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

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).