Enumeration of 2-(12, 3, 2) designs

A backtrack search with isomorph rejection is carried out to enumerate the 2-(12,3,2) designs. There are 242 995 846 such designs, which have automorphism groups whose size range from 1 to 1536. There are 88 616 310 simple designs. The number of resolvable designs is 62 929; these have 74 700 nonisomorphic resolutions. We use the following standard notations. A t-(v, k, >.) design is a family of k-subsets, called blocks, of a v-set such that each t-subset of the v-set is contained in exactly>. blocks. A design is simple if it has no repeated blocks. A 2-(v, k, >.) design is called a balanced incomplete block design (BIBD). Two more parameters that are related to a design are b, the number of blocks, and r, the number of blocks in which a point occurs. The values of band r can easily be determined from the values of the other parameters, as vr = bk,r(k-1) = >.(v-1). It has for a long time been known that 2-(12,3,2) designs exist; see [1]. One construction of such a design is as follows. A 2-(45,12,3) design can be constructed from a McFarland difference set [5]. Then we get a desired derived design by deleting any block in this design and deleting all points not in this block from the other blocks. Many non isomorphic 2-(12,3,2) designs are known; Royle [6, 11J quickly found one million with a hill-climbing computer algorithm. A complete enumeration of these designs is, however, yet to be carried out. This is the goal of our work.