  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.
