Local List-Decoding and Testing of Random Linear Codes from High Error

Abstract

In this paper, we give efficient algorithms for list-decoding and testing random linear codes. Our main result is that random sparse linear codes are locally list-decodable and locally testable in the high-error regime with only a constant number of queries. More precisely, we show that for all constants $c> 0$ and $\gamma > 0$, and for every linear code $\mathcal C \subseteq \{0,1\}^N$ which is (1) sparse: $|\mathcal C| \leq N^c$, and (2) unbiased: each nonzero codeword in $\mathcal C$ has weight $\in (\frac{1}{2} - N^{-\gamma}, \frac{1}{2} + N^{-\gamma})$, then $\mathcal C$ is locally testable and locally list-decodable from $(\frac{1}{2} - \epsilon)$-fraction worst-case errors using only ${ {poly}}(\frac{1}{\epsilon})$ queries to a received word. We also give subexponential time algorithms for list-decoding arbitrary unbiased (but not necessarily sparse) linear codes in the high-error regime. In particular, this yields the first subexponential time algorithm even for the problem of (unique) decoding ra...