标题： 同时有理函数码：超越最小距离一半的改进分析，考虑重数和极点 显示英文标题

标题： Simultaneous Rational Function Codes: Improved Analysis Beyond Half the Minimum Distance with Multiplicities and Poles

摘要： 在本文中，我们扩展了Abbondati等人（2024）关于同时有理函数码解码的工作，通过解决两个重要的场景：重数和极点（分母的零点）。首先，我们通过考虑多精度评估，将先前的结果推广到具有重数的有理码。然后，利用Guerrini等人（2023）的混合模型，我们将方法扩展到可能包含极点的有理函数向量。我们的贡献包括：对解码算法失败概率的严格分析，该分析推广并改进了若干先前结果；扩展到处理无法假设所有错误都是随机的情况的混合模型；以及在处理重数中的极点的更一般情境下的新改进分析。理论结果提供了这些更复杂场景中重建失败的全面概率分析，推动了有理函数码纠错领域的最新进展。 显示英文摘要

摘要： In this paper, we extend the work of Abbondati et al. (2024) on decoding simultaneous rational function codes by addressing two important scenarios: multiplicities and poles (zeros of denominators). First, we generalize previous results to rational codes with multiplicities by considering evaluations with multi-precision. Then, using the hybrid model from Guerrini et al. (2023), we extend our approach to vectors of rational functions that may present poles. Our contributions include: a rigorous analysis of the decoding algorithm's failure probability that generalizes and improves several previous results, an extension to a hybrid model handling situations where not all errors can be assumed random, and a new improved analysis in the more general context handling poles within multiplicities. The theoretical results provide a comprehensive probabilistic analysis of reconstruction failure in these more complex scenarios, advancing the state of the art in error correction for rational function codes.