Generalized Bhaskar Rao designs of block size three (Q1062065)

W is a generalized Bhaskar Rao design with parameters (v,b,r,k,\(\lambda\) ;G) if W has entries from G or 0 - the zero of the group ring - so that if \((a_{i1},...,a_{ib})\) and \((b_{j1},...,B_{jb})\) are distinct rows of W then the scalar product \((WW^+)_{ij}=(a_{i1},...,a_{ib})\cdot (b_{i1},...,b_{jb})\) should give each element of G the same number of times (\(\lambda\) /\(| G|)\). If A is obtained from W by making each non-zero element 1 then A should be the incidence of a BIBD (v,b,r,k,\(\lambda)\). This paper finishes the proof of \textit{C. Lam} and the author [ibid. 10, 83-95 (1984; Zbl 0552.05012)] and the author [ibid. 10, 69-82 (1984; Zbl 0552.05011)] showing that ''The necessary conditions \(\lambda \equiv 0\quad (mod | G|),\) \(\lambda (v-1)\equiv 0\quad (mod 2),\) \(\lambda v(v- 1)\equiv 0\quad (mod 6)\) for \(| G|\) odd, \(\lambda v(v-1)\equiv 0\quad (mod 24)\) for \(| G|\) even, are sufficient for the existence of a generalized Bhaskar Rao design GBRD (v,b,r,3,\(\lambda\) ;G) for the elementary abelian group G, of each order \(| G|\).