Codes, designs and graphs obtained from some projective symplectic group

Abstract

After the classification of finite simple groups, there is still much work to be done to give a clear geometric identification of the finite simple groups. There are also many problems in enumerating and characterizing a structure which either has a particular group acting on it or which has some degree of symmetry from a group action. It has been shown that there exists interplay between finite simple groups and codes. In this thesis we construct and enumerate binary linear codes for the projective symplectic group S8(2) from the permutation representations of degree 120, 136, 255, 2295, 5355, 5440 and 11475. We find that the support of codewords of a given weight in a code hold a combinatorial design, or represent points of a projective space PG(2m−1,q), or represent the rows of the adjacency matrix of a graph or equivalently are the incidence vectors of the blocks of a design. Through coding theory, the interplay between the combinatorial objects is enhanced and the internal structures of the group characterized.