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

On Polynomial Remainder Codes

Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work. We explicitly allow the code symbols to be polynomials of different degrees, which leads to two different notions of weight and distance. Algebraic decoding ...

The Chinese Remainder Theorem

Chinese Remainder Codes: Using Lattices to Decode Error Correcting Codes Based on Chinese Remaindering Theorem

This report is an incomplete survey of Chinese Remaindering Codes. We study the work of Goldreich, Ron and Sudan [GRS00] and Boneh [B02] which give unique and list-decoding algorithms for an error correcting code based on the Chinese Remaindering Theorem. More specifically, we will look at a decoding algorithm from [GSM00] which uniquely decodes upto (n − k) log p1 log p1+log pn errors. We will...

Computing Hilbert class polynomials with the Chinese remainder theorem

We present a space-efficient algorithm to compute the Hilbert class polynomial HD(X) modulo a positive integer P , based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(|D|1/2+ log P ) space and has an expected running time of O(|D|1+ ). We describe practical optimizations that allow us to handle larger discriminants than othe...

Robustness in Chinese Remainder Theorem

Chinese Remainder Theorem (CRT) has been widely studied with its applications in frequency estimation, phase unwrapping, coding theory and distributed data storage. Since traditional CRT is greatly sensitive to the errors in residues due to noises, the problem of robustly reconstructing integers via the erroneous residues has been intensively studied in the literature. In order to robustly reco...