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 F Omega where Omega denotes the vertex set of the Higman-Sims graph. This module is semisimple if char F not equal 2,5 and absolutely indecomposable otherwise. Also if char F is an element of {2, 5} the submodule lattice is determined explicitly. The binary case F - F-2 is studied in detail under coding theoretic aspects. The HS-orbits in the subcodes of dimension <= 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 H-22 = Im v the HS-orbits are determined as v-images of HS-orbits of certain low weight vectors in F Omega which carry some special graph configurations. The second method consists in using the fact that H-23/H-21 is a Klein four group under addition, if H-23 denotes the code generated by H-22 and a "Higman vector" x(m) of weight 50 associated to a heptad m in the shortened Golay code G(22 ,) and H-21 denotes the doubly even subcode of H-22 <= H-78 = H-22(perpendicular to). Using the mentioned observation about H-23/H-21 and the results on the HS-orbits in H-23 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 H-23.