Self-dual binary codes from small covers and simple polytopes

Abstract

The work of Volker Puppe and Matthias Kreck exhibited some intriguing connections between the algebraic topology of involutions on closed manifolds and the combinatorics of self-dual binary codes. On the other hand, the work of Michael Davis and Tadeusz Januszkiewicz brought forth a topological analogue of smooth, real toric varieties, known as “small covers”, which are closed smooth manifolds equipped with some actions of elementary abelian $2$–groups whose orbit spaces are simple convex polytopes. Building on these works, we find various new connections between all these topological and combinatorial objects and obtain some new applications to the study of self-dual binary codes, as well as colorability of polytopes. We first show that a small cover $M^n$ over a simple $n$–polytope $P^n$ produces a self-dual code in the sense of Kreck and Puppe if and only if $P^n$ is $n$–colorable and $n$ is odd. Then we show how to describe such a self-dual binary code in terms of the combinatorics of $P^n$. Moreover, we can construct a family of binary codes ${\mathfrak{B}}_k\left(P^n\right)$, for $0\le k\le n$, from an arbitrary simple $n$–polytope $P^n$. Then we give some necessary and sufficient conditions for ${\mathfrak{B}}_k\left(P^n\right)$ to be self-dual. A spinoff of our study of such binary codes gives some new ways to judge whether a simple $n$–polytope $P^n$ is $n$–colorable in terms of the associated binary codes ${\mathfrak{B}}_k\left(P^n\right)$. In addition, we prove that the minimum distance of the self-dual binary code obtained from a $3$–colorable simple $3$–polytope is always $4$.