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

## TL;DR

This paper introduces a multi-level robust Chinese Remainder Theorem for polynomials that allows accurate reconstruction even with small degree residue errors, balancing polynomial degree range and error bounds.

## Contribution

It proposes a novel multi-level robust CRT for polynomials and provides a simple closed-form reconstruction algorithm for improved accuracy under residue errors.

## Key findings

- Effective polynomial reconstruction with bounded residue errors.
- Trade-off formulation between polynomial degree range and error bound.
- A simple closed-form algorithm for robust polynomial reconstruction.

## Abstract

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 polynomial as accurately as possible in the sense that a reconstructed polynomial is obtained with only the last $\tau$ number of coefficients being possibly erroneous, when the residues are affected by errors with degrees upper bounded by $\tau$. In this regard, we first propose a multi-level robust Chinese remainder theorem (CRT) for polynomials, namely, a trade-off between the dynamic range of the degree of the polynomial to be reconstructed and the residue error bound $\tau$ is formulated. Furthermore, a simple closed-form reconstruction algorithm is also proposed.