On the Classification of All Self-Dual Additive Codes over GF(4) of Length up to 12

Details On the Classification of All Self-Dual Additive Codes over GF(4) of Length up to 12 On the Classification of All Self-Dual Additive Codes over GF(4) of Length up to 12

2005-04-25

arxiv.org/abs/math/0504522v4

Journal of Combinatorial Theory, Series A 113(7), pp. 1351-1367, 2006

Mathematics

Combinatorics

18 pages, 4 figures

Scientific paper

10.1016/j.jcta.2005.12.004

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.

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