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We give graphical characterisation of the access structure to both classicaland quantum information encoded onto a multigraph defined for prime dimension$q$, as well as explicit decoding operations for quantum secret sharing basedon 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 $k\leq \alphan$ with high probability.

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