This paper is concerned with achieving optimal coherence for highly redundant real unit-norm frames. As the redundancy grows, the number of vectors in the frame becomes too large to admit equiangular arrangements. In this case, other geometric optimality criteria need to be identified. To this end, we use an iteration of the embedding technique by Conway, Hardin and Sloane. As a consequence of their work, a quadratic mapping embeds equiangular lines into a simplex in a real Euclidean space. Here, higher degree polynomial maps embed highly redundant unit-norm frames to simplices in high-dimensional Euclidean spaces. We focus on the lowest degree case in which the embedding is quartic.
The fundamental question concerning phase retrieval by projections in R<sup>m</sup> is what is the least number of projections needed and what dimensions can be used. We will look at recent advances concerning phase retrieval by orthogonal complements and phase retrieval by hyperplanes which raise a number of problems which would give a complete answer to this fundamental problem.
We study the problem of packing points in real or complex projective space so that the minimum distance is maximized. Existing bounds for this problem include the Welch and orthoplex bounds. This paper discusses the Levenstein bound, which follows from Delsarte's linear programming bound. We highlight the relationship between the Welch and Levenstein bounds.