After the broundbreaking works 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 somehow philosophical: for the rest of dimensions (except 2 and 3) there do not exists such structures. Is there however any extra obstruction (geometrical, algebraic)  which gives evidences that no such 'rigid' algebraic structure may exist?