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).

翻译：局部可解码码（LDC）是一种纠错码，它允许仅通过探测（可能损坏的）码字中的少量位置来恢复任意单个消息符号。松弛局部可解码码（RLDC）进一步允许解码器在损坏的码字上输出一个特殊的失败符号$\bot$。虽然已知的RLDC构造比标准LDC实现了更好的参数，但理解LDC与RLDC之间的关系是引人入胜的。对于$q=3$（Gur, Minzer, Weissenberg, and Zheng, arXiv:2512.12960, 2025）和$q \geq 15$（Grigorescu, Kumar, Manohar, and Mon, arXiv:2511.02633, 2025），已知存在分离结果（即存在不是$q$查询LDC的$q$查询RLDC），而任何$2$查询RLDC也给出一个$2$查询LDC（Block, Blocki, Cheng, Grigorescu, Li, Zheng, and Zhu, CCC 2023）。在本工作中，我们通过移除对线性码的要求，推广并强化了Grigorescu, Kumar, Manohar, and Mon（arXiv:2511.02633, 2025）的主要结果。具体而言，我们证明了任何健全性误差低于某个阈值$s(q)$的$q$查询RLDC也会产生一个具有可比参数的$q$查询LDC。即使RLDC具有不完美的完备性，但解码器是非自适应的，这一结论仍然成立。我们的结果也扩展到局部可校正码（LCC）和松弛局部可校正码（RLCC）的设置。利用我们的结果，我们进一步推导了任意RLDC和RLCC的改进下界，以及邻近概率可检查证明（PCPP）。