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A complete classification of quaternary complex Hadamard matrices of orders10, 12 and 14 is given, and a new parametrization scheme for obtaining newexamples of affine parametric families of complex Hadamard matrices isprovided. On the one hand, it is proven that all 10x10 and 12x12 quaternarycomplex Hadamard matrices belong to some parametric family, but on the otherhand, it is shown by exhibiting an isolated 14x14 matrix that there cannot be ageneral method for introducing parameters into these types of matrices.

The quaternary complex Hadamard matrices of orders 10, 12, and 14