非均匀有界余数误差下中国剩余定理的精确动态范围-鲁棒性权衡
Exact Dynamic Range--Robustness Tradeoff for Chinese Remainder Theorem under Non-Uniformly Bounded Remainder Errors
- University of Delaware（特拉华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对非均匀余数误差界下的中国剩余定理，推导了动态范围与鲁棒性的精确权衡条件，并给出解码算法，揭示余数间鲁棒性耦合。研究问题：非均匀误差下CRT的精确权衡；方法：标量充要条件与解码算法；贡献：刻画动态范围与误差界向量的关系。
AI中文摘要：
中国剩余定理（CRT）对余数误差高度敏感。鲁棒CRT通过从错误的余数中正确恢复折叠整数来解决此问题，从而使最终重建误差受限于余数误差水平。现有的动态范围-鲁棒性权衡结果主要假设均匀余数误差界，即所有余数共享一个标量误差界。本文研究了当余数误差界为向量时（即不同余数可能具有不同误差界）的精确权衡。对于任意固定的候选范围长度（该长度指定了未知整数要从其错误余数中确定的整数区间）和任意固定的余数误差界向量，我们推导出一个标量的充分必要条件以实现鲁棒确定。基于此条件，我们刻画了给定余数误差界向量下最大可允许范围长度（称为动态范围）。反之，对于给定范围长度，我们刻画了所有可容忍余数误差的余数误差界向量。我们还给出了一种精确的折叠整数向量解码算法。我们表明，所提出的结果在无误差情况下退化为经典CRT，并在所有余数误差界相等时与已知结果一致。数值结果表明，动态范围取决于完整的余数误差界向量。它们还表明，余数在鲁棒性上是耦合的，这无法由单一均匀误差界捕获。
英文摘要：
The Chinese remainder theorem (CRT) is highly sensitive to remainder errors. Robust CRT addresses this problem by recovering the folding integers correctly from erroneous remainders, so that the final reconstruction error is bounded by the remainder error level. The existing dynamic range--robustness tradeoff results mainly assume a uniform remainder error bound, where all remainders share one scalar error bound. This paper studies the exact tradeoff when the remainder error bound is a vector in the sense that different remainders may have different error bounds. For any fixed candidate range length, which specifies the integer interval over which an unknown integer is to be determined from its erroneous remainders, and any fixed remainder error bound vector, we derive a scalar necessary and sufficient condition for robust determination. Based on this condition, we characterize the largest admissible range length, called the dynamic range, for a given remainder error bound vector. Conversely, for a given range length, we characterize all remainder error bound vectors under which remainder errors can be tolerated. We also give an exact folding integer vector decoding algorithm. We show that the proposed results reduce to the classical CRT in the error-free case and coincide with the known result when all remainder error bounds are equal. Numerical results show that the dynamic range depends on the full remainder error bound vector. They also show that the remainders are coupled in the robustness, which cannot be captured by a single uniform error bound.
