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All generalized Hadamard matrices of order 18 over a group of order 3,H(6,3), are enumerated in two different ways: once, as class regular symmetric(6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with agroup of order 3 acting semi-regularly on points and blocks, and secondly, ascollections of full weight vectors in quaternary Hermitian self-dual codes oflength 18. The second enumeration is based on the classification of Hermitianself-dual [18,9] codes over GF(4), completed in this paper. It is shown that upto monomial equivalence, there are 85 generalized Hadamard matrices H(6,3), and245 inequivalent Hermitian self-dual codes of length 18 over GF(4).

Classification of generalized Hadamard matrices H(6,3) and quaternary Hermitian self-dual codes of length 18