首页/论文/2002.00087

arXiv 2002.000872020-02cs.ITcs.CRmath.ITmath.NT

Exact and Robust Reconstructions of Integer Vectors Based on Multidimensional Chinese Remainder Theorem (MD-CRT)

Li Xiao, Xiang-Gen Xia, Yu-Ping Wang

DOI：: 10.1109/TSP.2020.3023584

英文摘要：

The robust Chinese remainder theorem (CRT) has been recently proposed for robustly reconstructing a large nonnegative integer from erroneous remainders. It has found many applications in signal processing, including phase unwrapping and frequency estimation under sub-Nyquist sampling. Motivated by the applications in multidimensional (MD) signal processing, in this paper we propose the MD-CRT and robust MD-CRT for integer vectors. Specifically, by rephrasing the abstract CRT for rings in number-theoretic terms, we first derive the MD-CRT for integer vectors with respect to a general set of integer matrix moduli, which provides an algorithm to uniquely reconstruct an integer vector from its remainders, if it is in the fundamental parallelepiped of the lattice generated by a least common right multiple of all the moduli. For some special forms of moduli, we present explicit reconstruction formulae. Moreover, we derive the robust MD-CRT for integer vectors when the remaining integer matrices of all the moduli left divided by their greatest common left divisor (gcld) are pairwise commutative and coprime. Two different reconstruction algorithms are proposed, and accordingly, two different conditions on the remainder error bound for the reconstruction robustness are obtained, which are related to a quarter of the minimum distance of the lattice generated by the gcld of all the moduli or the Smith normal form of the gcld.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Multi-Stage Robust Chinese Remainder Theorem

It is well-known that the traditional Chinese remainder theorem (CRT) is not robust in the sense that a small error in a remainder may cause a large error in the reconstruction solution. A robust CRT was recently proposed for a special case when the greatest common divisor (gcd) of all the moduli is more than 1 and the remaining integers factorized by the gcd of all the moduli are co-prime. In this special case, a closed-form reconstruction from erroneous remainders was proposed and a necessary and sufficient condition on the remainder errors was obtained. It basically says that the reconstruc…

2013-03-13cs.CEcs.CRcs.IT

On Solving a Generalized Chinese Remainder Theorem in the Presence of Remainder Errors

In estimating frequencies given that the signal waveforms are undersampled multiple times, Xia et. al. proposed to use a generalized version of Chinese remainder Theorem (CRT), where the moduli are $M_1, M_2, \cdots, M_k$ which are not necessarily pairwise coprime. If the errors of the corrupted remainders are within $τ=\sds \max_{1\le i\le k} \min_{\stackrel{1\le j\le k}{j\neq i}} \frac{\gcd(M_i,M_j)}4$, their schemes can be used to construct an approximation of the solution to the generalized CRT with an error smaller than $τ$. Accurately finding the quotients is a critical ingredient in the…

2017-08-31cs.ITmath.IT

On Solving Ambiguity Resolution with Robust Chinese Remainder Theorem for Multiple Numbers

Chinese Remainder Theorem (CRT) is a powerful approach to solve ambiguity resolution related problems such as undersampling frequency estimation and phase unwrapping which are widely applied in localization. Recently, the deterministic robust CRT for multiple numbers (RCRTMN) was proposed, which can reconstruct multiple integers with unknown relationship of residue correspondence via generalized CRT and achieves robustness to bounded errors simultaneously. Naturally, RCRTMN sheds light on CRT-based estimation for multiple objectives. In this paper, two open problems arising that how to introdu…

2018-06-29cs.ITmath.IT

Robustness in Chinese Remainder Theorem

Chinese Remainder Theorem (CRT) has been widely studied with its applications in frequency estimation, phase unwrapping, coding theory and distributed data storage. Since traditional CRT is greatly sensitive to the errors in residues due to noises, the problem of robustly reconstructing integers via the erroneous residues has been intensively studied in the literature. In order to robustly reconstruct integers, there are two kinds of traditional methods: the one is to introduce common divisors in the moduli and the other is to directly decrease the dynamic range. In this paper, we take further…

2017-08-16cs.ITmath.IT

用于整数向量重构的鲁棒多维中国剩余定理

Robust Multidimentional Chinese Remainder Theorem for Integer Vector Reconstruction

本文研究一般模数集下的鲁棒多维中国剩余定理，提出余数误差的充要与充分条件及闭式重构算法，推广至实数矩阵，并验证其在亚奈奎斯特采样频率估计中的有效性。

2023-11-20cs.ITeess.SPmath.IT

使用平面阵列和多维中国剩余定理（MD-CRT）的运动目标SAR成像——第一部分：通用框架

Moving Target SAR Imaging Using Planar Arrays And Multidimensional Chinese Remainder Theorem (MD-CRT)--Part I: A General Framework

University of Delaware（特拉华大学）

针对三维地形上运动目标的SAR成像，提出基于平面阵列的多子阵框架，利用多维中国剩余定理（MD-CRT）从多个向量余数中恢复目标参数，扩大无模糊范围并给出误差界。

2026-06-16cs.ITeess.SPmath.IT

更多相关论文收起更多论文

A Robust Generalized Chinese Remainder Theorem for Two Integers

A generalized Chinese remainder theorem (CRT) for multiple integers from residue sets has been studied recently, where the correspondence between the remainders and the integers in each residue set modulo several moduli is not known. A robust CRT has also been proposed lately for robustly reconstruct a single integer from its erroneous remainders. In this paper, we consider the reconstruction problem of two integers from their residue sets, where the remainders are not only out of order but also may have errors. We prove that two integers can be robustly reconstructed if their remainder errors…

2015-10-09cs.ITmath.IT

基于平面阵列和多维中国余数定理（MD-CRT）的运动目标SAR成像——第二部分：两种子阵设计

Moving Target SAR Imaging Using Planar Arrays And Multidimensional Chinese Remainder Theorem (MD-CRT)--Part II: Two Subarray Designs

University of Delaware（特拉华大学）

本文基于第一部分框架，研究两种平面子阵设计用于运动目标SAR成像，分析性能并比较不同阵列几何，证明非可分离阵列在鲁棒性上优于可分离阵列。

2026-06-16cs.ITeess.SPmath.IT

Network Coding Based on Chinese Remainder Theorem

Random linear network code has to sacrifice part of bandwidth to transfer the coding vectors, thus a head of size k log|T| is appended to each packet. We present a distributed random network coding approach based on the Chinese remainder theorem for general multicast networks. It uses a couple of modulus as the head, thus reduces the size of head to O(log k). This makes it more suitable for scenarios where the number of source nodes is large and the bandwidth is limited. We estimate the multicast rate and show it is satisfactory in performance for randomly designed networks.

2012-08-20cs.ITmath.IT

线性化多项式中国剩余码

Linearized Polynomial Chinese remainder codes

XLIM, France（XLIM）；MATHIS/CRYPTIS；Inria Bordeaux - Sud-Ouest Research Center, France（法国波尔多-西南Inria研究中心）

介绍一类与秩和和秩度量相关的新码，基于有限域上线性化多项式的有效中国剩余定理，还为部分码提出解码算法。

2026-07-07cs.ITcs.SCmath.IT

基于CRT的整数环渐进理想分层秘密共享

Asymptotically Ideal Hierarchical Secret Sharing Based on CRT for Integer Ring

School of Mathematics and Big Data, Chaohu University（巢湖学院数学与大数据学院）；School of Mathematics and Statistics, Hefei University（合肥学院数学与统计学院）；School of Computer Science and Technology, University of Science and Technology of China（中国科学技术大学计算机科学与技术学院）

本文提出基于CRT和单向函数的分层秘密共享方案，实现渐进理想的安全性与信息率，解决传统分层共享方案的安全性和信息率不足问题。

2026-04-12cs.CRcs.ITmath.IT

Error Correction in Polynomial Remainder Codes with Non-Pairwise Coprime Moduli and Robust Chinese Remainder Theorem for Polynomials

This paper investigates polynomial remainder codes with non-pairwise coprime moduli. We first consider a robust reconstruction problem for polynomials from erroneous residues when the degrees of all residue errors are assumed small, namely robust Chinese Remainder Theorem (CRT) for polynomials. It basically says that a polynomial can be reconstructed from erroneous residues such that the degree of the reconstruction error is upper bounded by $τ$ whenever the degrees of all residue errors are upper bounded by $τ$, where a sufficient condition for $τ$ and a reconstruction algorithm are obtained.…

2014-12-31cs.ITmath.IT

Robust Polynomial Reconstruction via Chinese Remainder Theorem in the Presence of Small Degree Residue Errors

Based on unique decoding of the polynomial residue code with non-pairwise coprime moduli, a polynomial with degree less than that of the least common multiple (lcm) of all the moduli can be accurately reconstructed when the number of residue errors is less than half the minimum distance of the code. However, once the number of residue errors is beyond half the minimum distance of the code, the unique decoding may fail and lead to a large reconstruction error. In this paper, assuming that all the residues are allowed to have errors with small degrees, we consider how to reconstruct the polynomi…

2017-03-23cs.ITmath.IT

异步多通道USF：用于模展开的改进CRT

Asynchronous Multi-Channel USF: Modified CRT for Modulo Unfolding

Imperial College London（帝国理工学院）

提出一种无需同步的异步多通道USF架构，通过时空信号提升和图平滑性实现鲁棒重建，数值实验验证了准确性。

2026-06-23cs.ITeess.SPmath.IT

Towards Robustness in Residue Number Systems

The problem of robustly reconstructing a large number from its erroneous remainders with respect to several moduli, namely the robust remaindering problem, may occur in many applications including phase unwrapping, frequency detection from several undersampled waveforms, wireless sensor networks, etc. Assuming that the dynamic range of the large number is the maximal possible one, i.e., the least common multiple (lcm) of all the moduli, a method called robust Chinese remainder theorem (CRT) for solving the robust remaindering problem has been recently proposed. In this paper, by relaxing the a…

2016-02-09cs.ITmath.ITmath.NT