The dodecahedral theorem states that in a packing of unit spheres in R^3, the Voronoi cell of minimum possible volume is a regular dodecahedron with inradius one. The theorem was conjectured by Fejes Toth in 1943, and proved by Hales and McLaughlin in 1998 using techniques developed by Hales for his proof of the Kepler conjecture. The proof of Hales and McLauughlin, while apparently correct, is difficult to verify due to the many cases and extensive computations required. In his 1964 book Regular Figures, Fejes Toth suggested a possible proof for the dodecahedral theorem but was unable to verify a key inequality. We describe an approach to completing Fejes Toth's proof that uses strengthened bounds for spherical codes.