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 | ||
| ∄ | ? | ∃ | |||||
| 2 | 55 | 10 | 1 | 443161355 | 1,2 | 3,6,7 | 4,5,8,…,443161355 |
| 3 | 20 | 5 | 2 | 34 | 1 | 2,…,34 | |
| 3 | 21 | 7 | 15 | 102 | 1 | 2,…,102 | |
| 4 | 15 | 5 | 1 | 5 | 1 | 2,…,5 | |
| 4 | 16 | 8 | 15 | 16 | 1,2 | 3,…,16 | |
| 4 | 18 | 9 | 14 | 71 | 1,2 | 3,…,71 | |
| 4 | 19 | 9 | 21 | 71 | 1,2 | 3,…,71 | |
| 5 | 16 | 7 | 5 | 5 | 1 | 2,…,5 | |
| 5 | 17 | 8 | 20 | 5 | 1 | 2,…,5 | |
Vedran Krcadinac,