any code,

any received word

can be

replaced by a

“low complexity” received word

, which

is indistinguishable from the code perspective

•

Similar to the Frieze-Kannan weak regularity

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Regularity for list decoding

Lemma: for any

• Code

• Received word

• Error term

There exists a “low complexity” received word such that*

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(2) g’ is a function of codewords:

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Regularity for list decoding

Codeword Original

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Proof of regularity lemma

• Proof very similar to the proof of the Frieze-Kannan weak regularity

lemma

• Maintain a randomized function such that

• Initially, g’ is constant (the global distribution of g) • If false for some , refine g’ to take f into account

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Upshot

• Using the lemma, we may assume from now on that our

received word is of low complexity

where are low degree polynomials, and is some combiner function

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Overview

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A very special case

(which will turn out to be not so special)

• Received word

• _{Lets assume a very strong}_{structural property}_{: each f}_i

depends only on a few variables

• So:

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Rethinking minimal distance

• Very very special case: g=0

• If is such that then f=0

• Question: can we hope for a similar phenomena, when g

depends on a few coordinates?

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A structural lemma

• Let

• Lemma: if has , then

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Proof of structural lemma

• Proof very similar to the Schwarz-Zippel lemma

• _{Assume f depends on x}_n • Expand:

• _{g(x) independent of x}_n

•

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Overview

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Pseudo-random polynomials

• Definition: a polynomial f(x) is pseudo-random if it

cannot be approximated by any lower degree polynomial

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Pseudo-random polynomials: examples

• Linear functions: always pseudo-random, eg

• Quadratics: high-rank quadratics are pseudo-random, eg

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A dichotomy theorem

• Low degree polynomials exhibit a very strong dichotomy:

if they are not pseudo-random, they are very structured

• Theorem: If f(x) is a polynomial which can be (even slightly) be

approximated by a lower degree polynomial, then it can be decomposed as a function of a few lower degree polynomials

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Decompositions of polynomials

• The dichotomy theorem can be applied iteratively, to

decompose any low-degree polynomial as a function of a few polynomials which are pseudo-random

• To a large extent, pseudo-random polynomials behave as

“independent variables”

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Going back to the very special case

• Higher-order Fourier analysis allows us to assume that

where are pseudo-random polynomials.

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Overview

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Result

• Reed-Muller codes are special: can be list decoded up to

minimal distance (for constant degrees, fields)

• Proof relies on three ingredients:

1. Regularity for codes

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Follow up work

• We extend the current result to the case of large fields

• This requires a few new ingredients:

1. Optimizing the arguments, to get a polynomial dependency on the field size

2. Extending higher-order Fourier to large fields, with bounds independent of field size

3. “list decoding” pseudo-random decompositions:

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Take home message

• Notions of structure and pseudo-randomness are very powerful; dichotomy theorems make them universal

• This work: coding theory, applied to RM codes

• Other applications: math - graph theory, number theory,

ergodic theory, discrete geometry; CS - property testing, complexity, algorithms

• Question: do our techniques generalize to other codes?

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