Existence of near resolvable (v, k, k-1) BIBDs with k∈{9, 12, 16}

Existence of doubly near resolvable (v, 4, 3) BIBDs

The existence of doubly near resolvable (v, 2, 1)-BIBDs was established by Mullin and Wallis in 1975. For doubly near resolvable (v, 3, 2)-BIBDs, the existence problem was investigated by Lamken in 1994, and completed by Abel, Lamken and Wang in 2007. In this paper, we look at doubly near resolvable (v, 4, 3)-BIBDs; we establish that these exist whenever v ≡ 1 (mod 4) except for v = 9 and possi...

Some new near resolvable BIBDs with k=7 and resolvable BIBDs with k=5

The Existence of Kirkman Squares-Doubly Resolvable (v, 3, 1)-BIBDs

A Kirkman square with index λ, latinicity μ, block size k, and v points, KSk(v; μ, λ), is a t × t array (t = λ(v − 1)/μ(k − 1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V , and (3) the collection of blocks obtained from the non-empty cells of the array is a (...

The Existence of DNR ( mt + 1 , m , m − 1 ) - BIBDs

A (v, m, m − 1)-BIBD D is said to be near resolvable (NRBIBD) 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. If an (v, m, λ)-NRBIBD has a pair of orthogonal resolutions, it is said to be doubly resolvable and is denoted D...

On the coexistence of conference matrices and near resolvable 2-(2k+1, k, k-1) designs

We show that a near resolvable 2-(2k + 1, k, k − 1) design exists if and only if a conference matrix of order 2k+2 does. A known result on conference matrices then allows us to conclude that a near resolvable 2-(2k + 1, k, k − 1) design with even k can only exist if 2k + 1 is the sum of two squares. In particular, neither a near resolvable 2-(21, 10, 9) design nor a near resolvable 2-(33, 16, 1...