3-Query RLDCs are Strictly Stronger than 3-Query LDCs
Tom Gur, Dor Minzer, Guy Weissenberg, Kai Zhe Zheng
TL;DR
This work constructs 3-query RLDCs with constant alphabet and length roughly k^2 up to polylog factors, establishing a separation from 3-query LDCs by leveraging a new 2-query dPCP with quasi-linear size and very low soundness, which in turn yields a 3-query PCPP via a black-box transformation. Central to the approach is a novel, HDX-based 2-query dPCP, combined with a composition framework and a suite of PCP transformations (alphabet/decoding-degree reductions) that preserve query complexity while driving down alphabet size. The authors then translate PCPPs into RLDCs through a query-preserving transformation, achieving 3-query RLDCs with constant decoding radius and near-quadratic blow-up, thereby resolving an open problem on RLDC vs LDC separation. The results rely on route-through-HD X structures, direct product testing, and intricate PCP composition theorems to control length, alphabet, decoding degree, and soundness, culminating in a robust separation and several PSPACE-hardness implications. Overall, the paper advances the landscape of PCP-based coding by connecting low-query dPCPs, PCPPs, RLDCs, and LDCs to achieve unprecedented parameter regimes and new hardness consequences.
Abstract
We construct $3$-query relaxed locally decodable codes (RLDCs) with constant alphabet size and length $\tilde{O}(k^2)$ for $k$-bit messages. Combined with the lower bound of $\tildeΩ(k^3)$ of [Alrabiah, Guruswami, Kothari, Manohar, STOC 2023] on the length of locally decodable codes (LDCs) with the same parameters, we obtain a separation between RLDCs and LDCs, resolving an open problem of [Ben-Sasson, Goldreich, Harsha, Sudan and Vadhan, SICOMP 2006]. Our RLDC construction relies on two components. First, we give a new construction of probabilistically checkable proofs of proximity (PCPPs) with $3$ queries, quasi-linear size, constant alphabet size, perfect completeness, and small soundness error. This improves upon all previous PCPP constructions, which either had a much higher query complexity or soundness close to $1$. Second, we give a query-preserving transformation from PCPPs to RLDCs. At the heart of our PCPP construction is a $2$-query decodable PCP (dPCP) with matching parameters, and our construction builds on the HDX-based PCP of [Bafna, Minzer, Vyas, Yun, STOC 2025] and on the efficient composition framework of [Moshkovitz, Raz, JACM 2010] and [Dinur, Harsha, SICOMP 2013]. More specifically, we first show how to use the HDX-based construction to get a dPCP with matching parameters but a large alphabet size, and then prove an appropriate composition theorem (and related transformations) to reduce the alphabet size in dPCPs.