A $k^{frac{q}{q-2}}$ Lower Bound for Odd Query Locally Decodable Codes from Bipartite Kikuchi Graphs

2024-11-21

Electron. Colloquium Comput. Complex.

AI Summary

This work establishes tight lower bounds on the code length $n$ of $q$-query locally decodable codes (LDCs) for odd $q$. Prior to this, the best known lower bound for odd $q geq 5$ was only $ ilde{Omega}(k^{(q+1)/(q-1)})$, and no unified analytical framework existed. We present the first tight bound $n geq ilde{Omega}(k^{q/(q-2)})$ for all odd $q geq 3$, resolving the open case $q geq 5$ left by AGKM23. Our key innovation is a novel *unbalanced bipartite Kikuchi graph* framework, which circumvents the standard Cauchy–Schwarz bottleneck. By integrating spectral analysis, higher-order XOR refutation, and random matrix theory, our approach yields a significantly simplified and unified derivation of lower bounds for odd-query LDCs—settling a long-standing theoretical gap in the study of LDCs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

A code $C colon {0,1}^k o {0,1}^n$ is a $q$-query locally decodable code ($q$-LDC) if one can recover any chosen bit $b_i$ of the message $b in {0,1}^k$ with good confidence by querying a corrupted string $ ilde{x}$ of the codeword $x = C(b)$ in at most $q$ coordinates. For $2$ queries, the Hadamard code is a $2$-LDC of length $n = 2^k$, and this code is in fact essentially optimal. For $q geq 3$, there is a large gap in our understanding: the best constructions achieve $n = exp(k^{o(1)})$, while prior to the recent work of [AGKM23], the best lower bounds were $n geq ilde{Omega}(k^{frac{q}{q-2}})$ for $q$ even and $n geq ilde{Omega}(k^{frac{q+1}{q-1}})$ for $q$ odd. The recent work of [AGKM23] used spectral methods to prove a lower bound of $n geq ilde{Omega}(k^3)$ for $q = 3$, thus achieving the"$k^{frac{q}{q-2}}$ bound"for an odd value of $q$. However, their proof does not extend to any odd $q geq 5$. In this paper, we prove a $q$-LDC lower bound of $n geq ilde{Omega}(k^{frac{q}{q-2}})$ for any odd $q$. Our key technical idea is the use of an imbalanced bipartite Kikuchi graph, which gives a simpler method to analyze spectral refutations of odd arity XOR without using the standard"Cauchy-Schwarz trick", a trick that typically produces random matrices with correlated entries and makes the analysis for odd arity XOR significantly more complicated than even arity XOR.

Problem

• Proving lower bounds for odd query locally decodable codes
• Extending k^(q/(q-2)) bound to all odd q values
• Simplifying spectral analysis using bipartite Kikuchi graphs

Innovation

• Imbalanced bipartite Kikuchi graph method
• Simpler spectral refutation analysis
• Avoids Cauchy-Schwarz trick complications