×

Frames with block size four. (English) Zbl 0769.05020

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.

MSC:

05B05 Combinatorial aspects of block designs
PDFBibTeX XMLCite
Full Text: DOI