All Directed BIBDs with k = 3 Exist

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...

The existence of directed BIBDs

For any positive integers k¿ 3 and , let cd(k; ) denote the smallest integer such that the necessary conditions 2 (v − 1) ≡ 0 (mod k − 1) and v(v − 1) ≡ 0 (mod ( k2 )) for the existence of a DB(k; ; v) are also su3cient for every v¿ cd(k; ). In this article we provide an estimate for cd(k; ) when k ≡ 0 (mod 4) and any . Combined with the results in (Discrete Math. 222 (2000) 27–40), we complete...

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

A necessary condition for existence of a (v, k, k−1) near resolvable BIBD is v ≡ 1 (mod k). In this paper, we update earlier known existence results when k ∈ {9, 12, 16}, and show this necessary condition is sufficient, except possibly for 26, 37 and 149 values of v for k = 9, 12, 16 respectively. Some new results for existence of (9, 8)-frames of type 9 are also obtained; in particular, we sho...

Kronecker Products and BIBDs

Recursive constructions are given which permit, under conditions described in the paper, a (v, b, r, k, lambda)configuration to be used to obtain a (v', b', r', k, lambda)-configuration. Although there are many equivalent definitions we will mean by a (v, b, r, k, lambda)-configuration or BIBD that (0, 1)-matrix A of size v x b with row sum r and column sum k satisfying AAT = (r lambda)I + lamb...

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