New constructions of optimal arrangements of $2d$ lines in $\mathbb{C}^d$

Abstract

In this paper we provide new constructions of equiangular tight frames of size $2d$ in $\mathbb{C}^d$. We generalize the doubling construction of Fallon and Iverson to a tensor multiplication construction based on a suitable pair consisting of a complex Hadamard matrix and an equiangular tight frame. In particular, such a pair always exists whenever there is an amicable pair of real Hadamard matrices. Most notably, amicable Hadamard pairs of order $q+1$ exist for all prime powers $q\equiv 3\pmod 4$. We also find specific constructions based on a family of pairs of order 6 and on pairs whose equiangular tight frames are defined by Paley conference matrices with $q\equiv 1\pmod 4$. Finally, we provide a power construction of equiangular tight frames that generalizes the construction of Turyn for conference matrices.

Disclosure

“6. Acknowledgments Alexey Glazyrin was partially supported by the NSF grants DMS-2054536, DMS-2349063 and the Simons Foundation’s Travel Support for Mathematicians program. 7. AI statement ChatGPT 4.6 Sol was used for literature search and light copyediting throughout the paper. The constructions in Subsections 3.3, 3.4 and the argument for Proposition 3 were found with AI assistance. All text in the paper was written by the author,”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T05:18:42.599742Z

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