We study uniquely decodable codes and list decodable codes in the high-noise regime, specifically codes that are uniquely decodable from $\frac{1-\varepsilon}{2}$ fraction of errors and list decodable from $1-\varepsilon$ fraction of errors. We present several improved explicit constructions that achieve near-optimal rates, as well as efficient or even linear-time decoding algorithms. Our contributions are as follows. 1. Explicit Near-Optimal Linear Time Uniquely Decodable Codes: We construct a family of explicit $\mathbb{F}_2$-linear codes with rate $\Omega(\varepsilon)$ and alphabet size $2^{\mathrm{poly} \log(1/\varepsilon)}$, that are capable of correcting $e$ errors and $s$ erasures whenever $2e + s < (1 - \varepsilon)n$ in linear-time. 2. Explicit Near-Optimal List Decodable Codes: We construct a family of explicit list decodable codes with rate $\Omega(\varepsilon)$ and alphabet size $2^{\mathrm{poly} \log(1/\varepsilon)}$, that are capable of list decoding from $1-\varepsilon$ fraction of errors with a list size $L = \exp\exp\exp(\log^{\ast}n)$ in polynomial time. 3. List Decodable Code with Near-Optimal List Size: We construct a family of explicit list decodable codes with an optimal list size of $O(1/\varepsilon)$, albeit with a suboptimal rate of $O(\varepsilon^2)$, capable of list decoding from $1-\varepsilon$ fraction of errors in polynomial time. Furthermore, we introduce a new combinatorial object called multi-set disperser, and use it to give a family of list decodable codes with near-optimal rate $\frac{\varepsilon}{\log^2(1/\varepsilon)}$ and list size $\frac{\log^2(1/\varepsilon)}{\varepsilon}$, that can be constructed in probabilistic polynomial time and decoded in deterministic polynomial time. We also introduce new decoding algorithms that may prove valuable for other graph-based codes.

翻译：我们研究高噪声机制下的唯一可译码与列表可译码，具体而言，即能从 $\frac{1-\varepsilon}{2}$ 比例错误中唯一译码以及能从 $1-\varepsilon$ 比例错误中列表译码的编码。我们提出了若干改进的显式构造，这些构造能达到近最优的码率，并具备高效甚至线性时间的译码算法。我们的贡献如下。1. 显式近最优线性时间唯一可译码：我们构造了一族显式的 $\mathbb{F}_2$ 线性码，其码率为 $\Omega(\varepsilon)$，字母表大小为 $2^{\mathrm{poly} \log(1/\varepsilon)}$，能够在 $2e + s < (1 - \varepsilon)n$ 时以线性时间纠正 $e$ 个错误和 $s$ 个删除。2. 显式近最优列表可译码：我们构造了一族显式的列表可译码，其码率为 $\Omega(\varepsilon)$，字母表大小为 $2^{\mathrm{poly} \log(1/\varepsilon)}$，能够在多项式时间内以列表大小 $L = \exp\exp\exp(\log^{\ast}n)$ 从 $1-\varepsilon$ 比例的错误中进行列表译码。3. 具有近最优列表大小的列表可译码：我们构造了一族显式的列表可译码，其列表大小达到最优的 $O(1/\varepsilon)$，尽管码率是次优的 $O(\varepsilon^2)$，能够在多项式时间内从 $1-\varepsilon$ 比例的错误中进行列表译码。此外，我们引入了一种称为多重集分散器的新组合对象，并利用它给出了一族列表可译码，其码率接近最优 $\frac{\varepsilon}{\log^2(1/\varepsilon)}$，列表大小为 $\frac{\log^2(1/\varepsilon)}{\varepsilon}$，该族码可在概率多项式时间内构造并在确定性多项式时间内译码。我们还引入了新的译码算法，这些算法可能对其他基于图的编码具有重要价值。