John Jasper

Kirkman Equiangular Tight Frames and Codes

An equiangular tight frame (ETF) is a set of unit vectors in a Euclidean space whose coherence is as small as possible, equaling the Welch bound. Also known as Welch-bound-equality sequences, such frames arise in various applications, such as waveform design and compressed sensing. At the moment, there are only two known flexible methods for constructing ETFs: harmonic ETFs are formed by carefully extracting rows from a discrete Fourier transform; Steiner ETFs arise from a tensor-like combination of a combinatorial design and a regular simplex. These two classes seem very different: the vectors in harmonic ETFs have constant amplitude, whereas Steiner ETFs are extremely sparse. We show that they are actually intimately connected: a large class of Steiner ETFs can be unitarily transformed into constant-amplitude frames, dubbed Kirkman ETFs. Moreover, we show that an important class of harmonic ETFs is a subset of an important class of Kirkman ETFs. This connection informs the discussion of both types of frames: some Steiner ETFs can be transformed into constant-amplitude waveforms making them more useful in waveform design; some harmonic ETFs have low spark, making them less desirable for compressed sensing. We conclude by showing that real-valued constant-amplitude ETFs are equivalent to binary codes that achieve the Grey-Rankin bound, and then construct such codes using Kirkman ETFs.

Equiangular Tight Frames From Hyperovals

An equiangular tight frame (ETF) is a set of equal norm vectors in a Euclidean space whose coherence is as small as possible, equaling the Welch bound. Also known as Welch-bound-equality sequences, such frames arise in various applications, such as waveform design, quantum information theory, compressed sensing, and algebraic coding theory. ETFs seem to be rare, and only a few methods of constructing them are known. In this paper, we present a new infinite family of complex ETFs that arises from hyperovals in finite projective planes. In particular, we give the first ever construction of a complex ETF of 76 vectors in a space of dimension 19. Recently, a computer-assisted approach was used to show that real ETFs of this size do not exist, resolving a longstanding open problem in this field. Our construction is a modification of a previously known technique for constructing ETFs from balanced incomplete block designs.

Characterization of sequences of frame norms

Abstract We show that frames with frame bounds A and B are images of orthonormal bases under positive operators with spectrum contained in . Then, we give an explicit characterization of the diagonals of such operators, which in turn gives a characterization of the sequences which are the norms of a frame. Our result extends the tight case result of Kadison [Proc. Natl. Acad. Sci. USA 99: 4178–4184, 2002], [Proc. Natl. Acad. Sci. USA 99: 5217–5222, 2002], which characterizes diagonals of orthogonal projections, to a non-tight case. We illustrate our main theorem by studying the set of possible lower bounds of positive operators with prescribed diagonal.

The Schur-Horn Theorem for operators with finite spectrum

We characterize the set of diagonals of the unitary orbit of a self-adjoint operator with a finite spectrum. Our result extends the Schur-Horn theorem from a finite dimensional setting to an infinite dimensional Hilbert space analogous to Kadisons theorem for orthogonal projections and the second authors result for operators with three point spectrum.

Tremain equiangular tight frames

Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph.

Constructive Proof of Carpenter's Theorem

We give a constructive proof of the carpenters theorem due to Kadison. Unlike the original proof, our approach also yields the real case of this theorem.

Existence of frames with prescribed norms and frame operator

In this chapter we survey several recent results on the existence of frames with prescribed norms and frame operator. These results are equivalent to Schur-Horn type theorems which describe possible diagonals of positive self-adjoint operators with specified spectral properties. The first infinite dimensional result of this type is due to Kadison who characterized diagonals of orthogonal projections. Kadison’s theorem automatically gives a characterization of all possible sequences of norms of Parseval frames. We present some generalizations of Kadison’s result such as (a) the lower and upper frame bounds are specified, (b) the frame operator has two point spectrum, and (c) the frame operator has a finite spectrum.

Packings in real projective spaces

This paper applies techniques from algebraic and differential geometry to determine how to best pack points in real projective spaces. We present a computer-assisted proof of the optimality of a particular 6-packing in

Steiner equiangular tight frames redux

\mathbb{R}\mathbf{P}^3

Quasi-symmetric designs and equiangular tight frames

, we introduce a linear-time constant-factor approximation algorithm for packing in the so-called Gerzon range, and we provide local optimality certificates for two infinite families of packings. Finally, we present perfected versions of various putatively optimal packings from Sloanes online database, along with a handful of infinite families they suggest, and we prove that these packings enjoy a certain weak notion of optimality.