  Additive codes over GF(9) that are self-dual with respect to the Hermitian
trace inner product have a natural application in quantum information theory,
where they correspond to ternary quantum error-correcting codes. However, these
codes have so far received far less interest from coding theorists than
self-dual additive codes over GF(4), which correspond to binary quantum codes.
Self-dual additive codes over GF(9) have been classified up to length 8, and in
this paper we extend the complete classification to codes of length 9 and 10.
The classification is obtained by using a new algorithm that combines two graph
representations of self-dual additive codes. The search space is first reduced
by the fact that every code can be mapped to a weighted graph, and a different
graph is then introduced that transforms the problem of code equivalence into a
problem of graph isomorphism. By an extension technique, we are able to
classify all optimal codes of length 11 and 12. There are 56,005,876
(11,3^11,5) codes and 6493 (12,3^12,6) codes. We also find the smallest codes
with trivial automorphism group.
