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. To the best of our knowledge, this is the first fully explicit linear time decodable code over an alphabet of size 2^{o(1/\varepsilon)}, that beats the O(\varepsilon^2) rate barrier.

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. To the best of our knowledge, this is the first fully explicit list decodable code with polynomial-time list decoding over an alphabet of size 2^{o(1/\varepsilon)}, that beats the O(\varepsilon^2) rate barrier.

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.

Our techniques are based on plurality analysis and graph-concatenated codes, which are widely used in the literature. We also introduce new decoding algorithms that may prove valuable for other graph-based codes.

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. To the best of our knowledge, this is the first fully explicit linear time decodable code over an alphabet of size 2^{o(1/\varepsilon)}, that beats the O(\varepsilon^2) rate barrier.

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. To the best of our knowledge, this is the first fully explicit list decodable code with polynomial-time list decoding over an alphabet of size 2^{o(1/\varepsilon)}, that beats the O(\varepsilon^2) rate barrier.

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.

Our techniques are based on plurality analysis and graph-concatenated codes, which are widely used in the literature. We also introduce new decoding algorithms that may prove valuable for other graph-based codes.