Supermultiplets in particle physics, stripped of their spatial dimensions, can be described in terms of a class of N-edge regular bipartite decorated graphs known as Adinkras. We will start by explaining how to construct all of these from quotients of the 1-skeleton of binary N-dimensional hypercubes by doubly even binary linear error correcting codes. We then geometrize these graphs, realizing them on Riemann surfaces arising as rigid branched covers of the thrice-punctured sphere. The additional markings on an Adinkra can then be reinterpreted as providing special spin structures and divisors coming from discrete Morse functions. Applying Kitaev's surface code construction yields an associated quantum error-correcting code. The supersymmetric origin and extra markings suggest ingredients for a supersymmetric extension of the surface code as well as a form of mirror symmetry.