Locally correctable codes from lifting
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Transcript of 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!