6,641 research outputs found
Centrally symmetric polytopes with many faces
We present explicit constructions of centrally symmetric polytopes with many
faces: first, we construct a d-dimensional centrally symmetric polytope P with
about (1.316)^d vertices such that every pair of non-antipodal vertices of P
spans an edge of P, second, for an integer k>1, we construct a d-dimensional
centrally symmetric polytope P of an arbitrarily high dimension d and with an
arbitrarily large number N of vertices such that for some 0 < delta_k < 1 at
least (1-delta_k^d) {N choose k} k-subsets of the set of vertices span faces of
P, and third, for an integer k>1 and a>0, we construct a centrally symmetric
polytope Q with an arbitrary large number N of vertices and of dimension
d=k^{1+o(1)} such that least (1 - k^{-a}){N choose k} k-subsets of the set of
vertices span faces of Q.Comment: 14 pages, some minor improvement
Geometric Reasoning with polymake
The mathematical software system polymake provides a wide range of functions
for convex polytopes, simplicial complexes, and other objects. A large part of
this paper is dedicated to a tutorial which exemplifies the usage. Later
sections include a survey of research results obtained with the help of
polymake so far and a short description of the technical background
Quasi-Cyclic Complementary Dual Code
LCD codes are linear codes that intersect with their dual trivially. Quasi
cyclic codes that are LCD are characterized and studied by using their
concatenated structure. Some asymptotic results are derived. Hermitian LCD
codes are introduced to that end and their cyclic subclass is characterized.
Constructions of QCCD codes from codes over larger alphabets are given
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