On weak isometries of Preparata codes

Two codes $C_1$ and $C_2$ are said to be weakly isometric if there exists a mapping $J\colon C_1\to C_2$ such that for all $x,y$ in $C_1$ the equality $d(x,y)=d$ holds if and only if $d(J(x),J(y))=d$, where $d$ is the code distance of $C_1$. We prove that Preparata codes of length $n\ge2^{12}$ are weakly isometric if and only if the codes are equivalent. A similar result is proved for punctured Preparata codes of length at least $2^{10}-1$.