New designs with prescribed automorphism groups

The following table contains some new information on the existence of designs with small parameters. Each row contains at least one set of parameters t-(v,k,λ) for which designs have previously been unknown, according to the Handbook of Combinatorial Designs [1]. The new constructions are described in the papers [3] and [4]. The first row of our table improves upon Table 1.35 in [5], and the other rows upon Table 4.46 in [2].

Given t, v, and k, there is a least integer λmin such that the parameters t-(v,k,λmin) are admissible, i.e. satisfy the necessary divisibility conditions. All other admissible λ are of the form λ = m·λmin, for m ∈ N. The largest λ for which a simple design exists is λmax = (v-t over k-t). Because of complementation, it suffices to consider λ ≤ λmax/2. We denote by M the largest integer m such that m·λmin ≤ λmax/2. In the table integers m ∈ {1,…,M} are divided into columns according to whether designs do not exist (∄), are unknown (?), or exist (∃). The last column is linked to files containing prescribed automorphism groups and base blocks in GAP format, from which the designs can be constructed.

 t  v   k  λmin   M m
∄?∃
255101  443161355  1,2  3,6,7   4,5,8,…,443161355 
3205234 12,…,34
321715102 12,…,102
415515 12,…,5
41681516 1,23,…,16
41891471 1,23,…,71
41992171 1,23,…,71
516755 12,…,5
5178205 12,…,5

References

  1. C.J. Colbourn, J.H. Dinitz (eds.), Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, 2007.
  2. G.B. Khosrovshahi, R. Laue, t-designs with t≥3, in: The Handbook of Combinatorial Designs, Second Edition (eds. C.J. Colbourn and J.H. Dinitz), CRC Press, 2007, pp. 79-101.
  3. V. Krcadinac, Some new designs with prescribed automorphism groups, to appear in the Journal of Combinatorial Designs [preprint].
  4. V. Krcadinac, M.O. Pavcevic, New small 4-designs with nonabelian automorphism groups, in preparation.
  5. R. Mathon, A. Rosa, 2-(v,k,λ) designs of small order, in: The Handbook of Combinatorial Designs, Second Edition (eds. C.J. Colbourn and J.H. Dinitz), CRC Press, 2007, pp. 25-58.

Vedran Krcadinac,