On Z4-linear Preparata-like and Kerdock-like codes

We say that a binary code of length n is additive if it is isomorphic to a subgroup of Z₂^α × Z₄^β, where the quaternary coordinates are transformed to binary by means of the usual Gray map and hence α + 2β = n. In this paper, we prove that any additive extended Preparata (1968) -like code always verifies α = 0, i.e., it is always a Z₄-linear code. Moreover, we compute the rank and the dimension of the kernel of such Preparata-like codes and also the rank and the kernel of the Z₄-dual of these codes, i.e., the Z₄-linear Kerdock-like codes.