Hadamard full propelinear codes with associated group $C_{2t}\times C_2$; rank and kernel

We introduce the Hadamard full propelinear codes that factorize as direct product of groups such that their associated group is $C_{2t}\times C_2$. We study the rank, the dimension of the kernel, and the structure of these codes. For several specific parameters we establish some links from circulant Hadamard matrices and the nonexistence of the codes we study. We prove that the dimension of the kernel of these codes is bounded by 3 if the code is nonlinear. We also get an equivalence between circulant complex Hadamard matrix and a type of Hadamard full propelinear code, and we find a new example of circulant complex Hadamard matrix of order 16.