Self-dual binary codes from small covers and simple polytopes

Chen, Bo¹ ; Lü, Zhi² ; Yu, Li³

¹ School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China
² School of Mathematical Sciences, Fudan University, Shanghai, China
³ Department of Mathematics and IMS, Nanjing University, Nanjing, China

Algebraic and Geometric Topology, Tome 18 (2018) no. 5, pp. 2729-2767

Résumé

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 Mn over a simple n–polytope Pn produces a self-dual code in the sense of Kreck and Puppe if and only if Pn 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 Pn. Moreover, we can construct a family of binary codes Bk(Pn), for 0 ≤ k ≤ n, from an arbitrary simple n–polytope Pn. Then we give some necessary and sufficient conditions for Bk(Pn) to be self-dual. A spinoff of our study of such binary codes gives some new ways to judge whether a simple n–polytope Pn is n–colorable in terms of the associated binary codes Bk(Pn). 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.

Détail

DOI : 10.2140/agt.2018.18.2729

Classification : 57M60, 57R91, 57S25, 94B05
Keywords: self-dual code, polytope, small cover