Motivated by recent developments in coding theory, particular in list-decoding, we introduce a new error model which we call semi-adversarial errors. This error model bridges between fully random errors and fully adversarial errors by allowing some symbols of a message to be corrupted by an adversary while others are replaced with uniformly random symbols. As our main quest, we seek to understand optimal efficient unique decoding algorithms in the semi-adversarial model. For interleaved Reed--Solomon (IRS), folded Reed--Solomon (FRS) and univariate multiplicity codes, we design decoding algorithms running in near-linear time for most mixtures of random and adversarial errors. Our analysis matches the information-theoretic optimum for semi-adversarial errors. Our algorithm for interleaved Reed--Solomon codes is an improved implementation of the decoding algorithm by Bleichenbacher--Kiayias--Yung (BKY) for fully random errors. We use a novel monomial-tracking technique to analyze its performance in this new semi-adversarial errors. Inspired by the BKY algorithm, we use novel interpolations to extend our approach to the settings of folded Reed--Solomon and multiplicity codes, resulting in fast algorithms for unique decoding against semi-adversarial errors. Our new decoders for FRS and multiplicity codes replace the sophisticated root-finding step in traditional algorithms, such as the Guruswami--Wang algorithm, with a straightforward polynomial long division. Analysis of these algorithms requires more robust monomial-tracking arguments than IRS codes.

翻译：受编码理论最新进展（特别是列表译码）的启发，我们引入了一种新的错误模型——半对抗性错误。该错误模型通过允许消息中部分符号被对手破坏，而其他符号被均匀随机符号替换，从而在完全随机错误与完全对抗性错误之间建立桥梁。作为核心目标，我们致力于理解半对抗性模型下的最优高效唯一译码算法。针对交织Reed–Solomon码、折叠Reed–Solomon码及单变量重数码，我们设计了在近线性时间内运行、可处理大部分随机与对抗性错误混合情形的译码算法。我们的分析达到了半对抗性错误的信息论最优界。针对交织Reed–Solomon码的算法是对Bleichenbacher–Kiayias–Yung（BKY）完全随机错误译码算法的改进实现。我们采用一种新颖的单项式追踪技术分析其在新半对抗性错误模型中的性能。受BKY算法启发，我们通过新型插值方法将思路拓展至折叠Reed–Solomon码与重数码场景，从而得到针对半对抗性错误实现唯一译码的快速算法。针对折叠Reed–Solomon码与重数码的新译码器以简洁的多项式长除法替代传统算法（如Guruswami–Wang算法）中复杂的求根步骤。对这些算法的分析需要比交织Reed–Solomon码更鲁棒的单项式追踪论证。