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Self-dual binary codes
from small covers and simple polytopes
Bo Chen, Zhi Lü and Li Yu |
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Algebraic & Geometric Topology 18 (2018) 2729–2767
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Abstract |
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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 –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 over a simple –polytope produces a self-dual code in the sense of Kreck and Puppe if and only if is –colorable and is odd. Then we show how to describe such a self-dual binary code in terms of the combinatorics of . Moreover, we can construct a family of binary codes , for , from an arbitrary simple –polytope . Then we give some necessary and sufficient conditions for to be self-dual. A spinoff of our study of such binary codes gives some new ways to judge whether a simple –polytope is –colorable in terms of the associated binary codes . In addition, we prove that the minimum distance of the self-dual binary code obtained from a –colorable simple –polytope is always . |
Keywords
self-dual code, polytope, small cover
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Mathematical Subject Classification 2010
Primary: 57M60, 57R91, 57S25, 94B05
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References |
Publication
Received: 27 March 2017
Revised: 25 February 2018
Accepted: 23 March 2018
Published: 22 August 2018
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Authors |
| Bo Chen | |
| School of Mathematics and
Statistics Huazhong University of Science and Technology Wuhan China |
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| Zhi Lü | |
| School of Mathematical
Sciences Fudan University Shanghai China |
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| Li Yu | |
| Department of Mathematics and
IMS Nanjing University Nanjing China |
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