| Abstract: | The interrelations between finite geometries (finite incidence structures) and linear codes over finite fields are discussed under some special fundamental aspects. For any incidence structure \({\mathcal{I}}\) block codes, block-difference codes and co-block codes over finite fields of characteristic p are discussed resp. introduced; correspondingly p-modular co-blocks are defined for \({\mathcal{I}}\). Orthogonality modulo p is introduced as a concept relating different geometries having the same point set. Conversely three types of block-tactical geometries may be derived from vector classes of fixed Hamming weight in a given linear code. These geometries are tactical configurations if the given code admits a transitive permutation group. A combination of both approaches leads to the concept of p-closure of a finite geometry and to the notions of p-closed, weakly p-closed and p-dense incidence structures. These geometric concepts are applied to simple or directed graphs via their natural “adjacency geometry”. Here the above mentioned code theoretic treatment leads to the concept of p-modular co-adjacent vertex sets. As instructive examples the Petersen graph, its complemetary graph and the Higman-Sims graph are considered. |
