6 research outputs found
Application of Constacyclic codes to Quantum MDS Codes
Quantum maximal-distance-separable (MDS) codes form an important class of
quantum codes. To get -ary quantum MDS codes, it suffices to find linear MDS
codes over satisfying by the
Hermitian construction and the quantum Singleton bound. If
, we say that is a dual-containing code. Many new
quantum MDS codes with relatively large minimum distance have been produced by
constructing dual-containing constacyclic MDS codes (see \cite{Guardia11},
\cite{Kai13}, \cite{Kai14}). These works motivate us to make a careful study on
the existence condition for nontrivial dual-containing constacyclic codes. This
would help us to avoid unnecessary attempts and provide effective ideas in
order to construct dual-containing codes. Several classes of dual-containing
MDS constacyclic codes are constructed and their parameters are computed.
Consequently, new quantum MDS codes are derived from these parameters. The
quantum MDS codes exhibited here have parameters better than the ones available
in the literature.Comment: 16 page
A Class of Constacyclic Codes Containing Formally Self-dual and Isodual Codes
In this paper, we investigate a class of constacyclic codes which contains isodual codes and formally self-dual codes. Further, we introduce a recursive approach to obtain the explicit factorization of , where are positive integers and is an element of order in . Moreover, we give many examples of interesting isodual and formally self-dual constacyclic codes
An Algorithm to find the Generators of Multidimensional Cyclic Codes over a Finite Chain Ring
The aim of this paper is to determine the algebraic structure of
multidimensional cyclic codes over a finite chain ring . An
algorithm to find the generator polynomials of dimensional (D) cyclic
codes of length over has been developed
using the generator polynomials of cyclic codes over .
Additionally, the generators of D cyclic codes with length over have been obtained as separable polynomials for the
case , where is the cardinality of
residue field of
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