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

Kuan Cheng, Xin Li, Songtao Mao

TL;DR

This work generalize and strengthen the main result in Grigorescu, Kumar, Manohar, and Mon, and shows that any $q-query RLDC with soundness error below some threshold also yields a $q-query LDC with comparable parameters, even if the RLDC has imperfect completeness but with a non-adaptive decoder.

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).

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

TL;DR

This work generalize and strengthen the main result in Grigorescu, Kumar, Manohar, and Mon, and shows that any q-query LDC with comparable parameters, even if the RLDC has imperfect completeness but with a non-adaptive decoder.

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 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 -query RLDCs that are not -query LDCs) are known for (Gur, Minzer, Weissenberg, and Zheng, arXiv:2512.12960, 2025) and (Grigorescu, Kumar, Manohar, and Mon, arXiv:2511.02633, 2025), while any -query RLDC also gives a -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 -query RLDC with soundness error below some threshold also yields a -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).
Paper Structure (20 sections, 22 theorems, 75 equations)

This paper contains 20 sections, 22 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. Techniques overview
  4. Summary of techniques in grigorescu2025relaxed.
  5. Smoothness relative to a codeword
  6. Outline of the Paper
  7. Preliminaries
  8. Basic notation
  9. Locally decodable/correctable codes and their relaxed variants
  10. From RLDCs to LDCs in the Low-Soundness Regime
  11. Characterization of RLDC decoders
  12. Heavy and light indices
  13. Proof of Lemma \ref{['lem:ns_mass_bound-again']}: Bounding the probability of nonsmoothable queries
  14. Proof of Lemma \ref{['lem:corrupted-light']}: Bounding the probability of light error
  15. Tradeoff between Query Complexity and Soundness Error
  16. ...and 5 more sections

Key Result

Theorem 1.3

Let $C:\{0,1\}^k\to\{0,1\}^n$ be a linear $(q,\delta,1,s)$-RLDC with a (possibly adaptive) decoder and $s<2^{-\lfloor q/2\rfloor}$, then, for every radius $r\in \bigl(0,\frac{\delta}{2q}(1-s2^{\lfloor q/2\rfloor}) \bigr)$, $C$ is a Moreover, the analogous implication holds in the RLCC/LCC setting.

Theorems & Definitions (40)

  • Definition 1.1: Locally decodable codes (binary LDCs)
  • Definition 1.2: Relaxed LDCs (binary RLDCs)
  • Theorem 1.3: grigorescu2025relaxed
  • Theorem 1.4: Informal, binary case of Theorem \ref{['thm:derived_ldc']}
  • Theorem 1.5: Informal, binary case of Theorem \ref{['thm:rldc-to-ldc-imperfect-general']}
  • Theorem 1.6: Informal, binary case of Theorem \ref{['thm:main_tq']}
  • Corollary 1.7
  • Definition 1.8: CSP
  • Definition 1.9: PCPP
  • Corollary 1.10
  • ...and 30 more