On the Construction of the Steiner System S(5, 8, 24)

Let VI = GF(211), regarded as an 11-dimensional vector space over GF(2), and form V = V, @ (71,). Define vi = 01~ + v, (i = O,..., 22). Since f(X) is irreducible over GF(2), {v. ,..., v 1o, v,} is a basis for V, and so for 11 < i < 22, we can write vi = CzEr, v,, for certain sets Xi _C X = (0 ,..., 10, w}. Moreover, the sets X, can very quickly be computed explicitly by calculating the powers of 01 in terms of the polynomial basis 1, a,..., oll” of GF(2n). Setting Y; = X,“, we compute that Yi is the set of faces of the dodecahedron of Fig. 1 surrounding some fixed face fi .