Exponential Lower Bounds for 2-query Relaxed Locally Decodable Codes

2026-02-23

AI Summary

This work investigates lower bounds on the blocklength of two-query (adaptive) relaxed locally decodable codes (RLDCs) over a binary alphabet under the standard Hamming error model. By reducing RLDCs to standard locally decodable codes (LDCs) and performing a refined combinatorial analysis of the message bits associated with fixed codeword positions, the authors establish—for the first time—an exponential lower bound on the blocklength for constant-query RLDCs. This result reveals a sharp “phase transition” in achievable code length at constant query complexity, resolving an open problem posed by Gur and Lachish. It also stands in stark contrast to the nearly linear-length constructions of Ben-Sasson et al., thereby delineating the fundamental theoretical limits of such codes.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Locally Decodable Codes (LDCs) are error-correcting codes $C\colon\Sigma^n\rightarrow \Sigma^m,$ encoding \emph{messages} in $\Sigma^n$ to \emph{codewords} in $\Sigma^m$, with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length $m$ that is super-polynomial in $n$, for codes with constant query complexity and constant alphabet size. In a very surprising result, Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (SICOMP 2006) show how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs. In this work, we study RLDCs in the standard Hamming-error setting. We prove an exponential lower bound on the length of Hamming RLDCs making $2$ queries (even adaptively) over the binary alphabet. This answers a question explicitly raised by Gur and Lachish (SICOMP 2021) and is the first exponential lower bound for RLDCs. Combined with the results of Ben-Sasson et al., our result exhibits a ``phase-transition''-type behavior on the codeword length for some constant-query complexity. We achieve these lower bounds via a transformation of RLDCs to standard Hamming LDCs, using a careful analysis of restrictions of message bits that fix codeword bits.

Problem

• Relaxed Locally Decodable Codes
• Hamming error
• exponential lower bound
• query complexity
• binary alphabet

Innovation

• relaxed locally decodable codes
• exponential lower bounds
• 2-query
• Hamming error
• code length