Group divisible designs with block size four and type g^u m^1 - III

TL;DR

This paper establishes the existence of certain group divisible designs with block size four and specific group types, expanding the known cases where necessary conditions are also sufficient.

Contribution

It proves the sufficiency of necessary conditions for 4-GDDs of type g^u m^1 for a wide range of g values, including many composite and power-of-two cases.

Findings

Necessary conditions are sufficient for g = 14, 20, 22, 26, 28, ... , 496.

Sufficiency is shown for g = 2^t q^s with specific primes q.

Possible exceptions include certain large composite g values.

Abstract

We deal with group divisible designs that have block size 4 and group type g^u m^1, where g = 2 or 4 (mod 6). We show that the necessary conditions for the existence of a 4-GDD of type g^u m^1 are sufficient when g = 14, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 58, 62, 68, 76, 88, 92, 100, 104, 116, 124, 136, 152, 160, 176, 184, 200, 208, 224, 232, 248, 272, 304, 320, 368, 400, 448, 464 and 496. Using these results we go on to show that the necessary conditions are sufficient for g = 2^t q^s, q = 19, 23, 25, 29, 31, s, t = 1, 2, ..., as well as for g = 2^t q, q = 2, 5, 7, 11, 13, 17, t = 1, 2, ..., with possible exceptions 56^9 m^1, 80^9 m^1 and 112^9 m^1 for a few large values of m.