A t -(v,k,2) design is a collection ~ of k-subsets (called blocks) of a v-element set X of points, that satisfies the property that each t-element subset of points is in exactly 2 blocks. We also require that ao block is repeated. A group G acting on X is an automorphism group of 8, if ~ is a Union of G-orbits of k-subsets.
The projective special linear group G = PSL(2, 31) acts 3-homogeneously on the projective line X =Z3, U {cยข} and is generated by C, : x --, x + 1 and C2 : x x/(x+l).(C, fixes oc and C2 : 30 ,-* oo ,-* l.)Ghas 15 orbits of 5-subsets and 83 orbits of 6-subsets. Let 2ij denote the number Of elements of the j-th orbit of 6-subsets, that contains an arbitrary fixed element of the i-th orbit of 5-subsets. (The 15 by 83 matrix (2i,j) is denoted by A(G;H;5,6) in [I] and as A5,6 in .) The matrix (2i, j) has uniform row sum 27. To construct a 5-(32,6,2) design with G as an automorphism group, we find a proper subset S of the columns of (2i, j) with uniform row sum 2. Heuristic search was used and orbit representatives for the resulting designs are given in the T~.hle 1. (The 3-homogenicity of the group enables the orbit representatives to be supersets of {0,1,o0}.) The necessary conditions for the existence of a 5-(32,6,2) design show that 3 must divide 2 and it is elementa W to show from the matrix (2i,j) that 2 = 3 and 2 = 9 are impossible to obtain with this group. Using the fact that a complement of a design is also a design we have:
Theorem. There exists a 5-(32,6,2) desion, with 2 E {6,12,15,21} and with automorphism group PSL(2,31 ). Direct action of the group PSL(2,31) on the projective line does not give 5-(32,6,2) designs with other values of 2.