The antipodal code of the truncated icosahedron

Let Γ be the truncated icosahedron graph (the graph of the C60 fullerene) and A its adjacency matrix. We study the binary code C = ker_F2(A), the space of null patterns of the σ-game on Γ. It is a self-orthogonal [60,4,24] code with weight enumerator 1 + 5z^24 + 10z^36. As a module for A5 it is the sum-zero submodule of the natural permutation module F2^5. Every codeword is fixed by the antipodal map of Γ, which is the central inversion of Aut(Γ) ≅ Ih; in particular the dual code has minimum distance 2. We show that the five minimum-weight words are exactly the maximum independent sets of Γ that are even covers (every vertex outside the set has an even number of neighbours in it). Among the 126,715 maximum independent sets of Γ, these five also form the unique Aut(Γ)-orbit of size five, the sets with stabiliser Th; they are the addition pattern of the bromofullerene C60Br24. Finally, C is the binary member of a family of eigenspace codes ker_Fp(A − λI), which we tabulate for p ≤ 13, and we record an exact reflection identity for Z2-signings of Γ. Every numbered claim is checked by an accompanying script (included) that rebuilds Γ from A5.