Two-circulant equiangular tight frames in every dimension

Determine whether the multiplying and power constructions for equiangular tight frames can produce 2-circulant ETFs of size d × 2d.

Background

The paper mentions the medium version of a conjecture concerning the existence of 2-circulant ETFs with 2d vectors in complex dimension d. It leaves unresolved whether the two constructions developed in the paper—the multiplying construction and the Kronecker power construction—can realize ETFs having this additional 2-circulant structure.

References

The medium version of Conjecture \ref{conj:double} Conjecture 20 states the existence of 2-circulant ETFs of size $d\times 2d$. It would be interesting to check whether the multiplying and power constructions can produce 2-circulant ETFs.

— New constructions of optimal arrangements of $2d$ lines in $\mathbb{C}^d$  (2608.16116 - Glazyrin, 17 Aug 2026) in Section 6, Discussion