Every binary self-dual code arises from Hilbert symbols

摘要

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.