458 research outputs found
Quantum Goethals-Preparata Codes
We present a family of non-additive quantum codes based on Goethals and
Preparata codes with parameters ((2^m,2^{2^m-5m+1},8)). The dimension of these
codes is eight times higher than the dimension of the best known additive
quantum codes of equal length and minimum distance.Comment: Submitted to the 2008 IEEE International Symposium on Information
Theor
Quantum Error Correction via Codes over GF(4)
The problem of finding quantum error-correcting codes is transformed into the
problem of finding additive codes over the field GF(4) which are
self-orthogonal with respect to a certain trace inner product. Many new codes
and new bounds are presented, as well as a table of upper and lower bounds on
such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information
Theory. Replaced Sept. 24, 1996, to correct a number of minor errors.
Replaced Sept. 10, 1997. The second section has been completely rewritten,
and should hopefully be much clearer. We have also added a new section
discussing the developments of the past year. Finally, we again corrected a
number of minor error
From Skew-Cyclic Codes to Asymmetric Quantum Codes
We introduce an additive but not -linear map from
to and exhibit some of its interesting
structural properties. If is a linear -code, then is an
additive -code. If is an additive cyclic code then
is an additive quasi-cyclic code of index . Moreover, if is a module
-cyclic code, a recently introduced type of code which will be
explained below, then is equivalent to an additive cyclic code if is
odd and to an additive quasi-cyclic code of index if is even. Given any
-code , the code is self-orthogonal under the trace
Hermitian inner product. Since the mapping preserves nestedness, it can be
used as a tool in constructing additive asymmetric quantum codes.Comment: 16 pages, 3 tables, submitted to Advances in Mathematics of
Communication
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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