Locally correctable codes from lifting Alan Guo MIT CSAIL Joint work with Swastik Kopparty (Rutgers) and Madhu Sudan (Microsoft Research)

Locally correctable codesfrom lifting

Alan GuoMIT CSAIL

Joint work withSwastik Kopparty (Rutgers) and Madhu Sudan (Microsoft Research)

Talk outline

• Error correcting codes• Locally correctable codes• Our contributions– New high rate LCCs– General framework of “lifting” codes– New lower bounds for Nikodym sets

Talk outline

• Error correcting codes• Locally correctable codes• Our contributions– New high rate LCCs– General framework of “lifting” codes– New lower bounds for Nikodym sets

Error correcting codes

• Encoding , Code • Rate = • Distance = minimum pairwise Hamming

distance between codewords• Example: Reed-Solomon code– Message: polynomial of degree – Encoding: evaluations at distinct points

Talk outline

• Error correcting codes• Locally correctable codes• Our contributions– New high rate LCCs– General framework for “lifting” codes– New lower bounds for Nikodym sets

Locality

• Would like to do certain tasks while making sublinear number of queries to symbols of received word

• Testing: decide if or if is far from • Decoding: recover a particular symbol of

message corresponding to nearest codeword• Correcting: recover a particular symbol of the

nearest codeword

Bivariate polynomial codes

• Message: bivariate polynomial of degree

• Encoding: Evaluations on every point on plane

• Schwartz-Zippel Lemma

• ; worse than RS! Why bother?• Advantage: locality - queries to correct a

symbol

Local correctability

A brief history of LCCs

• Want high rate with sublinear query complexity for constant fraction errors

• Bivariate polynomial codes– queries, but rate – More generally, -variate polynomial codes get us

queries, but rate • Multiplicity codes (Kopparty, Saraf, Yekhanin 2010)– Encode polynomial evaluations as well as derivatives– Can achieve queries with rate close to 1

Talk outline

• Error correcting codes• Locally correctable codes• Our contributions– New high rate LCCs– General framework for “lifting” codes– New lower bounds for Nikodym sets

Our contributions

• New codes with queries and rate close to 1• General study of “lifted codes”• New lower bounds for Nikodym sets

Talk outline

• Error correcting codes• Locally correctable codes• Our contributions– New high rate LCCs– General framework for “lifting” codes– New lower bounds for Nikodym sets

Main idea

• New code (lifted RS code)– Codewords = {bivariate polynomials whose

restrictions to lines are polynomials of deg }– Contains bivariate polynomials of deg , but

sometimes many more codewords• Code has basis of monomials • Characterize which belong in code• Lower bound rate of code by lower bounding

number of such

Main idea

• Example:, has degree but on each line looks like degree

because in , i.e. polynomials are only distinguishable modulo by looking at evaluations in

Main idea

• New code (lifted RS code)– Codewords = {bivariate polynomials whose

restrictions to lines are polynomials of deg }– Contains bivariate polynomials of deg , but

sometimes many more codewords• Code has basis of monomials • Characterize which belong in code• Lower bound rate of code by lower bounding

number of such

Dimension of lifted RS code

• Shadows, and Lucas’ Theorem– Let denote base expansion– Shadow: if for every – Lucas’ Theorem only if which implies

Dimension of lifted RS code

• Example:

– Over field of characteristic 2,

Dimension of lifted RS code

• When is in lifted code?• Expand:

• So is in lift iff for every and , where

Dimension of lifted RS code

• ,

Reed-Muller Lifted Reed-Solomon

𝑖 𝑖

𝑗 𝑗

Dimension of lifted RS code

• ,

Reed-Muller Lifted Reed-Solomon

𝑖 𝑖

𝑗 𝑗

Dimension of lifted RS code

• ,

Reed-Muller Lifted Reed-Solomon

𝑖 𝑖

𝑗 𝑗

Dimension of lifted RS code

• ,

Reed-Muller Lifted Reed-Solomon

𝑖 𝑖

𝑗 𝑗

Talk outline

• Error correcting codes• Locally correctable codes• Our contributions– New high rate LCCs– General framework for “lifting” codes– New lower bounds for Nikodym sets

General results

• Affine-invariant codes– for affine permutation

• Lifts– Restrictions to low-dim affine subspaces are

codewords in “base code”– Good distance– Good locality– Only need to analyze dimension

Talk outline

• Error correcting codes• Locally correctable codes• Our contributions– New high rate LCCs– General framework for “lifting” codes– New lower bounds for Nikodym sets

Application to Nikodym sets

• Multivariate polynomials outside of coding theory• Polynomial method (Dvir’s analysis of Kakeya sets)• Nikodym set

– For every point , there is a line through which is contained in the set, except possibly

– Can get lower bound of using polynomial method– Using multiplicity codes, can get bound – Using lifted codes, can get bound

Application to Nikodym sets

• Polynomial method– Assume dimension of

{-variate polynomial code of deg }– Exists nonzero vanishing identically on – actually vanishes everywhere!• Let • Exists line through that intersects in points• vanishes at points, but has deg • , so

Application to Nikodym sets

• Improved polynomial method– Assume dimension of

{lifted RS code of deg }– Exists nonzero vanishing identically on – actually vanishes everywhere!• Let • Exists line through that intersects in points• vanishes at points, but has deg • , so

Summary

• Lifting– Natural operation– Build longer codes from short ones– Preserve distance– Gain locality– Can get high rate

• Applications outside of coding theory– Improve polynomial method (e.g. Nikodym sets)

Thank you!