On irreducible polynomial remainder codes

IEEE Trans. Circuits Syst. II

2018

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 τ number of coefficients being possibly erroneous, when the residues are affected by errors with degrees upper bounded by τ . In this regard, we first propose a multilevel 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 τ is formulated. Furthermore, a simple closed-form reconstruction algorithm is also proposed.

“…(ii) If deg(q 21 (x)) ≥ deg(m 1 (x)), from Lemma 2 we get deg(a 2 (x)) ≥ deg(m 1 (x)). When i = 1, i.e., deg(a(x)) < deg(M (x)) − deg(σ 1 (x)), we obtain from (10) and (11)…”

Section: Multi-level Robust Crt For Polynomialsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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

IEEE Trans. Circuits Syst. II

2018

“…Recently, different from the Hamming distance, another type of distance called degree-weighted distance for polynomial residue codes is defined, and accordingly, a coding framework based on the degree-weighted distance has been developed for polynomial residue codes with pairwise coprime moduli in [17], [18]. In this paper, with regard to this degree-weighted distance, we naturally study polynomial residue codes with non-pairwise coprime moduli.…”

Section: Introductionmentioning

confidence: 99%

Minimum Degree-Weighted Distance Decoding for Polynomial Residue Codes With Non-Coprime Moduli

IEEE Wireless Commun. Lett.

2017

This paper presents a new decoding for polynomial residue codes, called the minimum degree-weighted distance decoding. The newly proposed decoding is based on the degreeweighted distance and different from the traditional minimum Hamming distance decoding. It is shown that for the two types of minimum distance decoders, i.e., the minimum degree-weighted distance decoding and the minimum Hamming distance decoding, one is not absolutely stronger than the other, but they can complement each other from different points of view.

“…Moreover, residue codes with non-pairwise coprime moduli may be quite effective in providing a wild coverage of "random" errors [24]- [26]. In order to perform reliably polynomial-type operations (e.g., cyclic convolution, correlation, DFT and FFT computations) with reduced complexity in digital signal processing systems, residue codes over polynomials (called polynomial remainder codes in this paper) with pairwise or non-pairwise coprime polynomial moduli have been investigated as well [29]- [36], where codewords are residue vectors of polynomials with degrees in a certain range modulo the moduli and all polynomials are defined over a Galois field. Polynomial remainder codes are a large class of codes that include BCH codes and Reed-Solomon codes as special cases [27], [28].…”

Section: Introductionmentioning

confidence: 99%

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

2015

IEEE Trans. Commun.

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 τ whenever the degrees of all residue errors are upper bounded by τ , where a sufficient condition for τ 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.