Binary Linear Codes and Combinatorial Designs Constructed from the Alternating Groups A5, A7, and A9

The action of a finite simple group on the cosets of a maximal subgroup yields combinatorial designs whose incidence matrices span binary linear codes admitting the group as a group of automorphisms. This paper applies the point-stabiliser, conjugacy-class, and fixed-point construction methods of Key and Moori and of Moori uniformly to the alternating groups A5, A7, and A9, and characterises the parameters of the resulting codes and designs in terms of the group invariants. Complete catalogues are obtained for every maximal subgroup and every non-identity conjugacy class of each group; the catalogues for A9 are new. The length, block sizes, and replication numbers are shown to be determined in closed form by the index, orbit, conjugacy-class, and centraliser data, while the dimension is the rank of the incidence matrix over the binary field. A duality is established for pairs of non-conjugate isomorphic maximal subgroups that are fused in the symmetric group: their point-stabiliser catalogues coincide, while their conjugacy-class catalogues differ. The construction is validated against the published results for A7, and the two non-conjugate PSL (2,8):C3 classes of A9 provide an explicit instance of the duality.