Balanced incomplete block designs with block size 9 and \(\lambda=2,4,8\)
The necessary conditions for the existence of a balanced incomplete block design on \(v\) points with index \(\lambda\) and block size \(k\), or \((v,k,\lambda)\) BIBD, are that (i) \(\lambda(v-1) \equiv 0 \bmod (k-1)\) and (ii) \(\lambda v(v-1) \equiv 0 \bmod k(k-1)\). In this paper the authors study the existence of \((v,9,\lambda)\) BIBDs for \(\lambda = 2, 4\) and \(8\). For \(k = 9\) the necessary conditions on \(v \geq 9\) reduce to \(v \equiv 1, 9 \bmod (72/\text{gcd}(72, \lambda))\). Using a variety of direct and recursive constructions they show that the necessary conditions are sufficient for \(\lambda = 8\). For \(\lambda = 2\) the conditions are sufficient with the possible exception of designs with \(v = 189, 253, 505, 765, 837, 1197, 1837\) and \(1845\). For \(\lambda = 4\) the conditions are sufficient with the possible exception of designs with \(v = 315, 459\) and \(783\). The authors also give some examples of group divisible designs derived from balanced ternary designs.
scientific article; zbMATH DE number 1785700