On Regular Schemes and Tight Frames

Malcolm Egan (1, 2, 3)

1 CITI - CITI Centre of Innovation in Telecommunications and Integration of services
2 SOCRATE - Software and Cognitive radio for telecommunications
3 MARACAS - Modèle et algorithmes pour des systèmes de communication fiables

Finite frames are sequences of vectors in finite dimensional Hilbert spaces that play a key role in signal processing and coding theory. In this paper, we study the class of tight unit-norm frames for C d that also form regular schemes, called tight regular schemes (TRS). Many common frames that arise in applications such as equiangular tight frames and mutually unbiased bases fall in this class. We investigate characteristic properties of TRSs and prove that for many constructions, they are intimately connected to weighted 1-designs—arising from quadrature rules for integrals over spheres in C^d —with weights dependent on the Voronoi regions of each frame element. Aided by additional numerical evidence, we conjecture that all TRSs in fact satisfy this property.

Domaines: Théorie de l'information [cs.IT]; Combinatoire [math.CO]

Malcolm Egan. On Regular Schemes and Tight Frames. SETA 2018 - Sequences and Their Applications, Oct 2018, Hong Kong, Hong Kong SAR China. pp.1-12. ⟨hal-01839448⟩