PBDs, Frames, and Resolvability

Minimum embedding of P3‐designs into (K4—e)‐designs

“…There exists a 3-frame of type 16 n for all n 1 ðmod 3Þ (see [11], for example). Hence there exists a P 3 -frame of type 16…”

“…Starting with an f3g-GDD of type 1 9 or 1 15 (these are Steiner triple systems), and proceeding similarly, handles n ¼ 8 and 14. Using instead f3; 5g-GDDs of type 1 11 and 1 17 having a unique block of size 5, and deleting a point not on the block of size 5, handles n ¼ 10 and n ¼ 16. When n ¼ 11, delete one point from a TD(4; 3) and proceed similarly.…”

Balanced incomplete block designs with block size 8

“…Now give all points a weight of 28 in WFC, and use the ingredient designs from Theorem 2.4.b to get a 8Y 2 GDD on 28n points with groups of sizes 196, 224, and 252. We can ®ll these groups with 8Y 2 GDDs of types 7 28 , 7 32 , and 7 36 to obtain an 8Y 2 GDD of type 7 4n . To instead obtain 8Y 2 GDDs of type 7 4n1 we add w 7 new points and ®ll in the groups with 8Y 2 GDDs of types 7 29 , 7 33 Y and 7 37 ; thus we also have 8Y 2 GDDs of type 7 4n1 for all n b 581.…”

“…When the holey sets all span the point set (except for their hole) times, so forming holey -resolution sets, we will call the frame an -frame. Although -frames are not new, (see e.g., [36]), they do not seem to have been exploited previously in constructing BIBDs to the extent that we use them.…”

“…The ®fth solution has a line distribution of 8 114 9 10 12 11 , and a 12-line with point distribution T 2 1 T 10 2 , plus two 9-lines disjoint from this 12-line. Noting that 12-lines can't be disjoint, counting the points on the 12-lines for these two solutions gives T 36 1 T 36 2 and T 22 1 T 55 2 leaving just 21 and 16 points. However all points on the lines disjoint from the given 12-line have degree 13; hence they lie on 12 lines which are not disjoint from the given 12-line and just one line which is.…”