Representation of $\mathbb Z_4$-Linear Preparata Codes by Means of Vector Fields

A binary code is called $\mathbb Z_4$-linear if its quaternary Gray map preimage is linear. We show that the set of all quaternary linear Preparata codes of length $n=2^m$, $m$ odd, $m\ge3$, is nothing more than the set of codes of the form $\mathcal H_{\lambda,\psi}+\mathcal M$ with
$$ \mathcal H_{\lambda,\psi}=\{y+T_\lambda(y)+S_\psi(y)\mid y\in H^n\},\qquad \mathcal M=2H^n, $$
where $T_\lambda(\,\cdot\,)$ and $S_\psi(\,\cdot\,)$ are vector fields of a special form defined over the binary extended linear Hamming code $H^n$ of length $n$. An upper bound on the number of nonequivalent quaternary linear Preparata codes of length $n$ is obtained, namely, $2^{n\log_2n}$. A representation for binary Preparata codes contained in perfect Vasil'ev codes is suggested.