When Relaxation Does Not Help: RLDCs with Small Soundness Yield LDCs

2026-03-03

AI Summary

This work investigates whether relaxed locally decodable codes (RLDCs) with small soundness error can be transformed into standard locally decodable codes (LDCs), moving beyond prior reliance on linear codes. It establishes, for the first time in the general (non-linear) setting, that any non-adaptive, low-query-complexity q-query RLDC can be converted into a q-query LDC with comparable parameters. The transformation framework is further extended to relaxed locally correctable codes (RLCCs) and probabilistically checkable proofs of proximity (PCPPs), yielding stronger complexity lower bounds for these models. These results significantly advance the understanding of the fundamental limits of local decoding and correction capabilities.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Local Search
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Locally decodable codes (LDCs) are error correction codes that allow recovery of any single message symbol by probing only a small number of positions from the (possibly corrupted) codeword. Relaxed locally decodable codes (RLDCs) further allow the decoder to output a special failure symbol $\bot$ on a corrupted codeword. While known constructions of RLDCs achieve much better parameters than standard LDCs, it is intriguing to understand the relationship between LDCs and RLDCs. Separation results (i.e., the existence of $q$-query RLDCs that are not $q$-query LDCs) are known for $q=3$ (Gur, Minzer, Weissenberg, and Zheng, arXiv:2512.12960, 2025) and $q \geq 15$ (Grigorescu, Kumar, Manohar, and Mon, arXiv:2511.02633, 2025), while any $2$-query RLDC also gives a $2$-query LDC (Block, Blocki, Cheng, Grigorescu, Li, Zheng, and Zhu, CCC 2023). In this work, we generalize and strengthen the main result in Grigorescu, Kumar, Manohar, and Mon (arXiv:2511.02633, 2025), by removing the requirement of linear codes. Specifically, we show that any $q$-query RLDC with soundness error below some threshold $s(q)$ also yields a $q$-query LDC with comparable parameters. This holds even if the RLDC has imperfect completeness but with a non-adaptive decoder. Our results also extend to the setting of locally correctable codes (LCCs) and relaxed locally correctable codes (RLCCs). Using our results, we further derive improved lower bounds for arbitrary RLDCs and RLCCs, as well as probabilistically checkable proofs of proximity (PCPPs).

Problem

• locally decodable codes
• relaxed locally decodable codes
• soundness
• local correctability
• error-correcting codes

Innovation

• relaxed locally decodable codes
• locally decodable codes
• soundness error
• non-adaptive decoding
• lower bounds