New Infinite Classes Of Uniform Candelabra Quadruple Systems

A candelabra quadruple system (CQS) of type (gn:s) is a quadruple (X, S, (?), A) satisfying the following properties:(1) X is a set of ng+s elements; (2) S is a subset of X of size s; (3) (?)={G1,G2,..., Gn} is a set of g-subsets of X\S, which partition X\S; (4) A is a family of 4-subsets (called blocks) of X; (5) every 3-subset T of X with |T∩(S∪Gi)|<3 for all i is contained in a unique block and no 3-subset of S∪Gi for all i is contained in any block. CQSs were first introduced by Hanani who used hem to determine the existence of Steiner quadruple systems (SQSs), a lot of recursive constructions for SQSs can be interpreted in terms of CQSs. They were also used to study large sets of triple systems and construct optimal constant weight codes of constant weight 1, and minimum Hamming distance 4. There are already a lot of results of CQS(gn:s) when n ∈{3,4,5} or s= 0, but there are not much for the general type before. This thesis mainly studies the existence of CQS(gn:s) with g= 0 (mod 6) and s≤g.With the aid of three wise balanced designs and Hartman’s fundamental construction for 3-designs, we prove that the existence of the desired CQS(gn:s) depends only on the existence of CQS(gm:s), where m ∈{6,8,14}. Then we construct a new class of partial CQSs with odd group size, which is a generalization of the even case. By using the new results of partial CQSs and other constructions for CQSs we determine the existence of CQS(gm:s) for m ∈{6,8,14}. Hence, there exists a CQS(gn:s) for all g≡ 0 (mod 6), s≡ 0 (mod 2), s≤ g and n≥3. We almost complete the existence of SQS(v) with two sub-designs S(2,4,v), which is a generalization of the SQS(v) with a sub-design S(2,4,v). Based on this result, we obtain some new CQS(gn:s) with s>g, i.e., for all g≡s≡0 (mod 2) and s≤4g, there is a CQS(g12m+4:s) when m(?){3,14,15,18,22,23,26,27,29}.