On the kernel and rank of Z 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 Z2 ×Zβ4 , 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-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. ∗Research partially supported by Spanish CICYT Grant TIC2000-0739-c04-01, by Catalan DURSI Grant 2001SGR 00219 and also by Ministerio de educación, cultura y deporte Grant SAB 2000-0058. †K.T. Phelps is with the Discrete & Statistical Sciences, Auburn University, Auburn, Al 36849-5307. USA. E-mail: [email protected]. ‡J. Borges and J. Rifà are with the Computer Science Department, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain. E-mail: {joaquim.borges, josep.rifa}@uab.es. §V.A. Zinoviev is with the Institute for Problems of Information Transmission of the Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 101447, Russia. E-mail: [email protected].