Existence of resolvable group divisible designs with block size four. I (Q1613563)

The authors prove, for \(m\) incongruent 0 modulo 12, that there exists a resolvable group divisible design of order \(v\), block size 4 and group size \(m\), if and only if \(v\equiv 0 \pmod 4\), \(v \equiv 0 \pmod m\), \(v - m\equiv 0 \pmod 3\), except when \((v, m)\) is equal to \((3, 12)\) and except possibly when \((v, m)\) is (3, 264), (3, 372), (8, 80), (8, 104), (9, 396), (40, 400) or (40, 520).