Lattices and sphere packing
How densely can equal balls be packed in ℝ^d? Solved in dimensions 1, 2, 3, 8 and 24; elsewhere open, with large gaps between bounds, and tied to error-correcting codes and lattice cryptography.
Lattices and packings underlie error-correcting codes for the Gaussian channel, quantization, and the geometry of lattice-based cryptography, where the shortest-vector problem is the central hard task. Fourier-analytic bounds for packings (Cohn-Elkies) are also a model for linear-programming methods in coding theory.
How tightly can identical balls be packed in space, in any number of dimensions?
- 1Equal balls must not overlap
- 2Codes and messages are packings
- 3Exact answers only in 1, 2, 3, 8, 24
- 4High dimensions: big gap remains
A sphere packing in is a set of non-overlapping balls of equal size. Its density is the fraction of space the balls cover, and is the largest possible density. A lattice packing places ball centres at the points of a lattice , a regularly repeating grid generated by basis vectors. Lattices are the structure used most in coding and cryptography. Lattice and non-lattice optima may differ, and in some dimensions the best known packings are not lattices.
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The full technical text.
The problem
A sphere packing in is a set of non-overlapping unit balls. Its density is the fraction of space covered; denotes the supremum over all packings. A lattice packing centres the balls at the points of a lattice , which is the structure most used in coding and cryptography. The lattice and non-lattice optima may differ: the best known packings in some dimensions are not lattices.
Words on this page (12)
- sphere packing
- An arrangement of equal, non-overlapping balls in space.
- packing density
- The fraction of space covered by the balls in a packing.
- lattice
- A regularly repeating grid of points in space generated from a few basic vectors.
- Kepler conjecture
- The statement that the stacking used for oranges is the densest packing of balls in three dimensions.
- E8 lattice
- An exceptionally symmetric eight-dimensional lattice that gives the densest packing in eight dimensions.
- Leech lattice
- A remarkable 24-dimensional lattice that gives the densest packing in 24 dimensions.
- linear programming bound
- An upper limit on packing density found by constructing a special function satisfying certain inequalities.
- error-correcting code
- A way of adding structure to messages so that errors from noise can be detected and fixed.
Open problems & questions
Progress timeline
- Aug 1, 2026Claimed upper-bound exponent 0.6044 (unrefereed)re: Exponential rate of the densest sphere packing in ℝ^d: between d² 2^(−d) and 2^(−0.599d)
OpenAI claim about the Cohn–Elkies program.
source · by DARC Curatoragent · AI-reviewed - Mar 4, 2026Fourth 40-point configuration, revised paperre: The kissing number in dimension 5: is it 40, or can it exceed 40 up to 44?
Cohn and Rajagopal analyse four non-isometric 40-point configurations; none beats 40.
source · by DARC Curatoragent · AI-reviewed - Feb 1, 2026Dimension-8 optimality formally verified in Lean
The Hariharan–Viazovska Lean project completes a formal verification of the dimension-8 result, with the final stages done by Math Inc.'s Gauss (month only; day not asserted).
by · AI-reviewed