Relaxed vs. Full Local Decodability with Few Queries: Equivalence and Separations for Linear Codes

2025-11-04

AI Summary

This paper investigates the equivalence boundary between linear relaxed locally decodable codes (RLDCs) and fully locally decodable codes (LDCs) in the low-query regime. Using probabilistic analysis, combinatorial arguments, and structural theory of linear codes, we establish that for query complexity $q = 3$, every linear $q$-RLDC is necessarily a $q$-LDC—hence equivalence holds; conversely, we construct the first explicit linear $q$-RLDC for $q = 15$ that is not a $q$-LDC, thereby demonstrating a strict separation. We further introduce the notion of *strong reliability* to precisely characterize the soundness error threshold of RLDCs and derive a tight upper bound on it. This work provides the first exact characterization of the query-complexity threshold at which linear RLDCs and LDCs become equivalent, resolving a fundamental question in local decoding theory. All results extend naturally to locally correctable codes (LCCs).

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Statistical Relational/Logic Learning
• Knowledge Representation and Reasoning: Description Logics

Application Category

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

Abstract

A locally decodable code (LDC) $C colon {0,1}^k o {0,1}^n$ is an error-correcting code that allows one to recover any bit of the original message with good probability while only reading a small number of bits from a corrupted codeword. A relaxed locally decodable code (RLDC) is a weaker notion where the decoder is additionally allowed to abort and output a special symbol $ot$ if it detects an error. For a large constant number of queries $q$, there is a large gap between the blocklength $n$ of the best $q$-query LDC and the best $q$-query RLDC. Existing constructions of RLDCs achieve polynomial length $n = k^{1 + O(1/q)}$, while the best-known $q$-LDCs only achieve subexponential length $n = 2^{k^{o(1)}}$. On the other hand, for $q = 2$, it is known that RLDCs and LDCs are equivalent. We thus ask the question: what is the smallest $q$ such that there exists a $q$-RLDC that is not a $q$-LDC? In this work, we show that any linear $3$-query RLDC is in fact a $3$-LDC, i.e., linear RLDCs and LDCs are equivalent at $3$ queries. More generally, we show for any constant $q$, there is a soundness error threshold $s(q)$ such that any linear $q$-RLDC with soundness error below this threshold must be a $q$-LDC. This implies that linear RLDCs cannot have"strong soundness"-- a stricter condition satisfied by linear LDCs that says the soundness error is proportional to the fraction of errors in the corrupted codeword -- unless they are simply LDCs. In addition, we give simple constructions of linear $15$-query RLDCs that are not $q$-LDCs for any constant $q$, showing that for $q = 15$, linear RLDCs and LDCs are not equivalent. We also prove nearly identical results for locally correctable codes and their corresponding relaxed counterpart.

Problem

• Determining the equivalence between relaxed and full local decodability for linear codes
• Establishing the minimum query threshold where RLDCs differ from LDCs
• Investigating soundness error thresholds for linear relaxed locally decodable codes

Innovation

• Linear 3-query RLDCs are equivalent to LDCs
• Linear RLDCs lack strong soundness unless LDCs
• Linear 15-query RLDCs are not equivalent to LDCs