Super-simple designs with v⩽32

Super‐simple resolvable balanced incomplete block designs with block size 4 and index 3

“…The result is a f5; 6; 7g-GDD of type 5 7 3 1 . Applying Construction 2.2 to this GDD with weight 4, we obtain a super-simple ð4; 3Þ-frame of 20 7 12 1 . Here, the input super-simple ð4; 3Þ-frames of types 4 5 , 4 6 , and 4 7 all come from Lemma 2.1.…”

“…A recent survey on super-simple ðv; 4; Þ-BIBDs with v 32 appeared in [7]. We summarize these known results in the following theorem.…”

“…. ; 20g, the hole set be f1; 2; 3; 4g, and the automorphism group be generated by ¼ ð1Þð2Þð3Þð4Þð5 7 Here, the base blocks in each of the first four rows form an initial parallel class. The development of the first two blocks in the last row, which are a short orbit, forms a holey parallel class missing the hole f1; 2; 3; 4g.…”

Orthogonal Decomposition and Packing of Complete Graphs

“…For r=4, it is known whenever a CRSS(n, 4, 2) design exists, and whenever a SS(n, 4, 4) design exists [1]. Several other sporadic results involving the case r=4 also appear in [11,18,23,29]. The main theorem of this paper solves the CRSS(n, r, k) existence problem completely, for all n>N(k, r).…”

“…In case that a SS(t, v, k, *) design splits into * copies of a SS(t, v, k, 1)-design, the design is called a completely reducible super-simple design, denoted by CRSS(t, v, k, *) or simply CRSS(v, k, *) if t=2. Recent results on super-simple and completely reducible super-simple designs can be found in [1,11,23,28,29]. The requirement that any two blocks have at most one pair in common is called the orthogonality property.…”