Error-correcting codes are one of the most fundamental objects in pseudorandomness, with applications in communication, complexity theory, and beyond. Codes are useful because of their ability to support decoding, which is the task of recovering a codeword from its noisy copy. List decoding is a relaxation where the decoder is allowed to output a list of codewords, and has seen tremendous progress over the last 25 years. In this thesis, we prove new algorithmic and combinatorial results about list decoding. We describe a list decoding framework that reduces the task of efficient decoding to proving distance in certain restricted proof systems. We then instantiate this framework for Tanner codes of Sipser and Spielman [IEEE Trans. Inf. Theory 1996] and Alon-Edmonds-Luby (AEL) distance amplification [FOCS 1995] of unique decodable base codes to get the first polynomial time list decoding algorithms for these codes. We also show extensions to the quantum version of AEL distance amplification to get polynomial-time list decodable quantum LDPC codes. We next give an alternate viewpoint of the above framework based on abstract regularity lemmas. We show how to efficiently implement the regularity lemma for the case of Ta-Shma's explicit codes near the Gilbert-Varshamov bound [STOC 2017]. This leads to a near-linear time algorithm for unique decoding of Ta-Shma's code. We also give new combinatorial results that improve known list sizes beyond the Johnson bound. Firstly, we adapt the AEL amplification to construct a new family of explicit codes that can be combinatorially list decoded to the optimal error correction radius. This is the first example of such a code that does not use significant algebraic ingredients. Secondly, we present list size improvements for Folded Reed-Solomon codes, improving the state of the art list size for explicit list decoding capacity achieving codes.

翻译：纠错码是伪随机性中最基本的对象之一，在通信、复杂性理论等领域具有广泛应用。编码的有用性源于其支持解码的能力，即从含噪副本中恢复码字的任务。列表解码是一种放宽条件的方法，允许解码器输出一个码字列表，在过去25年中取得了巨大进展。本论文证明了关于列表解码的新算法与组合结果。我们描述了一个列表解码框架，将高效解码任务简化为在某些受限证明系统中证明距离。随后，我们将该框架实例化应用于Sipser和Spielman的Tanner码[IEEE Trans. Inf. Theory 1996]以及Alon-Edmonds-Luby（AEL）距离放大技术[FOCS 1995]对唯一可解码基码的扩展，首次为这些编码实现了多项式时间列表解码算法。我们还展示了将AEL距离放大技术扩展至量子版本，从而获得多项式时间列表可解码量子LDPC码。接下来，我们基于抽象正则性引理提出了上述框架的替代视角。针对Ta-Shma在吉尔伯特-瓦尔沙莫夫界附近构建的显式编码[STOC 2017]，我们展示了如何高效实现正则性引理，从而为Ta-Shma编码提供了近乎线性时间的唯一解码算法。我们还提出了新的组合结果，改进了约翰逊界之外的已知列表规模上限。首先，我们采用AEL放大技术构建了新的显式编码族，可通过组合方法列表解码至最优纠错半径。这是首个无需使用复杂代数工具实现该特性的编码实例。其次，我们改进了折叠里德-所罗门码的列表规模上限，提升了当前达到列表解码容量的显式编码的最佳列表规模记录。