After the groundbreaking work of Viazovska, now we have a proof for the optimal density of sphere packings in dimensions 8 and 24. Both packings emerge from very particular algebraic lattice structures on the 8th and 24th dimensional space.

My question is somewhat philosophical:  for the rest of dimensions (except 2 and 3) there do not exist such structures. Is there however any extra obstruction (geometrical, algebraic) which gives evidence that no such 'rigid' algebraic structure may exist?