Almost impossible E_8 and Leech lattices

We start this short note by introducing two remarkable mathematical objects: the E8E8​ root lattice Lambda8Lambda8 in 8-dimensional Euclidean space and the Leech lattice Lambda24Lambda24 in 24-dimensional space. These two lattices stand out among their lattice sisters for several reasons. The first reason is that these both lattices are related to other unique and exceptional mathematical objects. The E8E8​ lattice is the root lattice of the semisimple exceptional Lie algebra E8E8​. The quotient of Lambda8Lambda8 by a suitable sublattice is isomorphic to the Hamming binary code of dimension 8 and minimum distance 4, which in its turn is an optimal error-correcting binary code with these parameters. The Leech lattice is famously connected to the exceptional finite simple groups, monstrous moonshine [7] and the monster vertex algebra [1]. Another reason is that Lambda8Lambda8 and Lambda24Lambda24 are solutions to a number of optimization problems. The E8E8​ and Leech lattice provide optimal sphere packings in their respective dimensions [5, 23]. Also both lattices are universally optimal, which means that among all point configurations of the same density, the Lambda8Lambda8 and Lambda24Lambda24 have the smallest possible Gaussian energy [6]. The third reason for our interest in these lattices is less obvious. The optimality of the E8E8​ and Leech lattices can be proven in a rather short way, while the solutions of analogous problems in other dimensions, even dimensions much smaller than 8 and 24, is still wide open. Finally, this last property seems to be inherited by other geometric objects obtained from Lambda8Lambda8 and Lambda24Lambda24, such as Hamming code, Golay code and the sets of shortest vectors of both lattices.