Homology, Homotopy and Applications

Volume 14 (2012)

Number 2

Every binary self-dual code arises from Hilbert symbols

Pages: 189 – 196

DOI: https://dx.doi.org/10.4310/HHA.2012.v14.n2.a11

Authors

Ted Chinburg (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Ying Zhang (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Abstract

In this paper we construct binary self-dual codes using the étale cohomology of $\mu_2$ on the spectra of rings of $S$-integers of global fields. We will show that up to equivalence, all self-dual codes of length at least $4$ arise from Hilbert pairings on rings of $S$-integers of $\mathbb{Q}$. This is an arithmetic counterpart of a result of Kreck and Puppe, who used cobordism theory to show that all self-dual codes arise from Poincaré; duality on real three manifolds.

Keywords

binary self-dual code, $S$-integer, étale cohomology

2010 Mathematics Subject Classification

11T71, 14F20, 14G50, 94B05

Published 4 December 2012