333 research outputs found
\u3ci\u3es\u3c/i\u3e-Extremal Additive \u3cb\u3eF\u3csub\u3e4\u3c/sub\u3e\u3c/b\u3e Codes
Binary self-dual codes and additive self-dual codes over F4 have in common interesting properties, for example, Type I, Type II, shadows, etc. Recently Bachoc and Gaborit introduced the notion of s-extremality for binary self-dual codes, generalizing Elkies\u27 study on the highest possible minimum weight of the shadows of binary self-dual codes. In this paper, we introduce a concept of s-extremality for additive self-dual codes over F4, give a bound on the length of these codes with even distance d, classify them up to minimum distance d = 4, give possible lengths and (shadow) weight enumerators for which there exist s-extremal codes with 5 ≤ d ≤ 11 and give five s-extremal codes with d = 7. We construct four s-extremal codes of length n = 13 and minimum distance d = 5. We relate an s-extremal code of length 3d to another s-extremal code of that length, and produce extremal Type II codes from s-extremal codes
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
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
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
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