Uniform frames with block size four and index one or three (Q2708217)

A \((4,1)\)-frame of type \(h^u\) is a \((4,1)\) group divisible design with \(u\) groups of size \(h\) whose blocks can be partitioned into holey parallel classes each of which partitions the difference between the set of vertices of the design and some of the groups. A \((4,1)\)-frame of type \(h^u\) is known to exist if and only if \(u \geq 5\), \(h \equiv 0 \pmod{3}\), and \(h(u-1)\equiv 0\pmod{4}\), unless \(h\) and \(u\) fall into one of several classes of pairs for which a corresponding \( (4,1) \)-frame has not yet been found. The author constructs a \( (4,1) \)-frame of type \( h^u \) for several classes of the ``unresolved'' parameters. As a consequence of these constructions, the only possible pairs \( (h,u) \) that satisfy the necessary conditions and for which there still may not exist a \( (4,1) \)-frame of type \( h^u \) are the pairs \( (36,8), (36,12), (24,12) \), pairs with \( h \equiv 6 \pmod{12} \), \( h \neq 18 \), and \( u \in\{7, 19, 23, 27, 35, 39, 43, 47, 63, 67\}\), and pairs with \( h=18 \) and \( u \in\{15, 17, 19, 23, 27, 39\}\). Similar results for the case of \( (4,3) \)-frames of type \( h^u \) yield the existence of a \( (4,3) \)-frame for all pairs \( (h,u) \) satisfying the necessary conditions for the existence of a \( (4,3) \)-frame of type \( h^u \) except possibly for the pairs with \( h \equiv 2 \pmod{4} \) and \( u \in\{19, 23, 27, 39\}\) or \( h=6 \) and \( u \in\{7, 43, 47, 67\}\).