بر تعداد فریم های هارمونیک

There is a finite number hn,d of tight frames of n distinct vectors for Cd which are the orbit of a vector under a unitary action of the cyclic group Zn. 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 hn,d grows like nd−1. By using a result of Laurent which describes the set of solutions of algebraic equations in roots of unity, we prove the asymptotic estimatehn,d≈ndφ(n)≥nd−1,n→∞. By using a group theoretic approach, we also give some exact formulas for hn,d, and estimate the number of cyclic harmonic frames up to projective unitary equivalence.