Every binary self-dual code arises from Hilbert symbols (Q1935104)

A \textit{self-dual binary code} is a triple \((W,V,E)\) where \(W\) is a \(\mathcal F_2\)-vector space of even dimension, \(E\) is an ordered basis of \(W\) and \(V\) is a subspace which is its own orthogonal complement with respect to the bilinear form defined at \(W\) which makes \(E\) an orthogonal basis and \(W\) an Euclidean space. Hence the dimension of \(V\) is half the dimension of \(W\). In this paper a construction is given using the spectra of \(S\)-integers rings of global fields. Let \(K\) be a global field, for a given place \(v\) at \(K\), let \({\mathcal F}\) be a sheaf on the small étale site of \(\text{Spec}(K_v)\) where \(K_v\) is the completion of \(K\) at \(v\), and consider reduced étale cohomology group \(H_{\text{et}}^r(K_v,{\mathcal F})\). In particular, let \(\mu_2\) be the group of square roots of unit, let \(S\) be a finite set of places containing all the Archimedean places and let \(U\) be the open, in the Zariski topology, complement of \(S\). Then the image of the restriction homomorphism \(\Phi:H_{\text{ et}}^r(U,\mu_2)\to\bigoplus_{v\in S}H_{\text{ et}}^r(K_v,\mu_2)\) is its own orthogonal complement with respect to a bilinear form at \(\bigoplus_{v\in S}H_{\text{ et}}^r(K_v,\mu_2)\) defined in terms of cup-product, and whenever \(-1\) is not a square at \(K_v\) then Euclidean basis will exist. Thus, by considering primes congruent with 3 mod 4 it is shown that all self-dual binary codes of length at least 4 at spaces of the form \(\bigoplus_{v\in S}\mathbb{Q}^*_p/(\mathbb{Q}^*_p)^2\) arise from bilinear forms determined by Hilbert symbols at the multiplicative groups \(\mathbb{Q}^*_p/(\mathbb{Q}^*_p)^2\). The authors introduce also a procedural construction in terms of boxed matrices.