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

TL;DR

This paper develops advanced error correction methods for polynomial remainder codes with non-pairwise coprime moduli, introducing a robust Chinese Remainder Theorem for polynomials that improves error correction and burst error correction capabilities.

Contribution

It presents a new robust Chinese Remainder Theorem for polynomials that handles multiple unrestricted errors and small degree errors, enhancing error correction in polynomial remainder codes.

Findings

Improved error correction capability for polynomial remainder codes.

Enhanced burst error correction in data transmission.

Stronger residue error correction using redundancy in moduli.

Abstract

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 $\tau$ whenever the degrees of all residue errors are upper bounded by $\tau$, where a sufficient condition for $\tau$ and a reconstruction algorithm are obtained. By releasing the constraint that all residue errors have small degrees, another robust reconstruction is then presented when there are multiple unrestricted errors and an arbitrary number of errors with small degrees in the residues. By making full use of redundancy in moduli, we obtain a stronger residue error correction capability in the sense that apart from the number of errors that can be corrected in the previous existing result, some errors with small degrees can be also corrected in the residues. With this newly obtained result, improvements in uncorrected error probability and burst error correction capability in a data transmission are illustrated.