Realization and classification of Hamiltonian-circle multisigns

We investigate the multisigns of Hamiltonian circles in the multisigned complete graph $\Sigma_n := (K_n, \sigma, \mathbb{F}_2^m)$. The \emph{multisign} of a circle $C$ is defined as the sum $$\sigma(C) := \sum_{e \in E(C)} \sigma(e).$$ For a fixed $m$ and sufficiently large $n$, we show that the set of multisigns of Hamiltonian circles $$\{\sigma(H) : H \text{ is a Hamiltonian circle of } \Sigma_n)\}$$ forms either a subspace, an affine subspace, or the entire space $\mathbb{F}_2^m$, except in certain exceptional cases.