Locally Decodable Codes of fixed number of queries and Subexponential Length

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Locally Decodable Codesof fixed number of
queries and Sub-exponential Length

• Article By Klim Efremenko
• Presented by Inon Peled
• 30 November 2008

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Intuition Locally Decodable Code
LDCs have applications in cryptography,
complexity theory. Example - Public Key
Cryptograohy http//www.cs.ucla.edu/rafail/PUBLI
C/95.pdf
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Structure of Presentation

• The rest of the presentation is organized as
  follows
• Formal definition for LDC.
• Example LDC Hadamard code.
• Construction of LDCs with fixed queries and
  sub-exp. codeword length.
• Construction of such binary LDCs.

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Definition Locally Decodable Code
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Definition Locally Decodable Code
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Next Topic

• Formal definition for LDC.
• Example LDC Hadamard code.
• Construction of LDCs with fixed queries and
  sub-exp. codeword length.
• Construction of such binary LDCs.

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Example Hadamard Code
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Example Hadamard Code
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Example Hadamard Code
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Example Hadamard Code - decoding
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Example Hadamard Code - completeness
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Example Hadamard Code - parameters
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Non-adaptive, Linear
The Hadamard code, as well as every code
that we will present later is Non-adaptive
makes all queries at once. And so cannot
adapt its queries one after another. Linear
a linear transformation.
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Next Topic

• Formal definition for LDC.
• Example LDC Hadamard code.
• Construction of LDCs with fixed queries and
  sub-exp. codeword length.
• Construction of such binary LDCs.

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Stages of Constructing our LDC
The construction of our LDC begins with
fixing a constant m, such that
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Stages of Constructing our LDC
The construction of our LDC begins with
fixing a constant m, such that
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Definition 1 S-Matching Vectors
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Definition 1 S-Matching Vectors
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Definition 1 S-Matching Vectors
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Example S-Matching Vectors
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Definition 2 ?, group generator
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Definition 2 ?, group generator
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Recap
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At last, the LDC !
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At last, the LDC !
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At last, the LDC !
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Decoding
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Definition S-decoding Polynomial
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Example S-decoding Polynomial
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The LDC, Decoding
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The LDC, Decoding
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Perfectly Smooth Decoder
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Perfectly Smooth Decoder, Cont.
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Success Probability of C

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Success Probability of C

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A (3,d,3d)-LDC

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Building ui, h, n

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Theorem, Grolmusz 2000
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Theorem, Grolmusz 2000
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Grolmusz ? ui

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Grolmusz ? n, h

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Next Topic

• Formal definition for LDC.
• Example LDC Hadamard code.
• Construction of LDCs with fixed queries and
  sub-exp. codeword length.
• Construction of such binary LDCs.

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Extension to Binary LDCs
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Binary LDC - Encoding
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Binary LDC - Encoding
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Binary LDCs - Decoding
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Binary LDCs - Decoding
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Completeness of dibin
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Smoothness of dibin
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Smoothness of dibin Cont.
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Smoothness of dibin Cont.
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Smoothness of dibin Cont.
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LDC Parameters of Cbin
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LDC Parameters of Cbin
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Summary

• Weve presented
• What a locally decodable code (LDC) is .
• The famous and popular Hadamard code.
• How to construct LDCs with fixed queries and
  sub-exp. codeword length.
• How to extend the construction to binary LDCs.

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