5 research outputs found
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 is studied in detail. If the moduli are not irreducible,
the notion of an error locator polynomial is replaced by an error factor
polynomial. We then obtain a collection of gcd-based decoding algorithms, some
of which are not quite standard even when specialized to Reed-Solomon codes
On the Joint Error-and-Erasure Decoding for Irreducible Polynomial Remainder Codes
A general class of polynomial remainder codes is considered. Such codes are
very flexible in rate and length and include Reed-Solomon codes as a special
case.
As an extension of previous work, two joint error-and-erasure decoding
approaches are proposed. In particular, both the decoding approaches by means
of a fixed transform are treated in a way compatible with the error-only
decoding. In the end, a collection of gcd-based decoding algorithm is obtained,
some of which appear to be new even when specialized to Reed-Solomon codes.Comment: Submitted (on 03/Feb/2012) to 2012 IEEE International Symposium on
Information Theor
Minimum Degree-Weighted Distance Decoding for Polynomial Residue Codes with Non-Pairwise Coprime Moduli
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 degree-weighted 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.Comment: 4 page
Error Correction in Polynomial Remainder Codes with Non-Pairwise Coprime Moduli and Robust Chinese Remainder Theorem for Polynomials
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 whenever the
degrees of all residue errors are upper bounded by , where a sufficient
condition for 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.Comment: 12 pages, 2 figure
Robust Polynomial Reconstruction via Chinese Remainder Theorem in the Presence of Small Degree Residue Errors
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 number of coefficients being possibly erroneous, when the
residues are affected by errors with degrees upper bounded by . In this
regard, we first propose a multi-level 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 is
formulated. Furthermore, a simple closed-form reconstruction algorithm is also
proposed.Comment: 5 page