Core Concepts

The paper establishes tight lower bounds on the block length of binary 3-query locally correctable codes (3-LCCs). It proves a sharp lower bound for design 3-LCCs that matches the best known construction up to a constant factor. It also obtains superpolynomial lower bounds for smooth 3-LCCs with high completeness.

Abstract

The paper studies the problem of proving lower bounds on the block length of binary 3-query locally correctable codes (3-LCCs). It makes the following key contributions:
Tight Lower Bounds for Design 3-LCCs:
The paper proves that for any binary linear design 3-LCC, the block length n satisfies n ≥ 2^((1-o(1))√k), where k is the message length.
This bound is tight up to a √8 factor in the exponent compared to the best known construction of design 3-LCCs.
As a corollary, the paper confirms the Hamada conjecture for 2-(n, 4, 1) designs up to a factor of 8 in the co-dimension.
Superpolynomial Lower Bounds for Smooth 3-LCCs:
The paper proves that for any δ-smooth (possibly non-linear) 3-LCC with completeness 1-ε, where ε ≤ γ/√log^2 n for some constant γ, the block length n satisfies n ≥ (k')^Ω(log k'), where k' = δ^3 k/log(1/δ).
When ε is a small constant, this implies a lower bound for general non-linear 3-LCCs that beats the prior best n ≥ ~Ω(k^3) lower bound by a polynomial factor.
The key technical ideas include:
A fine-grained analysis of the Kikuchi matrix method for design 3-LCCs to obtain the tight lower bound.
A new reduction from non-linear smooth 3-LCCs to a system of "chain polynomial equations" and spectral refutation via Kikuchi matrices to prove the superpolynomial lower bounds.

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by Pravesh K. K... at **arxiv.org** 04-10-2024

Deeper Inquiries

The tight lower bounds for design 3-LCCs have significant implications for the construction of optimal binary 3-LCCs. By proving that the block length of any binary linear 3-LCC with design properties must be at least 2(1−o(1))√k, the authors have essentially shown that Reed-Muller codes, which are design 3-LCCs, are optimal up to a factor of √8 in the exponent. This result confirms that Reed-Muller codes achieve the smallest possible block length for binary 3-LCCs, highlighting their optimality in terms of correcting errors locally. The implications of this tight lower bound suggest that further improvements in constructing binary 3-LCCs would need to focus on enhancing the design properties to achieve better performance in terms of block length and error correction capabilities.

The techniques used to prove superpolynomial lower bounds for smooth 3-LCCs may not directly translate to obtaining improved lower bounds for general (non-smooth) 3-LCCs. The key distinction lies in the properties and characteristics of smooth codes compared to general codes. Smooth codes have the additional property of near-perfect completeness and no codeword bit being queried with a high probability, making them more structured and predictable in their behavior. The reduction from smooth codes to a system of "chain polynomial equations" relies on the specific structure and adaptiveness of smooth codes, which may not be applicable to general codes that do not exhibit the same level of smoothness and completeness. Therefore, extending these techniques to general 3-LCCs would require a different approach that accounts for the lack of smoothness and adaptiveness in non-smooth codes.

The structural properties of design 3-LCCs, compared to general 3-LCCs, play a crucial role in enabling the authors to obtain such tight lower bounds. Design 3-LCCs have specific properties that make them more constrained and structured in their correction capabilities. For example, in a design 3-LCC, the correcting sets for every codeword bit form a perfect matching, and every pair of codeword bits is queried an equal number of times across all matchings. These properties create a more uniform and balanced structure within the code, allowing for a more precise analysis and tighter lower bounds. By leveraging the unique characteristics of design 3-LCCs, the authors were able to conduct a fine-grained analysis using the Kikuchi matrix method and derive the sharp lower bounds for these codes. In contrast, general 3-LCCs lack such specific structural constraints, making it more challenging to achieve the same level of precision in lower bound proofs.

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