Lattice Packings: An Upper Bound on the Number of Perfect Lattices

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Gives a geometric proof that the number of non-similar perfect lattices PD has an upper bound exp(O(d^2 log d)) improving previous exp(O(d^3 log d)). Uses Voronoi domains (outer normal cones), caps the cone by bounding trace to get finite volume, lowers each domain’s volume by taking integer XX^T rays whose determinant ≥1, and controls minimal vector size by lattice reduction/transference so normalized simplices have polynomial factors. Notes enumeration explosion (dimension 9 exceeded 23 million found so far) and uses a trace cap to compare totals.

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" chapters (19)

1. 0:00Intro
2. 0:31Sphere Packing Problem
3. 1:28Lattice Packing Problem
4. 2:19Solution space
5. 2:41Ryshkov Polyhedron
6. 7:50Space of quadratic forms
7. 9:03Hermite Constant
8. 10:13Ryshkov Polyhedra
9. 12:46Similarity
10. 17:19An improved upper bound
11. 18:53Outer Normal Cones
12. 19:47Subdivision for d = 2
13. 21:02Proof strategy
14. 21:59Volumetric argument
15. 22:55Volume simplex
16. 23:13Volume Voronoi domain
17. 23:47Lower bound
18. 24:40Short minimal vectors
19. 25:31Conclusion