# Game of Sloanes

Emily King recently launched an online competition to find the best packings of points in complex projective space. The so-called Game of Sloanes is concerned with packing $n$ points in $\mathbf{CP}^{d-1}$ for $d\in\{2,\ldots,7\}$ and for $n\in\{d+2,\ldots,49\}$. John Jasper, Emily King and I collaborated to make the baseline for this competition by curating various packings from the literature and then numerically optimizing sub-optimal packings. See our paper for more information:

J. Jasper, E. J. King, D. G. Mixon, Game of Sloanes: Best known packings in complex projective space

If you have a packing that improves upon the current leader board, you can submit your packing to the following email address:

asongofvectorsandangles [at] gmail [dot] com

In this competition, you can win money if you find a new packing that achieves equality in the Welch bound; see this paper for a survey of these so-called equiangular tight frames (ETFs).

The Jasper Prize. John Jasper will pay US\$100 to the first person who submits an ETF of $n$ vectors in $\mathbb{C}^d$ such that

(i) $d\in\{2,\ldots,7\}$, $n\in\{d+2,\ldots,49\}$, and furthermore

(ii) no ETF of $n$ vectors in $\mathbb{C}^d$ is currently known to exist.

To be explicit, there are 22 pairs $(d,n)$ in the “Game of Sloanes range” for which an ETF is currently known to exist:

• $d=2, n=4$
• $d=3, n\in\{6,7,9\}$
• $d=4, n\in\{7,8,13,16\}$
• $d=5, n\in\{10,11,21,25\}$
• $d=6, n\in\{9,11,12,16,31,36\}$
• $d=7, n\in\{14,15,28,49\}$

Note: Fickus and Jasper conjecture that ETFs exist for both $(d,n)=(7,22)$ and $(7,43)$; see Conjecture 5.1 in this paper. Can you find either of these ETFs?

The goal of this competition is to find nice packings that generate new conjectures for the community to eventually prove (akin to Neil Sloane’s table of packings in real Grassmannian spaces). As an example, when building up the baseline for this competition, we happened upon a particularly nice packing of 5 points in $\mathbf{CP}^2,$ which is the subject of the following conjecture:

Conjecture. The columns of the following matrix span optimally packed points in $\mathbf{CP}^2$:

$\displaystyle \left[\begin{array}{lllll}a&b&b&c&c\\b&a&b&cw&cw^2\\b&b&a&cw^2&cw\end{array}\right],$

$\displaystyle a=\frac{\sqrt{13}+\sqrt{2+\sqrt{13}}-1}{3\sqrt{3}}, \quad b=\sqrt{\frac{1-a^2}{2}}, \quad c=\frac{1}{\sqrt{3}}, \quad w=e^{2\pi i/3}.$

(This is Conjecture 8 in the paper.) If you need motivation to prove this conjecture, consider the following:

King’s Coffee Prize. Emily King will buy a coffee for the first person to prove the above conjecture.

(FYI – Emily is a coffee snob, so the coffee you get from her will be worth, like, two or three normal coffees.)

We look forward to your contributions to the Game of Sloanes!