Every Binary Self-Dual Code Arises From Hilbert Symbols

TL;DR

This paper demonstrates that all binary self-dual codes of length at least 4 can be constructed from Hilbert symbols associated with rings of integers in global fields, linking coding theory with arithmetic geometry.

Contribution

It establishes that every binary self-dual code arises from Hilbert pairings on rings of integers, providing an arithmetic interpretation of self-dual codes.

Findings

All self-dual codes of length ≥ 4 are derived from Hilbert pairings.

Constructs binary self-dual codes using étale cohomology of Z/2.

Connects coding theory with arithmetic geometry via Hilbert symbols.

Abstract

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.