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arXiv · 2508.05284

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

Matteo Abbondati·Eleonora Guerrini·Romain Lebreton
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Abstract

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.

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Matteo Abbondati, Eleonora Guerrini, Romain Lebreton. 2025-08-07. Simultaneous Rational Function Codes: Improved Analysis Beyond Half the Minimum Distance with Multiplicities and Poles. https://arxiv.org/abs/2508.05284

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