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 in gzipped format.

    More recently Mario Pavcevic and I investigated symmetric designs with lambda = 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 a recent paper (Some New Symmetric Designs, Journal of Combinatorial Designs, 7 1999, 426-430, with M. -O. Pavcevic), 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.

    If you wish to have copies simply click on the appropriate name in the table below. Alternatively, you may wish to have the designs in octal form. This is obtained by concatenating the rows of the incidence matrix and writing the resulting binary integer in octal form. In this case the order of the automorphism group of the design is also given. {(v,k,lambda) design <-> v-k-lambda.gz}:

    31-10-3.gz (zero-one) 31-10-3.gz (octal form)
    23-11-5.gz (zero-one) 23-11-5.gz (octal form)
    25-9-3.gz (zero-one) 25-9-3.gz (octal form)
    15-3-1 (zero-one) 15-3-1 (octal form)
    61-25-10 (zero-one) 66-26-10 (zero-one)
    113-49-21 (zero-one) 61-25-10.no2 (zero-one)