[Paper Review] Equiangular lines, Incoherent sets and Quasi-symmetric designs

Neil I. Gillespie|arXiv (Cornell University)|Sep 15, 2018

Mathematical Approximation and Integration27 references5 citations

TL;DR

This paper establishes that equiangular lines in $×^d$ saturating both the absolute bound $d(d+1)/2$ and the incoherence bound $d$ exist if and only if $d = 2, 3, 7,$ or $23$, resolving a long-standing classification problem. It links such sets to tight spherical 5-designs and quasi-symmetric designs, proving that only these dimensions support configurations saturating both bounds, with a surprising result that the maximal 16-line set in $\mathbb{R}^6$ is the only additional example under further assumptions.

ABSTRACT

The absolute upper bound on the number of equiangular lines that can be found in $\mathbf{R}^d$ is $d(d+1)/2$. Examples of sets of lines that saturate this bound are only known to exist in dimensions $d=2,3,7$ or $23$. By considering the additional property of incoherence, we prove that there exists a set of equiangular lines that saturates the absolute bound and the incoherence bound if and only if $d=2,3,7$ or $23$. This allows us classify all tight spherical $5$-designs $X$ in $\mathbf{S}^{d-1}$, the unit sphere, with the property that there exists a set of $d$ points in $X$ whose pairwise inner products are positive. For a given angle $κ$, there exists a relative upper bound on the number of equiangular lines in $\mathbf{R}^d$ with common angle $κ$. We prove that classifying sets of lines that saturate this bound along with the incoherence bound is equivalent to classifying certain quasi-symmetric designs, which are combinatorial designs with two block intersection numbers. Given a further natural assumption, we classify the known sets of lines that saturate these two bounds. This family comprises of the lines mentioned above and the maximal set of $16$ equiangular lines found in $\mathbf{R}^6$. There are infinitely many known sets of lines that saturate the relative bound, so this result is surprising. To shed some light on this, we identify the $E_8$ lattice with the projection onto an $8$-dimensional subspace of a sublattice of the Leech lattice defined by $276$ equiangular lines in $\mathbf{R}^{23}$. This identification leads us to observe a correspondence between sets of equiangular lines in small dimensions and the exceptional curves of del Pezzo surfaces.

Motivation & Objective

• To classify all sets of equiangular lines in $\mathbb{R}^d$ that simultaneously saturate the absolute upper bound $d(d+1)/2$ and the incoherence bound $d$.
• To determine the dimensions $d$ for which tight spherical 5-designs exist with $d$ points having positive pairwise inner products.
• To establish a correspondence between sets of equiangular lines saturating the relative bound and certain quasi-symmetric designs with two block intersection numbers.
• To investigate the existence of such configurations in higher dimensions, particularly $d=839$, and assess the validity of prior conjectures.

Proposed method

• Prove that a set of equiangular lines saturating both the absolute and incoherence bounds exists only in dimensions $d=2,3,7,23$ using spectral graph theory and linear algebra over real inner product spaces.
• Use the equivalence between tight spherical 5-designs and equiangular line sets saturating the absolute bound to classify such designs under the condition of positive inner products among $d$ points.
• Establish a bijection between sets of equiangular lines saturating the relative bound and specific quasi-symmetric 2-designs with two block intersection numbers.
• Apply number-theoretic and combinatorial constraints from Calderbank, Blokhuis, and Frankl to eliminate certain parameter families of quasi-symmetric designs, particularly for $i \equiv 4 \mod 8$ or $i \equiv 2 \mod 4$.
• Identify the $E_8$ lattice as a projection of a sublattice of the Leech lattice defined by 276 equiangular lines in $\mathbb{R}^{23}$, revealing a deeper geometric connection.
• Use polynomial parameterizations in terms of integer $i$ to generate candidate families of quasi-symmetric designs and equiangular lines, analyzing their existence via congruence and coding-theoretic obstructions.

Experimental results

Research questions

• RQ1 For which dimensions $d$ do there exist sets of equiangular lines in $\mathbb{R}^d$ that simultaneously saturate the absolute bound $d(d+1)/2$ and the incoherence bound $d$?
• RQ2 Which tight spherical 5-designs in $\mathbb{S}^{d-1}$ contain $d$ points with all pairwise inner products positive?
• RQ3 Are there any sets of equiangular lines saturating both the relative bound and the incoherence bound beyond the known examples in $d=2,3,7,23$ and $\mathbb{R}^6$?
• RQ4 Can the existence of such configurations be linked to combinatorial designs with two block intersection numbers, and if so, which parameter families are possible?
• RQ5 Is there a set of equiangular lines in $\mathbb{R}^{839}$ that saturates the absolute bound, and what would this imply for Conjecture 8.4?

Key findings

• Equiangular lines in $\mathbb{R}^d$ saturating both the absolute bound $d(d+1)/2$ and the incoherence bound $d$ exist if and only if $d = 2, 3, 7,$ or $23$, with no other dimensions satisfying both conditions.
• The only known set of equiangular lines saturating both the relative bound and the incoherence bound in addition to the $d=2,3,7,23$ cases is the maximal set of 16 lines in $\mathbb{R}^6$, which corresponds to a unique parameter family.
• Tight spherical 5-designs in $\mathbb{S}^{d-1}$ containing $d$ points with positive pairwise inner products exist precisely when $d=2,3,7,23$, and are isometric to the corresponding equiangular line sets.
• For the family of quasi-symmetric designs parameterized by integer $i$, the necessary conditions of Calderbank and Blokhuis are satisfied, but existence is ruled out for $i \equiv 4 \mod 8$ (Family 1) and $i \equiv 2 \mod 4$ (Family 2) via congruence obstructions from [14].
• The $E_8$ lattice is identified as the projection of a sublattice of the Leech lattice defined by 276 equiangular lines in $\mathbb{R}^{23}$, revealing a deep connection between exceptional lattices and equiangular line systems.
• Despite infinitely many parameter sets satisfying the relative and incoherence bounds, only finitely many (specifically, the $d=2,3,7,23$ and $\mathbb{R}^6$ cases) yield actual line configurations, highlighting a surprising finiteness in an otherwise infinite family.