Constructions of equi-norm tight frames with minimal completion

In this paper, we present explicit constructions of equi-norm tight frame matrices that include a prescribed set of unit-norm vectors. Our method provides the deterministic procedures for completing a given equi-norm matrix with the minimal number of additional vectors, thereby avoiding the computational complexity associated with the Bendel–Mickey algorithm. Depending on the available spectral information, either the full decomposition or only the largest singular value, we propose efficient strategies for the frame completion. The explicit character of the proposed constructions further enables analytical investigation and facilitates extensions to frame matrices with additional structural constraints.