Dimension-counting bounds for equi-isoclinic subspaces

TL;DR

The work addresses how to bound and realize optimal packings of $r$-dimensional subspaces in $\mathbb{F}^d$ by dimension-count arguments. It develops a quartet of results: a sharper lower bound on block coherence, an exact $\alpha$-EI count in a specific even-dimension regime, a new universal upper bound on the maximum number of equi-isoclinic subspaces, and infinite families where the bound is saturated via dimension counting and Radon–Hurwitz structure. The approach hinges on dimension counts of nested corner spaces $\mathcal{K}_j$ and normalization techniques that reduce the problem to lower ambient dimension when possible. The findings connect equi-isoclinic subspaces to Radon–Hurwitz numbers, Naimark complements, and tight fusion-frame theory, yielding both nonexistence results and constructive saturations in infinite families. These results advance the theory of spectral packings and have implications for fusion-frame design and high-dimensional subspace arrangements.

Abstract

We make four contributions to the theory of optimal subspace packings and equi-isoclinic subspaces: (1) a new lower bound for block coherence, (2) an exact count of equi-isoclinic subspaces of even dimension $r$ in $\mathbb{R}^{2r+1}$ with parameter $α\neq \tfrac{1}{2}$, (3) a new upper bound for the number of $r$-dimensional equi-isoclinic subspaces in $\mathbb{R}^d$ or $\mathbb{C}^d$, and (4) a proof that when $d=2r$, a further refinement of this bound is attained for every $r$ in the complex case and every $r=2^k$ in the real case. For each of these contributions, the proof ultimately relies on a dimension count.

Figures (1)

• Figure 1: The spark bound from Theorem \ref{['thm: spark bound for coherence']} provides a lower bound for block coherence that exceeds the Welch bound \ref{['eq:Welch']} in some cases, as shown wherever a black line hovers over a colored line above. Where the spark and Welch bounds coincide, a point is marked. Filled circles represent known subspace packings, while open circles represent packings whose existence or nonexistence is unknown by the authors. At the points marked with an x, no $\operatorname{EITFF}_{\mathbb{F}}(d,r,n)$ exists since $d/r > n-2$ and $d/r \notin\{n,n-1\}$.

Theorems & Definitions (50)

• Lemma 2.1: LemmensS:73
• Remark 2.2
• Remark 2.3
• Theorem 3.1: Spark bound for coherence
• Proposition 3.2: Coherence bound for spark
• proof : Proof of Proposition \ref{['prop: coherence bound for spark']}
• proof : Proof of Theorem \ref{['thm: spark bound for coherence']}
• Example 3.3
• Corollary 3.4
• proof : Proof of Corollary \ref{['cor: EITFF nonexistence']}