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

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, the robust Chinese Remainder Theorem (CRT) for polynomials. It basically says that a polynomial can be reconstructed from its erroneous residues such that the degree of the reconstruction error is upper bounded by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">$\tau$</tex-math></inline-formula> whenever the degrees of all residue errors are upper bounded by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">$\tau$</tex-math></inline-formula> , where a sufficient condition for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">$\tau$</tex-math></inline-formula> and a reconstruction algorithm are obtained. By relaxing 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. We finally 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 data transmission are illustrated.