Does the Steiner system $S(2,11,1331)$ exist? I think it exists because $S(2, q, q^n)$ exists when $q$ is a prime power, $n\ge 2$. Confirmation will be very useful to me. A Steiner system forms a cover-free family where each block has at least one element not covered by the union of D other blocks. I am working on topology transparent scheduling in wireless networks [1]. A schedule is a collection of time slots where a node can transmit. One can use Steiner systems to come up with such schedules. Basically, a block in a Steiner system defines a schedule. The cover-free property ensures existence of a conflict-free slot in a block where a node's transmission will not fail. The number of blocks equals the number of nodes N. But the problem is that we may not have Steiner systems for a given (N, D). We have to use some Steiner system that is close enough at the loss of some efficiency. In this context, I was looking for some Steiner systems.
For $S(2,11,1331)$, the number of blocks = $ b = vr/q$ where $v = 1331$, $q=11$, $r=(v-1)/(q-1)=133$ giving $b = 16093$.
- Colbourn, Charles J., Alan CH Ling, and Violet R. Syrotiuk. "Cover-free families and topology-transparent scheduling for MANETs." Designs, Codes and Cryptography 32.1-3 (2004): 65-95.
Yes, it exists. Consider a 3-dimensional space $V$ over the field $K=\Bbb{F}_{11}$. There are $11^3=1331$ points in $V$. Any two points in $V$ determine a unique line (= a coset of a 1-dimensional subspace). The collection of all the lines in $V$ is thus the desired Steiner system.