Simultaneous rational number codes: Decoding beyond half the minimum distance with multiplicities and bad primes

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation Pub Date : 2025-08-06 DOI:10.1016/j.jsc.2025.102481

Matteo Abbondati, Eleonora Guerrini, Romain Lebreton

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引用次数: 0

Abstract

In the previous work of Abbondati et al. (2024), we extended the decoding analysis of interleaved Chinese remainder codes to simultaneous rational number codes. In this work, we build on Abbondati et al. (2024) by addressing two important scenarios: multiplicities and the presence of bad primes (divisors of denominators). First, we generalize previous results to multiplicity rational codes by considering modular reductions with respect to prime power moduli. Then, using hybrid analysis techniques, we extend our approach to vectors of fractions that may present bad primes.

Our contributions include: a decoding algorithm for simultaneous rational number reconstruction with multiplicities, a rigorous analysis of the algorithm's failure probability that generalizes several previous results, an extension to a hybrid model handling situations where not all errors can be assumed random, and a unified approach to handle bad primes 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 number codes.

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同时的有理数码：译码超过一半的最小距离与多重和坏素数

在Abbondati et al.（2024）之前的工作中，我们将交错中文剩余码的解码分析扩展到同时有理数码。在这项工作中，我们以Abbondati等人（2024）为基础，解决了两个重要的场景：多重性和坏素数（分母的除数）的存在。首先，我们通过考虑相对于素数幂模的模化，将之前的结果推广到多重有理码。然后，使用混合分析技术，我们将方法扩展到可能呈现坏素数的分数向量。我们的贡献包括：具有多重性的同时有理数重建的解码算法，对算法失效概率的严格分析，概括了以前的几个结果，扩展到处理并非所有错误都可以假设为随机的混合模型，以及处理多重性中的坏素数的统一方法。理论结果提供了在这些更复杂的情况下重构失败的全面概率分析，推进了有理数码的纠错技术的发展。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Symbolic Computation 工程技术-计算机：理论方法

CiteScore

2.10

自引率

14.30%

发文量

审稿时长

142 days

期刊介绍： An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects. It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.