Unique Decoding of Explicit $ε$-balanced Codes Near the Gilbert-Varshamov Bound

The Gilbert-Varshamov bound (non-constructively) establishes the existence of binary codes of distance $1/2 -\epsilon$ and rate $\Omega(\epsilon^2)$ (where an upper bound of $O(\epsilon^{2\log(1/\epsilon))$} is known). Ta-Shma [STOC 2017] gave an explicit construction of $\epsilon$-balanced binary codes, where any two distinct codewords are at a distance between $1/2 -\epsilon/2$ and $1/2+\epsilon/2$, achieving a near optimal rate of $\Omega(\epsilon^{2+\beta})$, where $\beta \to 0$ as $\epsilon \to 0$. We develop unique and list decoding algorithms for (essentially) the family of codes constructed by Ta-Shma. We prove the following results for $\epsilon$-balanced codes with block length $N$ and rate $\Omega(\epsilon^{2+\beta})$ in this family: - For all $\epsilon, \beta > 0$ there are explicit codes which can be uniquely decoded up to an error of half the minimum distance in time $N^{{O_{\epsilon,} \beta}(1)}$. - For any fixed constant $\beta$ independent of $\epsilon$, there is an explicit construction of codes which can be uniquely decoded up to an error of half the minimum distance in time $(\log(1/\epsilon))^{O(1)} \cdot N^{{O_\beta(1)}$.} - For any $\epsilon > 0$, there are explicit $\epsilon$-balanced codes with rate $\Omega(\epsilon^{2+\beta})$ which can be list decoded up to error $1/2 - \epsilon'$ in time $N^{{O_{\epsilon,\epsilon',\beta}(1)}$,} where $\epsilon', \beta \to 0$ as $\epsilon \to 0$. The starting point of our algorithms is the list decoding framework from Alev et al. [SODA 2020], which uses the Sum-of-Squares SDP hierarchy. The rates obtained there were quasipolynomial in $\epsilon$. Here, we show how to overcome the far from optimal rates of this framework obtaining unique decoding algorithms for explicit binary codes of near optimal rate. These codes are based on simple modifications of Ta-Shma's construction.