Related papers: Equiangular Tight Frames from Paley Tournaments

We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. In particular, we show that any finite tight frame can be written as a union of prime tight frames. We then characterize…

We introduce a new infinite family of $d\times 2d$ equiangular tight frames. Many matrices in this family consist of two $d\times d$ circulant blocks. We conjecture that such equiangular tight frames exist for every $d$. We show that our…

Finite tight frames for polynomial subspaces are constructed using monic Hahn polynomials and Krawtchouk polynomials of several variables. Based on these polynomial frames, two methods for constructing tight frames for the Euclidean spaces…

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…

We show that there does not exist a complex $d\times n$ equiangular tight frame with \[ d^2-d+1<n<d^2. \] The proof, which originated from an internal model at OpenAI, mimics the relationship between real equiangular tight frames and…

We introduce the notion of equiangular tight frames in real symplectic spaces and formulate a conjecture on their existence in terms of the dimension and number of vectors. Our main results shows the "symplectic Zauner's conjecture" is…

We derive easily verifiable conditions which characterize when complex Seidel matrices containing cube roots of unity have exactly two eigenvalues. The existence of such matrices is equivalent to the existence of equiangular tight frames…

Finite tight frames play an important role in miscellaneous areas, including quantum information theory. Here we apply a class of tight frames, equiangular tight frames, to address the problem of detecting the entanglement of bipartite…

We present all nontrivial real equiangular tight frames $\{\varphi_m\}_{m=1}^M$ in $\mathbb{R}^N$ obtained as spherical embeddings of primitive rank $3$ graphs on $M$ vertices, and those such that one of their associated $M$ strongly…

In this work, we show that a complex equiangular tight frame (ETF) composed by $N$ vectors in dimension $d$ exists if and only if a certain bistochastic matrix, univocally determined by $N$ and $d$, belongs to a special class of…

We derive various interesting properties of complex equiangular cyclic frames for many pairs (n, k) using Gauss sums and number theory. We further use these results to study the random and burst errors of some special cases of complex…

Two decades ago, Zauner conjectured that for every dimension $d$, there exists an equiangular tight frame consisting of $d^2$ vectors in $\mathbb{C}^d$. Most progress to date explicitly constructs the promised frame in various dimensions,…

We show that much of the theory of finite tight frames can be generalised to vector spaces over the quaternions. This includes the variational characterisation, group frames, and the characterisations of projective and unitary equivalence.…

Tight triangulations are exotic, but highly regular objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying…

Let $q\geq 2$ be an integer, and $\Bbb F_q^d$, $d\geq 1$, be the vector space over the cyclic space $\Bbb F_q$. The purpose of this paper is two-fold. First, we obtain sufficient conditions on $E \subset \Bbb F_q^d$ such that the inverse…

We study several interesting examples of Biangular Tight Frames (BTFs) - basis-like sets of unit vectors admitting exactly two distinct frame angles (ie, pairwise absolute inner products) - and examine their relationships with Equiangular…

We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular $(k,n)$ frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the…

Frames have become standard tools in signal processing due to their robustness to transmission errors and their resilience to noise. Equiangular tight frames (ETFs) are particularly useful and have been shown to be optimal for transmission…

This paper studies group frames ($G$-frames) where the unitary group representation can be projective. When the group is abelian, for most combinations $N, n$, we show that $ETF(N,n)$ can only exist for genuinely projective group…

We study the properties of a set of vectors called tight frames that obtained as the orthogonal projection of some orthonormal basis of $\R^n$ onto $\R^k.$ We show that a set of vectors is a tight frame if and only if the set of all cross…