| The paper is divided into five chapters. In chapter one the background of combinatorial design is introduced. In chapter two we focus our attention on a special kind of block design - doubly near resolvable balance incomplete block design (DNR(v, m, m -1)-BIBDs), and discuss the existence of DNR(v, m, m -1)-BIBDs for some special values of v. In chapter three we discuss the existence of DNR(v, 4, 3)-BIBDs with v a prime power = 5 (mod 8). In chapter four we discuss the existence of DNR(v, 4, 3)-BIBDs with v a prime power = 9 (mod 16). In chapter five a concluding remark and some problems for further research are presented.Block design is an important part in combinatorial design theory. Research of block design is focused on some block designs, such as BIBD, RBIBD, NRBIBD, SBIBD, PBIBD, GDD, TD ··· and so on.Nearly resolvable balance incomplete block design is a special kind of balance incomplete block design(BIBD). A (v, m, A)-BIBD D is said to be near resolvable and denoted by NR(v, m, A)-BIBD if the blocks of D can be partitioned into classes R1, R2,..., Rv such that for each point x of D, there is precisely one class having no block containing x and each class contains precisely v -1 points of the design. The classes R1, R2, ..., Rr form a resolutions of D. The necessary conditions for the existence of a NR(v, m, λ)-BIBD are v =1 (mod m) and λ = m. Suppose that R = {R1, R2,..., Rv} and R' = {R1', R2',..., Rv'} are two resolution of a NR(v, m, m -1)-BIBD D, then R and R' is said to be orthogonal provided that|Ri∩Rj'| ≤ 1, Ri∈ R, Rj' ∈ R'.If a NR(v, m, m -1)-BIBD has a pair of orthogonal near resolutions, it is said to be doubly resolvable and is denoted DNR(v, m, m -1)-BIBD. The existence of NR(v, m, m -1)-BIBD had been completely solved when m = 3,4 and 5. For m = 6, the necessary condition v = 1 (mod 6) is also sufficient with the possible exception of v = 55,145. For m ≥ 7, there are also some results. The existence of DNR(v, m, m -1)-BIBD had been completely solved... |
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