Resolvable Golf Designs And PBD Closure

:An idempotent quasigroup (IQ) possessing a unipotent orthogonal mate is called resolvable. A large set of idempotent symmetric quasigroup of order v is a partition of all3-vectors with3distinct components of a v-set into v-2IQ(v)s pairwise agreeing on only the idempotent symmetric rows, otherwise known as golf design, denote G(v). Resolvable golf design is that each of its members is resolvable idempotent symmetric quasigroups golf designs G(v). Zhou Junling, and Chang Yanxun has given the results of its progressive existence of RG(v), namely there exists a constant v0, such that for all odd v> v0there RG(v) exists.Let K be a set of positive integers, a pairwise balanced design PBD(v, K) of order v with block size from K is a. pair (V,B), where V is a finite set of cardinality v and B is a family of subsets of V, such that every unordered pair of distinct elements of V occurs in exactly one block of B and every block size comes from K. The necessary conditions for the existence of PBD(v, K) are (v-1)=0(modα) and v(v-1)=0(modβ), where α=gcd{k-1|k∈K} and β=gcd{k(k-1)|k∈K}. According to Wilson’s theory, there exists a constant v0such that, if v>v0, the necessary conditions of the existence of PBD(v, K) are also sufficient. We call K is a pairwise balanced design closure if B(K)=K where B(K)={v:there exists a PBD(v, K)}. To determine constant v0, this paper will investigate the PBD closure B(K) with block size set K={7,9,11,13,19,25,31,37,43,49,61,73,79,85,97}. In order to determine the PBD closure, we will flexibly use the Wilson fundamental construction, filling the holes construction, the constructor method and the direct product of singular indirect product in the combinatorial design theory, and so on. Given the following results, when v≥421513and v is odd, PBD(v, K) are exist. According to the results obtained, we can get the existence of resolvable golf designs such that, when v≥421513and v is odd, resolvable golf design RG(v) is exist.About resolvable golf design above delimitation is still relatively rough, yet many remain undetermined order paper for resolvable golf designs do further research on the one hand, by means of a suitable construction methods and computer searches, given a number of small order of the existence of the design; the other hand, the original recursive structure made improvements designed to reduce biodegradable golf research for the study of a special class of PBD parallel with the design of a part of, which is further research provides a viable new ideas.