Balanced Incomplete Block Designs with Block Size 9 and λe2,4,8

Abstract

The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: \begin{array}{rcl} \lambda (v - 1) &\equiv& 0\ \mod (k - 1) \\[2pt] \lambda v(v - 1 ) &\equiv& 0\ \mod k(k - 1) \end{array} In this paper we study ke9 with λe2,4 or 8. For λe8, we show these conditions on v are sufficient, and for λe2, 4 respectively there are 8 and 3 possible exceptions the largest of which are ve1845 and 783. We also give some examples of group divisible designs derived from balanced ternary designs.