OP-00199 Partially resolved

Complex equiangular tight frames with twice the dimension

Unit vectors $v_1, \dots, v_n \in \mathbb{C}^d$ with $n > d$ satisfy the Welch bound $\max_{i \ne j} |v_i^* v_j|^2 \ge \frac{n - d}{d(n - 1)}$, and the configurations attaining it are the equiangular tight frames; for $n = 2d$ the bound is $\frac{1}{2d - 1}$. A symmetric or skew-symmetric conference matrix of order $2d$ gives an equiangular tight frame of $2d$ vectors in $\mathbb{C}^d$; such matrices exist for instance when $2d - 1$ is a prime power. Fallon and Iverson constructed them whenever a skew Hadamard matrix of order $d + 1$ exists, and Iverson, Jasper and Mixon proved, with computer assistance, that they exist for every $d \le 165$.

Problem. Is there, for every integer $d \ge 2$, an equiangular tight frame of $2d$ vectors in $\mathbb{C}^d$, that is, unit vectors $v_1, \dots, v_{2d} \in \mathbb{C}^d$ with $$\sum_{i=1}^{2d} v_i v_i^* = 2 I_d \quad \text{and} \quad |v_i^* v_j|^2 = \frac{1}{2d - 1}$$ for all $i \ne j$?

Source: Fallon and Iverson (2025), Conjecture 4.

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: FA — Functional Analysis
Secondary: CO, MG
Keywords: conference matrices, equiangular tight frames, line packings, OpenProblemsInNLA
Literature checked: Oct. 8, 2026 by clemens-admin

References

@article{fallon2025,
  author        = {Fallon, Kean and Iverson, Joseph W.},
  title         = {On the optimal arrangement of $2d$ lines in $\mathbb{C}^d$},
  journal       = {Inf. Inference},
  volume        = {14},
  number        = {2},
  year          = {2025},
  pages         = {iaaf008},
  doi           = {10.1093/imaiai/iaaf008},
  eprint        = {2312.09975},
  archivePrefix = {arXiv},
}
@misc{iverson2024,
  author        = {Iverson, Joseph W. and Jasper, John and Mixon, Dustin G.},
  title         = {More on the optimal arrangement of $2d$ lines in $\mathbb{C}^d$},
  year          = {2024},
  eprint        = {2410.17379},
  archivePrefix = {arXiv},
}
@misc{glazyrin2026,
  author        = {Glazyrin, Alexey},
  title         = {New constructions of optimal arrangements of $2d$ lines in $\mathbb{C}^d$},
  year          = {2026},
  eprint        = {2608.16116},
  archivePrefix = {arXiv},
}
@misc{openproblemsinnla-fr09,
  author       = {Townsend, Alex},
  title        = {{FR-09}: Complex equiangular tight frames with twice the dimension},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/frames-and-matrix-designs/FR-09/README.md},
  note         = {Read at commit eb7c9d6, 2026-10-07},
}

Resolution claims

Rational contraction certificates for redundancy-two complex ETFs v1 GitHub ↗

Matthew J. Colbrook

This exposition, written by an AI agent at OpenProblemsInNLA from certificates and programs supplied by the author, proves with finite rational certificates that Hermitian conference matrices of order $2d$, and hence unit-norm complex equiangular tight frames of $2d$ vectors in $\mathbb{C}^d$, exist for $d=166$, $209$, $256$ and $1505$; it claims no priority for these dimensions. Two sequences of unimodular phases …

Partial resolution · not verified · fidelity not established · posted on GitHub 11 Sep 2026

Status history

When	Change	By	Reason
2026-10-08	Open → Partially resolved	The registry's curators	Rational contraction certificates for redundancy-two complex ETFs v1 recorded from OpenProblemsInNLA @ eb7c9d6, colbrook-frames-2026-09-11/conference-proof.md.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00199, "Complex equiangular tight frames with twice the dimension", clopen.solutions.