On /Zopf/(4)-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 /Zopf/(2)//alpha// x /Zopf/(4)//beta//, where the quaternary coordinates are transformed to binary by means of the usual Gray map and hence /alpha/ 2/beta/ = n. In this paper, we prove that any additive extended Preparata (1968) -like code always verifies /alpha/ = 0, i.e., it is always a /Zopf/(4)-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 /Zopf/(4)-dual of these codes, i.e., the /Zopf/(4)-linear Kerdock-like codes.