Equiangular lines, Incoherent sets and Quasi-symmetric designs

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 $\kappa$, there exists a relative upper bound on the number of equiangular lines in $\mathbf{R}^d$ with common angle $\kappa$. 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.