A testing set for Preparata-like codes

The reconstruction of an object of a given class by its intersection with some (so-called testing) set is studied. For the class, we consider Preparata-like codes, i. e. nonlinear codes of length $n=2^{2m}-1,$ $m=2,3,\dots,$ with code distance $5$ and twice the size of a linear code of the same length and distance. We determine conditions under which the union of a few concentric spheres forms the testing set for Preparata-like codes.