Coding theory and the Mathieu groups

Let ℳ denote the Mathieu group on 24 points. Let G be the subgroup of ℳ which has three sets of transitivity, the eight points on a Golay code-word (or Steiner octuple), one additional point, and the remaining 15 points. Using elementary results from the subject of algebraic coding theory, we present a new proof of the fact that G acts on the eight points as the alternating group, A8, and on the 15 points as the general linear group, G ℳ(4, 2). This result and other properties of the Mathieu groups obtained from it are then used to obtain the symmetry groups of the Nordstrom—Robinson nonlinear (15, 8) code and the linear, cyclic (15, 7) and (21, 12) BCH codes and the (21, 10) dual of a projective geometry code, all of which have distance 5