Approximate Locally Decodable Codes with Constant Query Complexity and Nearly Optimal Rate

We present simple constructions of good approxi-mate locally decodable codes (ALDCs) in the presence of a $\delta{-}$ fraction of errors for $\delta < 1/2$. In a standard locally decodable code $C:\Sigma_{1}^{k}\rightarrow\Sigma_{2}^{n}$, there is a decoder $M$ that on input $i\in[k]$ correctly outputs the i-th symbol of a message $x$ (with high probability) using only $q$ queries to a given string $w$ that is $\Delta$ -close to $C(x)$. In an ALDC, the decoder $M$ only needs to be correct on a 1 $-\varepsilon$ fraction of $i\in[k]$ for $\varepsilon$ much smaller than $\delta$. We present a construction of explicit ALDCs for all constants 1/2 $> \delta > \varepsilon$ with a constant number of queries $q$ and with constant, near-optimal rate. Standard LDCs with constant number of queries and any constant rate are known to be impossible. We additionally explore what is the lowest error probability $\varepsilon$ one can achieve for fixed $\delta$ and $q$. We show that for any ALDC, $\in=\Omega(\delta \mathrm{r}q/2\rceil)$. We then show that there exist explicit constant rate ALDCs for any constant $q$ that achieve $\varepsilon=O(\delta^{\lceil q/2\rceil})$. In particular, for $q=3$, we have a constant rate ALDC with error probability $\varepsilon=O(\delta^{2})$. A full version of this paper is available at https://eccc.weizmann.ac.il/report/2023/056/.