locally correctable codes from lifting

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!