Circulant Hadamard matrices as HFP-codes of type $C_{4n}\times C_2$

We prove that a circulant Hadamard code of length $4n$ can always be seen as an HFP-code (Hadamard full propelinear code) of type $C_{4n}\times C_2$, where $C_2=⟨u⟩$ or the same, as a cocyclic Hadamard code. We compute the rank and dimension of the kernel of these kind of codes.