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

TL;DR

This paper advances the decoding analysis of simultaneous rational function codes by incorporating multiplicities and poles, providing a more comprehensive probabilistic failure analysis for complex error scenarios.

Contribution

It generalizes previous decoding results to include multiplicities and poles, and extends the analysis to hybrid error models, improving understanding of failure probabilities.

Findings

Rigorous failure probability analysis for complex rational codes

Extension of decoding methods to handle poles and multiplicities

Improved theoretical bounds on decoding success rates

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.