Four different views of Glasgow University Tower

# Symmetric 2-designs

In recent years I have been interested in the classification of 2-designs, mainly, but not exclusively, symmetric. I was successful in determining all 2-(31,10,3) designs and using the same programs I was able to verify Denniston's results concerning the 2-(25,9,3) designs. Moreover, as a by-product of classifying Hadamard matrices of orders 28 and 24, I have been able to enumerate all 2-(23,11,5) designs as well as the 2-(27,13,6) designs. At present I have all designs coming from the first three of these cases on disc as zero-one matrices. Since there are 208310 designs with parameters (27-13-6), most of which have trivial automorphism group, the listing below only in includes those with non-trivial group.

In joint work with Mario Pavcevic [1] we investigated symmetric designs with λ = 10 and having an automorphism of order 5. In the case 2-(61,25,10) we determined all the designs with such an automorphism that fixes 11 points, but their numbers were so large as to make it impractical to store them. However, among these designs there are precisely 24 admitting the action of an elementary abelian group of order 25. Of these, 12 are self-dual, the other 12 being 6 pairs of dual designs. In the case of 2-(66,26,10) designs we discovered 558 admitting the action of the dihedral group of order 10 and exactly 22 having the elementary abelian group of order 25 as a group of automorphisms. These 22 comprise 10 self-dual designs and 6 pairs of dual designs. In the table below we list the dual designs and one from each of the dual pairs.

In another paper [2] a construction was given for a family of symmetric designs with paramters (2q^2+2q+1,q^2, q(q-1)/2), q a prime power. The designs with q = 3 are completely determined, while there are examples of such designs with q = 5 (corresponding to (61,25,10); see above), we list a few with the format described in the paper. There are many tens of thousands of these and the situation is probably similar for q = 7. For this reason we only give a few examples in this case also.

I also include the known designs with small parameters If you wish to have copies simply click on the appropriate link in the table below.

 7-3-1 11-5-2 13-4-1 15-7-3 16-6-2 19-9-4 21-5-1 23-11-5 25-9-3 27-13-6 31-6-1 31-10-3 37-9-2 61-25-10 61-25-10.no2 66-26-10 113-49-21

[1] M.-O. Pavcevic & E Spence, Some new symmetric designs with λ = 10 having an automorphism of order 5, Discrete Math. 196 (1999), 257-266.

[2] M.-O. Pavcevic & E Spence, Some new symmetric designs, J. Comb. Designs 7 (1999) 426-430