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 
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).
denote the field with 
elements, and let 
denote the multiplicative subgroup of quadratic residues. For each prime 
, we consider the 
with vertex set 
:
, and you are tasked with finding the partition 
that minimizes the k-means objective![\displaystyle{\frac{1}{|T|}\sum_{t\in[k]}\sum_{i\in C_t}\bigg\|x_i-\frac{1}{|C_t|}\sum_{j\in C_t}x_j\bigg\|^2\qquad(T\text{-IP})}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cfrac%7B1%7D%7B%7CT%7C%7D%5Csum_%7Bt%5Cin%5Bk%5D%7D%5Csum_%7Bi%5Cin+C_t%7D%5Cbigg%5C%7Cx_i-%5Cfrac%7B1%7D%7B%7CC_t%7C%7D%5Csum_%7Bj%5Cin+C_t%7Dx_j%5Cbigg%5C%7C%5E2%5Cqquad%28T%5Ctext%7B-IP%7D%29%7D&bg=ffffff&fg=303030&s=0 "\displaystyle{\frac{1}{|T|}\sum_{t\in[k]}\sum_{i\in C_t}\bigg\|x_i-\frac{1}{|C_t|}\sum_{j\in C_t}x_j\bigg\|^2\qquad(T\text{-IP})}")
for convenience later.) To do this, you will likely run 
of the data points (with an intelligent choice of random distribution), and then uses these data points as proto-centroids to initialize 
packing from 
packing involving the 
(this one beating the corresponding packing in 