24 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
On MDS Negacyclic LCD Codes
Linear codes with complementary duals (LCD) have a great deal of significance
amongst linear codes. Maximum distance separable (MDS) codes are also an
important class of linear codes since they achieve the greatest error
correcting and detecting capabilities for fixed length and dimension. The
construction of linear codes that are both LCD and MDS is a hard task in coding
theory. In this paper, we study the constructions of LCD codes that are MDS
from negacyclic codes over finite fields of odd prime power elements. We
construct four families of MDS negacyclic LCD codes of length
, and a family of negacyclic LCD codes
of length . Furthermore, we obtain five families of -ary
Hermitian MDS negacyclic LCD codes of length and four
families of Hermitian negacyclic LCD codes of length For both
Euclidean and Hermitian cases the dimensions of these codes are determined and
for some classes the minimum distances are settled. For the other cases, by
studying and -cyclotomic classes we give lower bounds on the minimum
distance
Quantum Error-Control Codes
The article surveys quantum error control, focusing on quantum stabilizer
codes, stressing on the how to use classical codes to design good quantum
codes. It is to appear as a book chapter in "A Concise Encyclopedia of Coding
Theory," edited by C. Huffman, P. Sole and J-L Kim, to be published by CRC
Press