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More on the optimal arrangement of $2d$ lines in $\mathbb{C}^d$

Joseph W. Iverson, John Jasper, Dustin G. Mixon

Abstract

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 conjecture holds for $d\leq 165$ by a computer-assisted application of a Newton-Kantorovich theorem. In addition, we supply numerical constructions that corroborate our conjecture for $d\leq 1500$.

More on the optimal arrangement of $2d$ lines in $\mathbb{C}^d$

Abstract

We introduce a new infinite family of equiangular tight frames. Many matrices in this family consist of two circulant blocks. We conjecture that such equiangular tight frames exist for every . We show that our conjecture holds for by a computer-assisted application of a Newton-Kantorovich theorem. In addition, we supply numerical constructions that corroborate our conjecture for .

Paper Structure

This paper contains 13 sections, 13 theorems, 86 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation and nomenclature
  4. Background
  5. Classical constructions
  6. Other ETFs of Paley size
  7. Recent breakthrough using doubling
  8. Doubling conference graphs
  9. Multi-generator harmonic collections of vectors
  10. Infinite families of complex $2$-circulant ETFs
  11. Continua of complex $2$-circulant ETFs
  12. Discussion
  13. Proof of Proposition \ref{['prop.paley conference matrices are the equivalent']}

Key Result

Proposition 4

Take any odd prime power $q$.

Figures (1)

  • Figure 1: Illustration of Example \ref{['ex.3x6']}. The vertices of a regular icosahedron partition into four lines of latitude. If we select coordinates so that the north pole points in the $(1,1,1)$ direction, then the vertices in the red and blue lines of latitude form a $2$-circulant representation of the real $3\times 6$ ETF.

Theorems & Definitions (47)

  • Conjecture 1: weak $d\times 2d$ conjecture; see FallonI:23
  • Example 2
  • Example 3
  • Proposition 4
  • Proposition 5: Theorem 6 in FallonI:23
  • Example 6
  • Example 7
  • Theorem 8
  • proof : Proof of Theorem \ref{['thm.doubling signature']}
  • Example 9
  • ...and 37 more