The existence of directed BIBDs. (Q1412663)

For any natural number \(k\geqq 3\) and any integer \(\lambda \), let c\( _{d}(k,\lambda )\) be the least integer such that the necessary conditions 2\( \lambda \)(\(\lambda -1)\equiv 0\pmod{(k-1)}\) and \(\lambda v(v-1)\equiv 0 \pmod{[k/2]}\) for the existence of a DB\((k,\lambda;v)\) are also sufficient, for all \(v \geqq c_{d}(k,\lambda ).\) In the paper, the estimate of c\(_{d}(k,\lambda )\) when \(k\) is a multiple of 4 and \(\lambda \) is arbitrary is given. Combining this result with that given by the authors in a previous paper [Discrete Math. 222, 27--40 (2000; Zbl 0967.05011)], one obtains an estimate of the number \(c_{d}(k,\lambda )\), for all \(k\geqq 3\) and all integers \( \lambda\).