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Tables of the existence of equiangular tight frames
Matthew Fickus, Dustin G. Mixon
TL;DR
The paper addresses the need to identify and organize equiangular tight frames, which constitute most explicit Grassmannian frames. It surveys known ETF constructions and builds existence tables, finding combinatorial constructions and sharply uneven knowledge of real and complex existence. The resulting resource records known families and dimensions while highlighting that complex nonexistence results remain limited.
Problem
The paper addresses the need for a comprehensive, updateable account of ETF constructions and existence across dimensions.
Method
The paper surveys known ETF families, uses construction results including strongly regular graphs and Steiner systems, and develops existence tables for small dimensions.
Results
Real ETF existence is characterized in part through strongly regular graphs, while known complex nonexistence beyond general bounds is limited to (M, N) = (3, 8) and (5, 8).
Takeaways & Limitations
The tables provide a living record of known ETF constructions and impossibility results, with complex tables listing only parameters known to admit ETFs.
Takeaways & Limitations
For complex ETFs, only one nonexistence result is known beyond the standard general condition, and its Gröbner-basis proof offers no evident generalization.
Abstract
from arXiv · showhide
A Grassmannian frame is a collection of unit vectors which are optimally incoherent. To date, the vast majority of explicit Grassmannian frames are equiangular tight frames (ETFs). This paper surveys every known construction of ETFs and tabulates existence for sufficiently small dimensions.
1 Introduction
The paper frames Grassmannian-frame construction through the Welch bound: equality requires equiangular tight frames, which are therefore accessible optimal configurations. It then presents a living survey of ETF constructions and existence tables.
- Grassmannian frames minimize worst-case coherence among finite unit-vector sequences on real or complex spheres.
- ETFs attain equality in the Welch bound and are consequently Grassmannian frames, while their structure makes them easier to identify.The paper contrasts their accessibility with the general difficulty of identifying Grassmannian frames.
- The survey is designed as a living document, updated periodically to incorporate new ETF constructions and impossibility results.
- The paper summarizes infinite ETF families, develops real and complex cases, examines redundancy near 2, and constructs existence tables for small dimensions.The tables appear at the end of the paper after the methodology discussion.
2 Infinite families of equiangular tight frames
Known nontrivial infinite ETF families are built from combinatorial designs rather than directly from their functional-analytic characterization. The section outlines constructions from strongly regular graphs, difference sets, and Steiner systems.
- The paper sets aside trivial ETFs, including orthonormal bases with N = M, regular simplices with N = M + 1, and one-dimensional frames.
- Known nontrivial infinite ETF families rely on combinatorial designs, despite ETFs being characterized through equality in the Welch bound.
- Strongly regular graphs can yield real ETFs with N = v + 1 vectors by appropriately manipulating their adjacency matrices.
- Difference sets construct ETFs with M = |D| and N = |G| by restricting group characters to the difference set before normalization.
- Steiner systems produce ETFs by embedding regular simplices on blocks containing each point, yielding v(r + 1) vectors in C^B and necessarily N > 2M.
3 Real equiangular tight frames
Real ETFs are constrained by dimension, integrality, and graph-theoretic conditions, while known constructions connect them to strongly regular graphs and tight spherical 5-designs.
- Real ETF existence implies the dimension bound N ≤ M(M + 1)/2, with the Naimark complement providing a complementary construction in dimension N − M.The rank-1 matrices impose the dimension bound, while completing suitably scaled ETF rows to an orthonormal basis yields another ETF.
- Real ETFs correspond exactly to strongly regular graphs with parameters determined by M and N.After sign normalization and removal of one vector, negative inner products define a strongly regular graph on N − 1 vertices.
- For N ≠ 2M, real ETF existence requires specified integrality conditions, including that two resulting quantities are odd integers.These conditions arise from the requirement that eigenvalues associated with the corresponding strongly regular graph be integral.
- Strongly regular graph tests add Krein, absolute bound, and multiplicity constraints, but none ruled out parameter pairs already allowed by earlier real-ETF conditions in the tested tables.The real-ETF specialization gives μ = k/2, and the tested necessary conditions did not eliminate plausible pairs.
- 3.1 Maximal real ETFs: Every tight spherical 5-design is the union Φ ∪ (−Φ) for an ETF Φ with M(M − 1)/2 elements, and conversely.Thus tight spherical 5-designs and these real ETFs characterize one another.
- 3.1 Maximal real ETFs: For maximal real ETFs with N = M(M + 1)/2, the dimension must lie in {3, 7, 23, 47, 79, 119, 167, . . .}, subject to further known construction and nonexistence results.For M ≠ 3, the necessary form is M = (2m + 1)^2 − 2; one listed result supplies additional constructions under square-free conditions.
4 Complex equiangular tight frames
Complex ETF theory has few general nonexistence tools, so progress relies chiefly on construction-based existence results. The section surveys difference-set families and maximal ETFs, including known evidence relevant to Zauner’s conjecture.
- 4 Complex equiangular tight frames: Complex ETF existence is governed mainly by constructions because only one general necessary condition and very few nonexistence results are known.Beyond the standard bounds, nonexistence is known for (M, N) = (3, 8) and (5, 8), while Gröbner-basis methods do not appear to generalize.
- 4.1 Difference sets: Difference sets in finite abelian groups provide a broad source of complex ETFs through character restrictions, with multiple parameterized families summarized by Theorem 8.The constructions include Paley and cyclotomic families, subject to conditions such as prime-power parameters.
- 4.1 Difference sets: The surveyed difference-set families impose arithmetic conditions including prime-power requirements, congruences, and twin-prime-power hypotheses.The source also corrects a condition on Hall difference sets and notes that the twin prime powers in one family must be odd.
- 4.2 Maximal ETFs: Maximal complex ETFs have N = M^2 vectors and correspond to SIC-POVMs in quantum mechanics, where they have applications in quantum Bayesianism and state tomography.Their existence is conjectured for every M ≥2 by Zauner’s conjecture.
- 4.2 Maximal ETFs: Maximal ETFs are known whenever M ≤17 or M ∈{19, 24, 28, 35, 48}, while no infinite family of maximal ETFs is currently known.Numerical tests suggest Zauner’s conjecture is likely true through dimension 67, at least within machine precision.
5 Redundancy 2, more or less
The redundancy-two regime connects ETFs to conference matrices and complex Hadamard matrices, yielding several arithmetic and matrix-based existence families. Related constructions also produce ETFs with one fewer vector, while broader Hadamard classifications extend the catalog.
- 5 Redundancy 2, more or less: Real redundancy-two ETFs require M to be odd and 2M −1 to be expressible as a sum of two squares.This is stated as a necessary condition rather than a construction theorem.
- 5 Redundancy 2, more or less: Conference-matrix constructions yield redundancy-two ETFs under several conditions, including prime-power congruences and specified matrix orders.Theorem 12 gives real ETFs with N = 2M for q + 1 when q ≡1 mod 4, while Theorem 12’s listed families include additional prime-power conditions.
- 5 Redundancy 2, more or less: Removing one vector from an ETF associated with an antisymmetric conference matrix and renormalizing produces another ETF with parameters (N/2, N −1).The operation uses Φ = (αΨΨ∗)^−1/2Ψ with α = M/(2M −1).
- 5 Redundancy 2, more or less: In the near-redundancy-two range, an ETF Gram matrix is equivalent to a complex Hadamard matrix with constant diagonal and self-adjoint off-diagonal.The relation is expressed through I + µQ and λI + Q, with the latter required to be complex Hadamard.
- 5 Redundancy 2, more or less: Self-adjoint complex Hadamard matrices with constant diagonal provide an infinite family of order n^2 for every n ≥2.These matrices satisfy the specifications needed for the near-redundancy-two ETF correspondence.
6 Table methodology
The paper constructs living existence tables by systematically testing parameter constraints and recording known ETF constructions and nonexistence results. Separate procedures cover real ETFs, complex ETFs, and the special case N = 2M under explicit dimension and source restrictions.
- Real ETFs: The real-ETF table searches admissible parameters, tests strongly regular graph conditions, records construction families, and marks certain nonexistence cases as DNE.The workflow includes integrality conditions, graph tests for (M, N) and (N −M, N), difference sets, Steiner systems, other constructions, nonexistence theorems, and strongly regular graphs.
- Scope and restrictions: The real tables focus on N > 2M, with N = 2M handled separately and N ≤1300 imposed alongside the standard upper bound N ≤M(M + 1)/2.The N = 2M case uses fundamentally different necessary conditions, and the N ≤1300 cutoff is described as somewhat arbitrary.
- Real ETFs: Strongly regular graph tables fill the remaining known real ETF constructions after the existing ETF literature is exhausted, yielding four constructions labeled SRG.The table uses Theorem 2 and the strongly regular graph table in [10].
- Complex ETFs: The complex table lists only known ETF parameters because no integrality conditions narrow the search, with only two stated nonexistence cases: (M, N) = (3, 8) and (5, 8).Within M ≤300 and N ≤1300, the table accounts for every known complex ETF in that range.
- Complex ETFs: The complex procedure gathers constructions from difference sets, Steiner systems, modified conference matrices, several design families, hyperovals, and maximal ETFs.The authors omit difficult-to-parameterize Davis–Jedwab–Chen and Hadamard difference sets, then consult the La Jolla repository to recover missed pairs in range.
- N = 2M: For N = 2M, the authors test Theorems 11 and 12, classify real-construction status, and identify every known conference graph tabulated in.They also note isolated non-conference-matrix constructions for (2, 4), (3, 6), and (5, 10), which are excluded from the table.