polytopes are constructed from generalized combinatorial cubes by taking quotients with respect to elementary Abelian subgroups (linear codes) of the automorphism group.
1. Iutroduction
Abstract polytopes are special kinds of partially ordered sets which share many properties with convex polytopes. In recent years there has been some interest in combinatorial generalizations of the classical theory of regular polytopes (cf. [4]) leading to the concept of regular incidence-polytopes;
cf.
[7] but see also [ll] and [a]. This concept provides a suitable setting for the study of structures resembling the classical regular polytopes. At the same time it allows interactions with the theory of buildings and diagram geometries (cf. [15, 11).
The purpose of this note is to construct abstract polytopes from binary codes. Generally these polytopes will not be regular but are derived from certain abstract regular polytopes .9 which are close analogues of the ordinary higherdimensional cubes. The group of 9 is a wreath-product C, 7. A(X), with A(X) the group of the vertex-figure X of 9. By taking quotients P/C of 9 by suitable binary linear codes C interesting new abstract polytopes arise. This method is well known for the construction of thin C,-geometries; this is the case where 9 is the ordinary n-cube (cf. [17]).
2. Polytopes and quotients
An incidence-polytope 9 of dimension d, or briefly, an (abstract) d-polytope, is a poset with the following properties (cf. ); the elements of 9 are called the faces of 9. 9 has a smallest face F_1 and a largest face F,; these are the improper faces of 9 (of dimension -1 and d, respectively) while the other faces are called the proper faces. The maximal chains of 9 called the frcrgs of P contain exactly d + 2 faces. Thus 9 has a rank function associating with each face F its dimension