Every binary self-dual code arises from Hilbert symbols

Ted Chinburg and Ying Zhang

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.

Homology, Homotopy and Applications, Vol. 14 (2012), No. 2, pp.189-196.

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