Circulant Self-dual Codes With Symplectic Inner Product

In this paper, we define symplectic inner product in linear codes, which is different from the usual Euclidean and Hermitian inner products, and consider circulant self-dual codes of small length over F3 and F5 with this inner product. We calculate the minimum weights for such codes by C++ programming. Meanwhile, we prove that the weight enumerator of self-dual codes is the invariant of a group. And we calculate the invariant rings. So we get the weight enumerator of circulant symplectic self-dual codes and give their complete classi-fications.