Every Binary Self-Dual Code Arises From Hilbert Symbols

Details Every Binary Self-Dual Code Arises From Hilbert Symbols Every Binary Self-Dual Code Arises From Hilbert Symbols

2012-04-13

arxiv.org/abs/1204.3115v1

Mathematics

Number Theory

7 pages, 2 tables

Scientific paper

In this paper we construct binary self-dual codes using the \'etale cohomology of $\mathbb{Z}/2$ 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 $\Q$. 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 Poincar\'e duality on real three manifolds.

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