# 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!

## 2 thoughts on “Game of Sloanes”

1. Great idea to have this competition! We have been using complex Grassmannian packings for a number of years in wireless communications. We have tabulated packings online https://engineering.purdue.edu/~djlove/grass.html though I admit most of them are from 2003. We also have an algorithm for finding complex packings and tables of packings we found in http://users.cms.caltech.edu/~jtropp/papers/DHST07-Constructing-Packings-preprint.pdf

We are currently looking into some new angles in closed form construction of ETFs. I look forward to contributing to this in the future.

Robert Heath – UT Austin

1. Hi Robert,

Thanks for your comment. It’s great to hear from one of the researchers who first brought Grassmannian packings to the attention of the frame theory community! We would also love to have a contribution to the Game of Sloanes of any kind from you.

We mentioned both Love’s website and the series of your alternating packing papers in the short article Dustin linked to above. The line packings on Love’s website were either already known or had suboptimal coherence. However, the packings which had suboptimal coherence seemed to be tight frames, so they might be 1-Grassmannian frames, i.e., have optimal coherence amongst all tight frames of the same parameters. A longer term plan of ours is to include 1-Grassmannian frames in the Game of Sloanes.

The alternating projection algorithm was one of the main ones we used to generate the packings we posted. In fact, we included our Matlab implementation of alternating projection on the website. We found we got better results by taking the output of one algorithm and feeding it to another. So many of the configurations we posted were worked on by alternating projection, even if that wasn’t the only method applied.

Another longer term plan of ours is to include Grassmannian packings of subspaces of dimension larger than 1. I expect that the alternating projection algorithm would again prove to be quite useful.

Best,
Emily