Parameters of the Binary Linear Codes and Combinatorial Designs from the Alternating Groups A5, A7, and A9 in Terms of the Group Invariants

When a finite simple group acts on the costs of a maximal subgroup, the combinatorial designs and binary linear codes obtained from the action carry parameters that are governed by the invariants of the group. This paper characterises the parameters of the codes and designs constructed from the alternating groups A_5, A_7, and A_9 in terms of those invariants, namely the index of the subgroup, the subdegrees of the action, the conjugacy-class lengths, the centraliser orders, and the fixed-point counts. The length of every code is shown to equal the index of the subgroup, and the block sizes and replication numbers of every design are expressed in closed form through the orbit and class data, so that all parameters except the dimension are determined by the group invariants without reference to the incidence matrix. A counting formula for the fixed points of a class representative on a closet space is established and used to give the block sizes of the fixed-point construction. For constructions that combine several conjugacy classes, the dimension and minimum distance are bounded through the sum of the constituent subspaces. The parameters are tested against the Singleton, Griesmer, and Assmus–Key bounds, and the tightness of the dimension bound is assessed across the catalogues. The dimension is identified as the one parameter not fixed by the invariants alone, equal to the rank of the incidence matrix over the binary field, and the way in which the invariants distinguish the two non-conjugate PSL (2,8):C_3 classes of A_9 is made explicit.