Rees, Rolf S.; Stinson, Douglas R. Frames with block size four. (English) Zbl 0769.05020 Can. J. Math. 44, No. 5, 1030-1049 (1992). Authors’ abstract: We investigate the spectrum for frames with block size four, and discuss several applications to the construction of other combinatorial designs. Our main result is that a frame of type \(h^ u\), having blocks of size four, exists if and only if \(u\geq 5\), \(h\equiv 0\bmod 3\) and \(h(u-1)\equiv 0\bmod 4\), except possibly where \[ h = 9\text{ and } u\in\{13,17,29,33,93,113,133,153,173,193\}; \tag{i} \]\[ h \equiv 0\bmod 12\text{ and } u\in\{8,12\},\tag{ii} \]\[ h = 36\text{ and } u\in\{7,18,23,28,33,38,43,48\}, \]\[ h = 24\text{ or } 120 \text{ and } u\in\{7\}, \]\[ h= 72\text{ and } u\in 2\mathbb{Z}^ +\cup\{n:n\equiv 3\bmod 4 \text{ and } n\leq 527\}\cup\{563\};\text{ or } \]\[ h \equiv 6\bmod 12\text{ and } u\in(\{17,29,33,563\}\cup\{n:n\equiv 3\text{ or } 11\bmod 12 \tag{iii} \]\[ \text{and } n\leq 527\}\cup\{n:n\equiv 7\bmod 12\text{ and } n\leq 259\}),\;h=18. \] Additionally, we give a new recursive construction for resolvable group-divisible designs from frames: if there is a resolvable \(k\)-GDD of type \(g^ u\), a \(k\)-frame of type \((mg)^ v\) where \(u\geq m+1\), and a resolvable TD\((k,mv)\) then there is a resolvable \(k\)-GDD of type \((mg)^{uv}\). We use this to construct some new resolvable GDDs with group size three and block size four. Reviewer: P.Reichensperger (Oberasbach) Cited in 39 Documents MSC: 05B05 Combinatorial aspects of block designs Keywords:spectrum; frames; block; combinatorial designs; resolvable group- divisible designs PDFBibTeX XMLCite \textit{R. S. Rees} and \textit{D. R. Stinson}, Can. J. Math. 44, No. 5, 1030--1049 (1992; Zbl 0769.05020) Full Text: DOI