Access structure in graphs in high dimension and application to secret sharing

We give graphical characterisation of the access structure to both classical and quantum information encoded onto a multigraph defined for prime dimension $q$, as well as explicit decoding operations for quantum secret sharing based on graph state protocols. We give a lower bound on $k$ for the existence of a $((k,n))_q$ scheme and prove, using probabilistic methods, that there exists $\alpha$ such that a random multigraph has an accessing parameter $k\leq \alpha n$ with high probability.