Lattice polytopes in coding theory

In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also prove a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations.Comment: 11 pages, 3 figure

Computational determination of the largest lattice polytope diameter

A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational framework to determine {\delta}(d, k) for small instances. We show that {\delta}(3, 4) = 7 and {\delta}(3, 5) = 9; that is, we verify for (d, k) = (3, 4) and (3, 5) the conjecture whereby {\delta}(d, k) is at most (k + 1)d/2 and is achieved, up to translation, by a Minkowski sum of lattice vectors

Computational determination of the largest lattice polytope diameter

A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational framework to determine {\delta}(d, k) for small instances. We show that {\delta}(3, 4) = 7 and {\delta}(3, 5) = 9; that is, we verify for (d, k) = (3, 4) and (3, 5) the conjecture whereby {\delta}(d, k) is at most (k + 1)d/2 and is achieved, up to translation, by a Minkowski sum of lattice vectors

Primitive Zonotopes

We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type $B_d$. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension $d$ whose coordinates are integers between $0$ and $k$, and with the computational complexity of multicriteria matroid optimization.Comment: The title was slightly modified, and the determination of the computational complexity of multicriteria matroid optimization was adde

Coxeter submodular functions and deformations of Coxeter permutahedra

We describe the cone of deformations of a Coxeter permutahedron, or equivalently, the nef cone of the toric variety associated to a Coxeter complex. This family of polytopes contains polyhedral models for the Coxeter-theoretic analogs of compositions, graphs, matroids, posets, and associahedra. Our description extends the known correspondence between generalized permutahedra, polymatroids, and submodular functions to any finite reflection group.Comment: Minor edits. To appear in Advances of Mathematic

Virtual polytopes

Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as the Grothendick group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. A selection of applications demonstrates their versatility

Virtual Polytopes

Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as elements of the Grothendieck group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. The present survey connects the theory of virtual polytopes with other geometrical subjects, describes a series of geometrizations together with relations between them, and gives a selection of applications