Computational Approaches to Lattice Packing and Covering Problems

by Achill Schürmann; Frank Vallentin


Preprint series: 04-07, Preprints

11H31 Lattice packing and covering, See also {05B40, 52C15, 52C17}


Abstract: We describe algorithms which solve two classical problems in lattice geometry for any fixed dimension d: the lattice covering and the simultaneous lattice packing-covering problem. Both algorithms involve semidefinite programming and are based on Voronoi\'s reduction theory for positive definite quadratic forms which describes all possible Delone triangulations of $Z^d$. Our implementations verify all known results in dimensions $d < = 5$. Beyond that we attain complete lists of all locally optimal solutions for $d = 5$. For $d = 6$ our computations produce new best known covering as well as packing-covering lattices which are closely related to the lattice $E_6^*$.

Keywords: lattice, packing, covering, quadratic forms, semidefinite programming

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