On the Codes Related to the Higman-Sims Graph

All linear codes of length 100 over a field F which admit the Higman-Sims simple group HS in its rank 3 representation are determined. By group representation theory it is proved that they can all be understood as submodules of the permutation module FΩ where Ω denotes the vertex set of the Higman-Sims graph. This module is semisimple if charF 6= 2, 5 and absolutely indecomposable otherwise. Also if charF ∈ {2, 5} the submodule lattice is determined explicitly. The binary case F = F2 is studied in detail under coding theoretic aspects. The HS-orbits in the subcodes of dimension 6 23 are computed explicitly and so also the weight enumerators are obtained. The weight enumerators of the dual codes are determined by MacWilliams transformation. Two fundamental methods are used: Let v be the endomorphism determined by an adjacency matrix. Then in H22 = Im v the HS-orbits are determined as v-images of HS-orbits of certain low weight vectors in FΩ which carry some special graph configurations. The second method consists in using the fact that H23/H21 is a Klein four group under addition, if H23 denotes the code generated by H22 and a “Higman vector” x(m) of weight 50 associated to a heptad m in the shortened Golay code G22, and H21 denotes the doubly even subcode of H22 6 H78 = H22⊥. Using the mentioned observation about H23/H21 and the results on the HS-orbits in H23 a model of G. Higman’s geometry is constructed, which leads to a direct geometric proof that G. Higman’s simple group is isomorphic to HS. Finally, it is shown that almost all maximal subgroups of the Higman-Sims group can be understood as stabilizers in HS of code words in H23.