On Locally Hoffman-Singleton Graphs

On the Graphs of Hoffman-Singleton and Higman-Sims

We propose a new elementary definition of the Higman-Sims graph in which the 100 vertices are parametrised with Z4 × Z5 × Z5 and adjacencies are described by linear and quadratic equations. This definition extends Robertson’s pentagonpentagram definition of the Hoffman-Singleton graph and is obtained by studying maximum cocliques of the Hoffman-Singleton graph in Robertson’s parametrisation. Th...

Hoffman-Singleton Graph

The Hoffman-Singleton Graph and its Automorphisms

We describe the Hoffman-Singleton graph geometrically, showing that it is closely related to the incidence graph of the affine plane over Z5. This allows us to construct all automorphisms of the graph.

Binary codes derived from the Hoffman-Singleton and Higman-Sims graphs

Some binary linear codes of length 50 and 100 are constructed using the adjacency matrices of the Hooman-Singleton graph and the Higman-Sims graph. Some of the codes are optimal or nearly optimal for the given length and dimension. The dual codes admit majority logic decoding.

The graphs of Hoffman-Singleton, Higman-Sims, McLaughlin and the Hermite curve of degree 6 in characteristic 5

We construct the graphs of Hoffman-Singleton, Higman-Sims and McLaughlin from certain relations on the set of non-singular conics totally tangent to the Hermitian curve of degree 6 in characteristic 5. We then interpret this geometric construction in terms of the subgroup structure of the automorphism group of this Hermitian curve.