On q-analogs of Steiner systems and covering designs

The q-analogs of covering designs, Steiner systems, and Turán designs are studied. It is shown that q-covering designs and q-Turán designs are dual notions. A strong necessary condition for the existence of Steiner structures (the q-analogs of Steiner systems) over F2 is given. No Steiner structures of strength 2 or more are currently known, and our condition shows that their existence would imply the existence of new Steiner systems of strength 3. The exact values of the q-covering numbers Cq(n, k, 1) and Cq(n, n−1, r) are determined for all q, n, k, r. Furthermore, recursive upper and lower bounds on the size of general q-covering designs and q-Turán designs are presented. Finally, it is proved that C2(5, 3, 2) = 27 and C2(7, 3, 2) 6 399. Tables of upper and lower bounds on C2(n, k, r) are given for all n 6 8.