3,109 research outputs found
On the Classification of All Self-Dual Additive Codes over GF(4) of Length up to 12
We consider additive codes over GF(4) that are self-dual with respect to the
Hermitian trace inner product. Such codes have a well-known interpretation as
quantum codes and correspond to isotropic systems. It has also been shown that
these codes can be represented as graphs, and that two codes are equivalent if
and only if the corresponding graphs are equivalent with respect to local
complementation and graph isomorphism. We use these facts to classify all codes
of length up to 12, where previously only all codes of length up to 9 were
known. We also classify all extremal Type II codes of length 14. Finally, we
find that the smallest Type I and Type II codes with trivial automorphism group
have length 9 and 12, respectively.Comment: 18 pages, 4 figure
Additive Asymmetric Quantum Codes
We present a general construction of asymmetric quantum codes based on
additive codes under the trace Hermitian inner product. Various families of
additive codes over \F_{4} are used in the construction of many asymmetric
quantum codes over \F_{4}.Comment: Accepted for publication March 2, 2011, IEEE Transactions on
Information Theory, to appea
Graph-Based Classification of Self-Dual Additive Codes over Finite Fields
Quantum stabilizer states over GF(m) can be represented as self-dual additive
codes over GF(m^2). These codes can be represented as weighted graphs, and
orbits of graphs under the generalized local complementation operation
correspond to equivalence classes of codes. We have previously used this fact
to classify self-dual additive codes over GF(4). In this paper we classify
self-dual additive codes over GF(9), GF(16), and GF(25). Assuming that the
classical MDS conjecture holds, we are able to classify all self-dual additive
MDS codes over GF(9) by using an extension technique. We prove that the minimum
distance of a self-dual additive code is related to the minimum vertex degree
in the associated graph orbit. Circulant graph codes are introduced, and a
computer search reveals that this set contains many strong codes. We show that
some of these codes have highly regular graph representations.Comment: 20 pages, 13 figure
New self-dual additive -codes constructed from circulant graphs
In order to construct quantum codes for ,
, , , , , , ,
, , , , we construct self-dual additive
-codes of length and minimum weight from circulant
graphs. The quantum codes with these parameters are constructed for the first
time.Comment: 11 page
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
- …