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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Self-dual binary codes from small covers and simple polytopes

Bo Chen, Zhi Lü and Li Yu

Algebraic & Geometric Topology 18 (2018) 2729–2767
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 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.

Keywords
self-dual code, polytope, small cover
Mathematical Subject Classification 2010
Primary: 57M60, 57R91, 57S25, 94B05
References
Publication
Received: 27 March 2017
Revised: 25 February 2018
Accepted: 23 March 2018
Published: 22 August 2018
Authors
Bo Chen
School of Mathematics and Statistics
Huazhong University of Science and Technology
Wuhan
China
Zhi Lü
School of Mathematical Sciences
Fudan University
Shanghai
China
Li Yu
Department of Mathematics and IMS
Nanjing University
Nanjing
China