10 research outputs found
MDS and MHDR cyclic codes over finite chain rings
In this work, a unique set of generators for a cyclic code over a finite
chain ring has been established. The minimal spanning set and rank of the code
have also been determined. Further, sufficient as well as necessary conditions
for a cyclic code to be an MDS code and for a cyclic code to be an MHDR code
have been obtained. Some examples of optimal cyclic codes have also been
presented
Recent progress on weight distributions of cyclic codes over finite fields
Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions
Several new classes of MDS symbol-pair codes derived from matrix-product codes
In order to correct the pair-errors generated during the transmission of
modern high-density data storage that the outputs of the channels consist of
overlapping pairs of symbols, a new coding scheme named symbol-pair code is
proposed. The error-correcting capability of the symbol-pair code is determined
by its minimum symbol-pair distance. For such codes, the larger the minimum
symbol-pair distance, the better. It is a challenging task to construct
symbol-pair codes with optimal parameters, especially,
maximum-distance-separable (MDS) symbol-pair codes. In this paper, the
permutation equivalence codes of matrix-product codes with underlying matrixes
of orders 3 and 4 are used to extend the minimum symbol-pair distance, and six
new classes of MDS symbol-pair codes are derived.Comment: 22 pages,1 tabl
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal