SOME LINEAR CODES, GRAPHS AND DESIGNS FROM MATHIEU GROUPS M24 AND M23

Abstract

In this thesis, we have used four steps to determine G-invariant codes from primitive permutation representations of Mathieu groups M24 and M23 . We constructed all G-invariant codes from primitive representations of degree 24, 276, 759, and 1288 from the simple group M24. We found one self dual [24, 12, 8] code, three irreducible codes; [276,11,128], [759,11,352] and [1288,11,648]. There were several decomposable, self orthogonal and projective linear binary codes. There were two strongly regular graphs from a representation of degree 276 and 759. These graphs are known. We determined designs from some binary codes using codewords of minimum weight. All the designs constructed were primitive. We constructed symmetric 1-designs from the primitive permutation representations of degree 24, 276, 759, 1771, 2024 and 3795 defined by the action of a group G on a set Ω = G/Gα. In most cases the full automorphism group of the design was M24 while in some cases the full automorphism group of the design was either S24 or S276. We also constructed all G-invariant codes from primitive representations of degree 23, 253, and 253 from the simple group M23. There was no self dual linear code. There were four irreducible codes [23,11,8], [253,11,112],[253,44] and [253,11,112] . There were several decomposable, self orthogonal and projective linear binary codes. There was no strongly regular graph from the three representations. We determined designs from some binary codes using codewords of minimum weight. All the designs constructed were primitive. We constructed symmetric 1-designs from the primitive permutation representations of degree 23, 253 and 253 defined by the action of a group G on a set Ω = G/Gα. In most cases the full automorphism group of the design was M23 while in some cases the full automorphism group of the design was either S23, S253 or S506