127 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
Quantum MDS Codes over Small Fields
We consider quantum MDS (QMDS) codes for quantum systems of dimension
with lengths up to and minimum distances up to . We show how
starting from QMDS codes of length based on cyclic and constacyclic
codes, new QMDS codes can be obtained by shortening. We provide numerical
evidence for our conjecture that almost all admissible lengths, from a lower
bound on, are achievable by shortening. Some additional codes that
fill gaps in the list of achievable lengths are presented as well along with a
construction of a family of QMDS codes of length , where , that
appears to be new.Comment: 6 pages, 3 figure
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