About the systems with full spark

Frames of a finite-dimensional Euclidean and unitary spaces composed of discrete Fourier transform matrices are considered. The relationship of phaseless reconstruction systems with the alternative completeness property is presented. In the complex case, alternative completeness is only a necessary condition for phaseless reconstruction. A system of vectors is constructed such that each of its subsystems with a volume equal to the dimension of space is linearly independent. These systems are called systems with full spark. In particular, such systems are optimal for phase retrieval.