In hypergraph theory, determining a good characterization of d, the degree sequence of an h-uniform hypergraph H, and deciding the complexity status of the reconstruction of H from d, are two challenging open problems. They can be formulated in the context of discrete tomography: asks whether there is a matrix A with nonnegative projection vectors H = (h,h,...,h) and V = (d1,d2,...,dn) with distinct rows. In this paper we consider the subcase where the vectors H and V are both homogeneous vectors, and we solve the related consistency and reconstruction problems in polynomial time. To reach our goal, we use the concepts of Lyndon words and necklaces of fixed density, and we apply some already known algorithms for their efficient generation.