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List decoding Reed-Muller codes up to minimal distance: Structure and pseudo- randomness in coding theory Abhishek Bhowmick (UT Austin) Shachar Lovett (UC San Diego)

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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. This talk shows this phenomena applied to coding theory

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Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

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Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

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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 This work: worst-case errors

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Unique decoding Codeword Received word

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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 Limits error to <½ the minimal distance

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Unique decoding Codeword Received word

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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 For special codes (hmmm…) can do better

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List decoding Codeword Received word

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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 Despite (or because) being basic, they are widely applied; however, they are far from fully understood

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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? Previous works: yes This work: YES

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Reed-Muller codes

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Minimal distance of Reed-Muller codes

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Hadamard codes Hadamard codes correspond to d=1 (linear functions) Minimal distance=1/|F| List decodable up to error 1/|F| [Goldreich-Levin’89, Goldreich-Rubinfeld-Sudan’00] Proof: Fourier analysis

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Large fields

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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 Johnson bound over binary fields GKZ conjecture: all RM codes are list decodable up to minimal distance (they are all “special”)

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GKZ conjecture

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Main result (this work)

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Extension to large fields (in progress)

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Proof idea

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Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

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The list decoding problem, revisited

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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 Viewpoint: codewords are “tests”

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Regularity for list decoding

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Codeword Original received word Low complexity received word

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Proof of regularity lemma

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Upshot

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Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

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A very special case (which will turn out to be not so special)

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Rethinking minimal distance

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A structural lemma

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Proof of structural lemma

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Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

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Pseudo-random polynomials

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Pseudo-random polynomials: examples

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A dichotomy theorem

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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” Made precise in higher-order Fourier analysis

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Going back to the very special case

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Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

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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 2.Structural property of RM codes (polynomials) 3.Pseudo-randomness for RM codes (polynomials)

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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

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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? Other domains?

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Thank you!

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