Description
This research looks for high density lattice sphere packings using a combination of theoretical results from linear algebra and coding theory together with a computer search that uses MAGMA and SAGE software packages. Dense sphere packings in high dimensions have many applications across different sciences such as determining ground states in interacting particle systems and signal set design. Given a cyclic extension L/Q of degree p with p an odd unramified prime in L/Q, the search space is comprised of lattices constructed from submodules, M ⊂ OL, where OL is the ring of integers of L. The submodules are described using transformations of the symmetric matrix of the trace form of L. The method builds on previous work characterizing the integral trace form of L in these such extensions. The search ranges over submodules of a specified index that is chosen to yield a high density. The search was able to find exceptionally dense sphere packings in dimensions 3, 5, 7, 11 and 13. Lattice packings with the highest possible density were found in dimensions 3, 5 and 7; a validation of the search.
