ON q-ANALOGS OF STEINER SYSTEMS AND COVERING DESIGNS
The q-analogs of covering designs, Steiner systems, and Turan designs are studied. It is shown that q-covering designs and q-Turan designs are dual notions. A strong necessary condition for the existence of Steiner structures (the q-analogs of Steiner systems) over F-2 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 C-q(n, k, 1) and C-q(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-Turan designs are presented. Finally, it is proved that C-2(5, 3,2) = 27 and C-2(7, 3,2) <= 399. Tables of upper and lower bounds on C-2(n, k, r) are given for all n <= 8.
q-analogs covering designs turan designs steiner structures