3-Query RLDCs are Strictly Stronger than 3-Query LDCs

2025-12-14

AI Summary

This work resolves the long-standing open problem—open since 2006—of separating 3-query relaxed locally decodable codes (RLDCs) from standard locally decodable codes (LDCs). We construct a 3-query RLDC over a constant-size alphabet with length $ ilde{O}(k^2)$, strictly improving upon the best-known lower bound $ ilde{Omega}(k^3)$ for 3-query LDCs. Our approach introduces three key technical innovations: (i) the first 3-query, quasilinear-length, constant-alphabet, perfectly complete probabilistically checkable proof of proximity (PCPP); (ii) a query-preserving reduction framework from PCPPs to RLDCs; and (iii) a novel synthesis of high-dimensional expander (HDX)-based dPCPs with Dinur–Harsha and Moshkovitz–Raz composition techniques. This yields the first asymptotic separation in code length between RLDCs and LDCs under identical query complexity, significantly advancing the theoretical frontiers of local decodability.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

We construct $3$-query relaxed locally decodable codes (RLDCs) with constant alphabet size and length $ ilde{O}(k^2)$ for $k$-bit messages. Combined with the lower bound of $ ildeΩ(k^3)$ of [Alrabiah, Guruswami, Kothari, Manohar, STOC 2023] on the length of locally decodable codes (LDCs) with the same parameters, we obtain a separation between RLDCs and LDCs, resolving an open problem of [Ben-Sasson, Goldreich, Harsha, Sudan and Vadhan, SICOMP 2006]. Our RLDC construction relies on two components. First, we give a new construction of probabilistically checkable proofs of proximity (PCPPs) with $3$ queries, quasi-linear size, constant alphabet size, perfect completeness, and small soundness error. This improves upon all previous PCPP constructions, which either had a much higher query complexity or soundness close to $1$. Second, we give a query-preserving transformation from PCPPs to RLDCs. At the heart of our PCPP construction is a $2$-query decodable PCP (dPCP) with matching parameters, and our construction builds on the HDX-based PCP of [Bafna, Minzer, Vyas, Yun, STOC 2025] and on the efficient composition framework of [Moshkovitz, Raz, JACM 2010] and [Dinur, Harsha, SICOMP 2013]. More specifically, we first show how to use the HDX-based construction to get a dPCP with matching parameters but a large alphabet size, and then prove an appropriate composition theorem (and related transformations) to reduce the alphabet size in dPCPs.

Problem

• Constructs 3-query RLDCs with constant alphabet and length O(k^2)
• Resolves separation between RLDCs and LDCs via lower bound
• Improves PCPP construction with 3 queries and quasi-linear size

Innovation

• 3-query RLDCs with constant alphabet size
• New PCPP construction with 3 queries and quasi-linear size
• Query-preserving transformation from PCPPs to RLDCs