On the number of harmonic frames

TL;DR

This paper studies the growth and enumeration of cyclic harmonic frames, showing they grow roughly like n^{d-1} and providing formulas and estimates for their count.

Contribution

It proves the asymptotic growth of the number of cyclic harmonic frames and offers exact formulas and estimates using algebraic and group theoretic methods.

Findings

Asymptotic estimate: h_{n,d} ≈ n^d / φ(n) as n→∞

Growth rate of cyclic harmonic frames is approximately n^{d-1}

Provides exact formulas and counts for cyclic harmonic frames

Abstract

There is a finite number $h_{n,d}$ of tight frames of $n$ distinct vectors for $\mathbb{C}^d$ which are the orbit of a vector under a unitary action of the cyclic group $\mathbb{Z}_n$. These cyclic harmonic frames (or geometrically uniform tight frames) are used in signal analysis and quantum information theory, and provide many tight frames of particular interest. Here we investigate the conjecture that $h_{n,d}$ grows like $n^{d-1}$. By using a result of Laurent which describes the set of solutions of algebraic equations in roots of unity, we prove the asymptotic estimate $$ h_{n,d} \approx {n^d \over \varphi(n)}\ge n^{d-1}, \qquad n\to\infty. $$ By using a group theoretic approach, we also give some exact formulas for $h_{n,d}$, and estimate the number of cyclic harmonic frames up to projective unitary equivalence.